IDEAL OPERATORS AND HIGHER INDESCRIBABILITY

Abstract

We investigate properties of the ineffability and the Ramsey operator, and a common generalization of those that was introduced by the second author, with respect to higher indescribability, as introduced by the first author. This extends earlier investigations on the ineffability operator by James Baumgartner, and on the Ramsey operator by Qi Feng, by Philip Welch et al., and by the first author.

1 Introduction

In the set theoretic literature, some of the most popular large cardinals have been equipped with canonical ideals, and sometimes also with certain operators on ideals. Early examples of such large cardinal operators are the ineffability operator $\mathcal I$ due to Baumgartner in [4], and the Ramsey operator $\mathcal R$ that was introduced and extensively studied by Feng in [8].

In the present paper, we want to analyze the interplay of certain large cardinal operators, and in particular the operators ${\mathcal I}$ and ${\mathcal R}$ , with a notion of higher indescribability that was introduced by the first author in [7], and which extends a notion of Bagaria from [2]. Let us note that Sharpe and Welch also introduced a notion of higher indescribability [15, Definition 3.21], but the relationship between the notion we use in the present paper and that of [15] is not currently known.

In the remainder of this section, we recall the definitions of the operators ${\mathcal I}$ and ${\mathcal R}$ , and the strongly Ramsey subset operator ${\mathcal S}$ (the latter has been first introduced in [12], where it is denoted by $\mathbf {T}_{\mathrm {cl}}$ ) that is associated with the large cardinal notion of strong Ramseyness, as introduced in [11]. In Section 2, we review infinitary second order formulas, and their associated notions of indescribability, and we show that many of their basic properties can be established by simple arguments that make use of generic ultrapowers. In Sections 3 and 4, we generalize results of Baumgartner [3, 4] to the context of higher indescribability. For example, we show that if $\kappa $ is a subtle cardinal then there are many cardinals $\alpha <\kappa $ which are $\Pi ^1_{\xi }$ -indescribable for all $\xi <\alpha ^+$ . Furthermore, if one assumes the ideal ${\mathcal I}^{\gamma +1}([\kappa ]^{<\kappa })$ associated with the $\gamma +1$ -ineffability of a cardinal $\kappa $ is nontrivial, where $\kappa \leq \gamma <\kappa ^+$ , then for any fixed bijection $b:\kappa \to \gamma $ the set

$$\begin{align*}\{\alpha<\kappa\mid \alpha\in{\mathcal I}^{\text{o.t.}(b[\alpha])}(\Pi^1_{\xi}(\alpha))^+\} \end{align*}$$

is in the filter dual to ${\mathcal I}^{\gamma +1}([\kappa ]^{<\kappa })$ (see Corollary 3.9). In Section 5, we provide two basic lemmas on iterations of the ineffability and the Ramsey operator, which can be viewed as generalizations of the following standard facts: whenever a set has stationarily many stationary initial segments it must be stationary and if a cardinal is weakly compact then the set of smaller cardinals which are not weakly compact is a weakly compact set. In Section 6, we show that the ideals associated with higher indescribability, and the results of applying our ideal operators to these ideals, can be described by certain infinitary second order formulas. In Section 7, we review a uniform framework for large cardinal operators from [12], that in particular includes the operators ${\mathcal I}$ , ${\mathcal R}$ , and ${\mathcal S}$ . ¹ In Section 8, we review the notion of pre-operators. In Section 9, we combine all of the ingredients to verify yet another generalization of results by Baumgartner, which we extend both to the context of higher indescribability and to the above-mentioned framework for large cardinal operators. For example, suppose $\xi <\kappa ^+$ and let ${\mathcal O}$ be the ineffability operator ${\mathcal I}$ or the Ramsey operator ${\mathcal R}$ . Then ${\mathcal O}^{\omega }(\Pi ^1_{\xi }(\kappa ))={\mathcal O}^{\omega }(\Pi ^1_{\xi +1}(\kappa ))$ , but ${\mathcal O}^n(\Pi ^1_{\xi }(\kappa ))\subsetneq {\mathcal O}^n(\Pi ^1_{\xi +1}(\kappa ))$ for all $n<\omega $ with $\kappa \in {\mathcal O}^n(\Pi ^1_{\xi +1}(\kappa ))^+$ (see Corollaries 9.7 and 9.9). In Section 10, we comment on some partially problematic results of the first author from [7]. In particular, let us point the reader to a simple question involving the Ramsey operator and $\Pi ^1_1$ -indescribability, namely Question 10.9, which has so far resisted all attempts at a resolution.

Without further mention, we will require all ideals to be ideals on some regular and uncountable cardinal $\kappa $ , and to be supersets of the bounded ideal on $\kappa $ . For any ideal I, $I^+$ denotes the collection of I-positive sets, that is, those subsets of $\kappa $ which are not in I, while $I^*$ denotes the filter that is dual to I, that is, the collection of complements of sets in I. We will often introduce ideals by defining the collection of their positive sets when this is more convenient.

The definition of the Ramsey operator that is provided below is not the original definition from [8], but a version that is known to be equivalent [6, Proposition 2.8]. Recall that for a set of ordinals A, an A-list is a sequence $\langle a_{\alpha }\mid \alpha \in A\rangle $ such that $a_{\alpha }\subseteq \alpha $ for any $\alpha \in A$ , and that a set $H\subseteq A$ is homogeneous for $\langle a_{\alpha }\mid \alpha \in A\rangle $ in case $a_{\alpha }=a_{\beta }\cap x$ whenever $\alpha <\beta $ are both in H.

Definition 1.1. Let I be an ideal on $\kappa $ .

• • Given a $\kappa $ -list $\vec a$ , we define the local instance of ${\mathcal I}$ at $\vec {a}$ ,

  $$\begin{align*}\mathcal I^{\vec a}(I)^+=\{x\subseteq\kappa\mid\exists H\in I^+\ H\subseteq x\textrm{ is homogeneous for }\vec a\}, \end{align*}$$
  and let $\mathcal I(I)^+=\bigcap \{\mathcal I^{\vec a}(I)^+\mid \vec a$ is a $\kappa $ -list $\}$ .

• • Given a regressive function $c\colon [\kappa ]^{<\omega }\to \kappa $ , we define the local instance of ${\mathcal R}$ at c,

  $$\begin{align*}\mathcal R^c(I)^+=\{x\subseteq\kappa\mid\exists H\in I^+\ H\subseteq x\textrm{ is homogeneous for }c\}, \end{align*}$$
  and let $\mathcal R(I)^+=\bigcap \{\mathcal R^c(I)^+\mid c\colon [x]^{<\omega }\to \kappa \textrm { regressive}\}$ .

Feng has shown that ${\mathcal R}(I)$ is a normal ideal on $\kappa $ for any ideal I on $\kappa $ , and an analogous result for the operator ${\mathcal I}$ is essentially due to Baumgartner (see also our Lemma 3.6). Some of the classical large cardinal ideals (see [3]) are directly generated by these two large cardinal operators.

Fact 1.2.

1. (1) If $\kappa $ is weakly ineffable, then $\mathcal I([\kappa ]^{<\kappa })$ is the weakly ineffable ideal on $\kappa $ .

2. (2) If $\kappa $ is ineffable, then $\mathcal I({\mathop {\mathrm {NS}}}_{\kappa })$ is the ineffable ideal on $\kappa $ .

3. (3) If $\kappa $ is Ramsey, then $\mathcal R([\kappa ]^{<\kappa })$ is the Ramsey ideal on $\kappa $ .

4. (4) If $\kappa $ is ineffably Ramsey, then $\mathcal R({\mathop {\mathrm {NS}}}_{\kappa })$ is the ineffably Ramsey ideal on $\kappa $ .

In order to define the strongly Ramsey subset operator ${\mathcal S}$ , let us recall that M is a $\kappa $ -model if $M\supseteq \kappa +1$ is a transitive model of $\mathrm { ZFC}^-$ of size $\kappa $ with $M^{<\kappa }\subseteq M$ . An M-ultrafilter U on $\kappa $ is a filter $U\subseteq P(\kappa )^M$ which measures all subsets of $\kappa $ in M. Also recall that an M-ultrafilter U on $\kappa $ is $\kappa $ -amenable for M if whenever $\mathcal A\in M$ is a $\kappa $ -sized collection of subsets of $\kappa $ in M, then $\mathcal A\cap U\in M$ .

Definition 1.3. Let I be an ideal on $\kappa $ . Given a set $a\subseteq \kappa $ , we define the local instance of ${\mathcal S}$ at a by letting $x\in {\mathcal S}^a(I)^+$ if and only if $x\subseteq \kappa $ and there is a $\kappa $ -model M with $a\in M$ and there is a $\kappa $ -amenable M-normal M-ultrafilter U on $\kappa $ such that $U\subseteq I^+$ and $x\in U$ . We let

$$\begin{align*}{\mathcal S}(I)^+=\bigcap\{{\mathcal S}^a(I)^+\mid a\subseteq\kappa\}. \end{align*}$$

It is easy to see that $\kappa $ is strongly Ramsey (as defined in [11]) if and only if $\kappa \in {\mathcal S}([\kappa ]^{<\kappa })^+$ , and furthermore, when $\kappa \in {\mathcal S}(I)^+$ , it follows that ${\mathcal S}(I)$ is a nontrivial normal ideal on $\kappa $ . If $\kappa $ is strongly Ramsey, then ${\mathcal S}([\kappa ]^{<\kappa })$ is the strongly Ramsey ideal on $\kappa $ , as introduced in [13].

In this paper, we will also investigate properties of iterated large cardinal operators. If $\mathcal O$ is a large cardinal operator, and I is an ideal, we define $\mathcal O^{\gamma }(I)$ inductively, setting $\mathcal O^{\gamma +1}(I)=\mathcal O(\mathcal O^{\gamma }(I))$ , and $\mathcal O^{\gamma }(I)=\bigcup _{\delta <\gamma }\mathcal O^{\delta }(I)$ when $\gamma $ is a limit ordinal.

2 Review of higher indescribability

2.1 On the notion of $\Pi ^1_{\xi }$ - and $\Sigma ^1_{\xi }$ -formulas

The following definition differs slightly from that of Bagaria [2, Definition 4.1] in that we allow for $\Pi ^1_{\xi }$ - and $\Sigma ^1_{\xi }$ -formulas to contain parameters of various kinds.

Definition 2.1. Suppose $\kappa $ is a regular cardinal. We define the notions of $\Pi ^1_{\xi }$ - and $\Sigma ^1_{\xi }$ -formula over $V_{\kappa }$ , for all ordinals $\xi $ as follows.

1. (1) A formula $\varphi $ is $\Pi ^1_0$ , or equivalently $\Sigma ^1_0$ , over $V_{\kappa }$ if it is a first order formula in the language of set theory; however, we allow for free variables and parameters from $V_{\kappa }$ of two types, namely of first and of second order.

2. (2) A formula $\varphi $ is $\Pi ^1_{\xi +1}$ over $V_{\kappa }$ if it is of the form $\forall X_{k_1}\cdots \forall X_{k_m}\psi $ where $\psi $ is $\Sigma ^1_{\xi }$ over $V_{\kappa }$ and $m\in \omega $ . Similarly, $\varphi $ is $\Sigma ^1_{\xi +1}$ over $V_{\kappa }$ if it is of the form $\exists X_{k_1}\cdots \exists X_{k_m}\psi $ where $\psi $ is $\Pi ^1_{\xi }$ over $V_{\kappa }$ and $m\in \omega $ . ²

3. (3) When $\xi $ is a limit ordinal, a formula $\varphi $ , with finitely many second-order free variables and finitely many second-order parameters, is $\Pi ^1_{\xi }$ over $V_{\kappa }$ if it is of the form

   $$\begin{align*}\bigwedge_{\zeta<\xi}\varphi_{\zeta}, \end{align*}$$
   where $\varphi _{\zeta }$ is $\Pi ^1_{\zeta }$ over $V_{\kappa }$ for all $\zeta <\xi $ . Similarly, $\varphi $ is $\Sigma ^1_{\xi }$ if it is of the form
   $$\begin{align*}\bigvee_{\zeta<\xi}\varphi_{\zeta}, \end{align*}$$
   where $\varphi _{\zeta }$ is $\Sigma ^1_{\zeta }$ over $V_{\kappa }$ for all $\zeta <\xi $ .

Before we can introduce the concept of higher indescribability, which is based on these formula classes, we will need to review a number of further preliminaries.

2.2 Canonical functions

For a regular uncountable cardinal $\kappa $ , the definition of the $\Pi ^1_{\xi }$ -indescribability of a set $S\subseteq \kappa $ , where $\kappa \leq \xi <\kappa ^+$ , that was introduced in [7], uses a sequence of functions $\langle F^{\kappa }_{\xi }\mid \xi <\kappa ^+\rangle $ , referred to as a sequence of canonical reflection functions at $\kappa $ , which is defined as follows. If $\xi <\kappa $ then we let $F^{\kappa }_{\xi }(\alpha )=\xi $ for all $\alpha \in \kappa $ . If $\xi \in \kappa ^+\setminus \kappa $ , fix a bijection $b_{\kappa ,\xi }:\kappa \to \xi $ and let $F^{\kappa }_{\xi }(\alpha )=b_{\kappa ,\xi }[\alpha ]$ for all $\alpha <\kappa $ . Notice that for each $\xi <\kappa ^+$ , the definition of the $\xi ^{th}$ canonical reflection function $F^{\kappa }_{\xi }$ is independent, modulo the nonstationary ideal, of which bijection $b_{\kappa ,\xi }$ is chosen. That is, if $b^1_{\kappa ,\xi },b^2_{\kappa ,\xi }:\kappa \to \xi $ are two bijections, then the set $\{\alpha <\kappa \mid b^1_{\kappa ,\xi }[\alpha ]=b^2_{\kappa ,\xi }[\alpha ]\}$ contains a club subset of $\kappa $ .

We obtain a sequence of canonical functions $\langle f^{\kappa }_{\xi }\mid \xi <\kappa ^+\rangle $ at $\kappa $ by letting $f^{\kappa }_{\xi }(\alpha )=\mathop {\mathrm { ot}}\nolimits (F^{\kappa }_{\xi }(\alpha ))$ be the transitive collapse of $F^{\kappa }_{\xi }(\alpha )$ , for all $\xi <\kappa ^+$ and all $\alpha <\kappa $ . For all such $\alpha $ and $\xi $ , let $\pi ^{\kappa }_{\xi ,\alpha }:F^{\kappa }_{\xi }(\alpha )\to f^{\kappa }_{\xi }(\alpha )$ be the transitive collapsing map of $F^{\kappa }_{\xi }(\alpha )$ . We will assume a fixed choice of these objects throughout the paper.

Intuitively, $\xi $ is to $\kappa $ as $f^{\kappa }_{\xi }(\alpha )$ is to $\alpha $ , and one can think of $f^{\kappa }_{\xi }(\alpha )$ as being $\alpha $ ’s version of $\xi $ in the sense that when some property involving $\kappa $ and $\xi $ is reflected down to $\alpha $ , the statement will be about $\alpha $ and $f^{\kappa }_{\xi }(\alpha )$ . Notice that (see [7, Proposition 2.1]) if I is a normal ideal on $\kappa $ , G is generic for $\mathcal P(\kappa )/I$ , and $j\colon V\to \mathop {\mathrm {Ult}}$ , with $\mathop {\mathrm {Ult}}=V^{\kappa }/G$ , is the corresponding generic ultrapower embedding, then, for all $\xi <\kappa ^+$ , the $\xi ^{th}$ canonical function $f^{\kappa }_{\xi }$ represents the ordinal $\xi $ in $\mathop {\mathrm {Ult}}$ , that is, $j(f^{\kappa }_{\xi })(\kappa )=[f^{\kappa }_{\xi }]_G=\xi $ . Similarly, for all $\xi <\kappa ^+$ , the $\xi ^{th}$ canonical reflection function $F^{\kappa }_{\xi }$ represents $j"\xi $ in $\mathop {\mathrm {Ult}}$ , that is, $j(F^{\kappa }_{\xi })(\kappa )=[F^{\kappa }_{\xi }]_G=j"\xi $ . Some background material on generic ultrapowers may be found in [9], but we will only need very little.

Throughout the rest of our paper, with respect to a regular and uncountable cardinal $\kappa $ , let G denote an arbitrary generic filter for $\mathcal P(\kappa )/{\mathop {\mathrm {NS}}}_{\kappa }$ over V, and let $j\colon V\to \mathop {\mathrm {Ult}}$ be the corresponding generic ultrapower embedding with critical point $\kappa $ . We may sometimes make the extra assumption that G contains some particular stationary subset of $\kappa $ as an element. Note that $\mathop {\mathrm { Ult}}$ may not be well-founded, but it is so up to $\kappa ^+$ (as calculated in V), and also that $H(\kappa ^+)\subseteq \mathop {\mathrm {Ult}}$ and that $H(\kappa )=H(\kappa )^{\mathop {\mathrm {Ult}}}$ in case $\kappa $ is inaccessible.

With respect to the objects that we have fixed at the beginning of this section, at the level of $j(\kappa )$ in $\mathop {\mathrm {Ult}}$ , we will always be using the sequence of bijections

$$\begin{align*}\langle b_{j(\kappa),\xi}\mid j(\kappa)\le\xi<j(\kappa)^+\rangle=j(\langle b_{\kappa,\xi}\mid\kappa\le\xi<\kappa^+\rangle)\end{align*}$$

to define the sequences of canonical (reflection) functions that we use.

The following proposition is an easy folklore observation, and will be used to show that the provability of certain statements about generic ultrapowers induces (ground model) statements about canonical (reflection) functions to hold on a club.

Proposition 2.2. Suppose $\kappa $ is a regular uncountable cardinal, $S\subseteq \kappa $ , and whenever G is generic for $P(\kappa )/{\mathop {\mathrm {NS}}}_{\kappa }$ it follows that $\kappa \in j(S)$ where $j:V\to \mathop {\mathrm {Ult}}$ is the corresponding generic ultrapower. Then S contains a club subset of $\kappa $ in V.

Proof For the sake of contradiction, suppose S does not contain a club subset of $\kappa $ in V. Then $T=\kappa \setminus S$ is stationary and we may let G be generic for $P(\kappa )/{\mathop {\mathrm {NS}}}_{\kappa }$ with $T\in G$ . Then $\kappa \in j(T)$ , but this contradicts our assumption that $\kappa \in j(S)$ .

We will need the following lemma later on, which was also presented in [7, Lemmas 2.7 and 2.8], together with easy elementary proofs. For the sake of completeness, and since we will often make use of similar more difficult arguments later on, we would like to provide an even easier proof that makes use of generic ultrapower representations. ³

Lemma 2.3. Suppose $\kappa $ is a regular cardinal. For all $\xi <\kappa ^+$ the following hold.

1. (1) If $\xi $ is a limit ordinal, then the set

   $$\begin{align*}D_0=\{\alpha<\kappa\mid f^{\kappa}_{\xi}(\alpha) \text{is a limit ordinal}\} \end{align*}$$
   is a club subset of $\kappa $ .

2. (2) The set

   $$\begin{align*}D_1 = \{\alpha<\kappa\mid f^{\kappa}_{\xi+1}(\alpha)=f^{\kappa}_{\xi}(\alpha)+1\} \end{align*}$$
   contains a club subset of $\kappa $ .

Proof

1. (1) It is easy to see that $D_0$ is closed below $\kappa $ . Let $j:V\to \mathop {\mathrm {Ult}}$ be any generic ultrapower obtained by forcing with $P(\kappa )/{\mathop {\mathrm {NS}}}_{\kappa }$ . Then $j(f^{\kappa }_{\xi })(\kappa )=\xi $ is a limit ordinal and hence $\kappa \in j(D_0)$ . By Proposition 2.2, $D_0$ is unbounded in $\kappa $ .

2. (2) Using that $j(f^{\kappa }_{\xi +1})(\kappa )=\xi +1$ , it follows that $\kappa \in j(D_1)$ , and the result thus follows by Proposition 2.2.

2.3 Restrictions of formulas

Let $\kappa $ be a regular and uncountable cardinal throughout. When defining the $\Pi ^1_{\xi }$ -indescribability of sets $S\subseteq \kappa $ where $\kappa \leq \xi <\kappa ^+$ , one cannot simply demand that every $\Pi ^1_{\xi }$ -sentence which is true in $V_{\kappa }$ must be true in $V_{\alpha }$ for some $\alpha \in S$ because, for example, there are $\Pi ^1_{\kappa }$ sentences with no first or second-order parameters that are true in $V_{\kappa }$ but which are false in $V_{\alpha }$ for all $\alpha <\kappa $ (see [7, Section 1]). However, one can demand that whenever a $\Pi ^1_{\xi }$ -sentence $\varphi $ holds in $V_{\kappa }$ , there must be some $\alpha \in S$ such that a canonically defined restriction $\varphi \mathrm {|}^{\kappa }_{\alpha }$ is true in $V_{\alpha }$ . Although we summarize the required background here, one may consult [7, Sections 3 and 4] for more information on such canonically defined restrictions of formulas.

In the following, when we talk about either $\Pi ^1_{\xi }$ -formulas or $\Sigma ^1_{\xi }$ -formulas over $V_{\kappa }$ , for some $\xi <\kappa ^+$ , we mean formulas which are of that exact complexity, and not any simpler one, and we say that these formulas are of complexity $\xi $ . We also treat such formulas as set theoretic objects, and thus we tacitly assume some reasonable and natural coding of these formulas, and interchangeably use these formulas on the meta level as well as the object level. In particular, we assume that any $\Pi ^1_{\xi }$ -formula or $\Sigma ^1_{\xi }$ -formula over $V_{\kappa }$ is coded as an element of $H(\kappa ^+)$ .

Definition 2.4. By induction on $\xi <\kappa ^+$ , we define $\varphi \mathrm {|}^{\kappa }_{\alpha }$ for all $\Pi ^1_{\xi }$ formulas $\varphi $ over $V_{\kappa }$ and all regular $\alpha <\kappa $ as follows. ⁴ First assume that $\xi <\kappa $ . If

$$\begin{align*}\varphi=\varphi(X_1,\ldots,X_m,A_1,\ldots,A_n),\end{align*}$$

with free second order variables $X_1,\ldots ,X_m$ and second order parameters $A_1,\ldots ,A_n$ , such that $\alpha>\xi $ , and all first order parameters of $\varphi $ are elements of $V_{\alpha }$ , then we define

$$\begin{align*}\varphi\mathrm{|}^{\kappa}_{\alpha}=\varphi(X_1,\ldots,X_m,A_1\cap V_{\alpha},\ldots,A_n\cap V_{\alpha}), \end{align*}$$

and we leave $\varphi \mathrm {|}^{\kappa }_{\alpha }$ undefined otherwise.

If $\xi =\zeta +1$ is a successor ordinal and $\varphi =\forall X_{k_1}\ldots \forall X_{k_m}\psi $ is $\Pi ^1_{\zeta +1}$ over $V_{\kappa }$ , then we define

$$\begin{align*}\varphi\mathrm{|}^{\kappa}_{\alpha}=\forall X_{k_1}\ldots\forall X_{k_m}(\psi\mathrm{|}^{\kappa}_{\alpha}) \end{align*}$$

in case $\psi \mathrm {|}^{\kappa }_{\alpha }$ is defined, and leave $\varphi \mathrm {|}^{\kappa }_{\alpha }$ undefined otherwise. We define $\varphi \mathrm {|}^{\kappa }_{\alpha }$ analogously when $\varphi $ is $\Sigma ^1_{\zeta +1}$ .

If $\xi \in \kappa ^+\setminus \kappa $ is a limit ordinal, and

$$\begin{align*}\varphi=\bigwedge_{\zeta<\xi}\psi_{\zeta} \end{align*}$$

is $\Pi ^1_{\xi }$ over $V_{\kappa }$ , then we define

$$\begin{align*}\varphi\mathrm{|}^{\kappa}_{\alpha}=\bigwedge_{\zeta\in f^{\kappa}_{\xi}(\alpha)}\psi_{(\pi^{\kappa}_{\xi,\alpha})^{-1}(\zeta)}\mathrm{|}^{\kappa}_{\alpha}\end{align*}$$

in case $\psi _{(\pi ^{\kappa }_{\xi ,\alpha })^{-1}(\zeta )}\mathrm {|}^{\kappa }_{\alpha }$ is a $\Pi ^1_{\zeta }$ -formula over $V_{\alpha }$ for every $\zeta \in f^{\kappa }_{\xi }(\alpha )$ . We leave $\varphi \mathrm {|}^{\kappa }_{\alpha }$ undefined otherwise. We define $\varphi \mathrm {|}^{\kappa }_{\alpha }$ similarly when $\xi \in \kappa ^+\setminus \kappa $ is a limit ordinal and $\varphi $ is $\Sigma ^1_{\xi }$ .

Note that by a simple induction on formula complexity, we obtain the following.

Observation 2.5. If $\varphi $ is a $\Pi ^1_{\xi }$ - or $\Sigma ^1_{\xi }$ -formula over $V_{\kappa }$ , and $\alpha <\kappa $ is regular, then whenever $\varphi \mathrm {|}^{\kappa }_{\alpha }$ is defined, it is a $\Pi ^1_{f^{\kappa }_{\xi }(\alpha )}$ - or $\Sigma ^1_{f^{\kappa }_{\xi }(\alpha )}$ -formula over $V_{\alpha }$ respectively.

Remark 2.6. We will need the following properties of our coding of formulas. We will leave it to our readers to check that any reasonable coding of formulas has these properties. Assume that $\varphi $ is either a $\Pi ^1_{\xi }$ - or $\Sigma ^1_{\xi }$ -formula over $V_{\kappa }$ for some $\xi <\kappa ^+$ .

1. (1) If $\xi <\kappa $ , and $A_1,\ldots ,A_n$ are all second order parameters appearing in $\varphi $ , then

   $$\begin{align*}j(\varphi(A_1,\ldots,A_n))=\varphi(j(A_1),\ldots,j(A_n)).\end{align*}$$

2. (2) $j(\forall X\,\varphi )=\forall X\,j(\varphi )$ .

3. (3) If $\xi \ge \kappa $ is a limit ordinal, and $\varphi $ is either of the form $\varphi =\bigwedge _{\zeta <\xi }\psi _{\zeta }$ , or of the form $\bigvee _{\zeta <\xi }\psi _{\zeta }$ , let $\vec \psi =\langle \psi _{\zeta }\mid \zeta <\xi \rangle $ . Then,

   $$\begin{align*}j(\varphi)=\bigwedge_{\zeta<j(\xi)}j(\vec\psi)_{\zeta}\quad\textrm{or}\quad j(\varphi)=\bigvee_{\zeta<j(\xi)}j(\vec\psi)_{\zeta}\end{align*}$$
   respectively.

We will need the following.

Observation 2.7. Let $\kappa $ be a regular uncountable cardinal, and let $\xi <\kappa ^+$ . Let $\vec \pi =\langle \pi ^{\kappa }_{\xi ,\alpha }\mid \alpha <\kappa \rangle $ . Then $j(\vec \pi )_{\kappa }^{-1}=j\upharpoonright \xi $ .

Proof Since each $\pi ^{\kappa }_{\xi ,\alpha }$ is the transitive collapse of $F^{\kappa }_{\xi }(\alpha )$ , it follows by elementarity that $j(\vec \pi )_{\kappa }$ is the transitive collapse of $j(F^{\kappa }_{\xi })(\kappa )=j"\xi $ . Hence, the image of $j(\vec \pi )_{\kappa }$ is $\xi $ , and its inverse is thus clearly identical to $j\upharpoonright \xi $ .

The proof of the next lemma is essentially the same as the first part of the proof of [7, Proposition 3.8]. Regarding the assumption of the next lemma, and also of some later results, note that $\kappa $ is regular in $\mathop {\mathrm {Ult}}$ if and only if G contains the set of regular cardinals below $\kappa $ . This is of course only possible if that latter set is a stationary subset of $\kappa $ , i.e., if $\kappa $ is weakly Mahlo.

Lemma 2.8. If $\varphi $ is either a $\Pi ^1_{\xi }$ - or $\Sigma ^1_{\xi }$ -formula over $V_{\kappa }$ for some $\xi <\kappa ^+$ and $\kappa $ is regular in $\mathop {\mathrm {Ult}}$ , then in $\mathop {\mathrm {Ult}}$ ,

$$\begin{align*}j(\varphi)\mathrm{|}^{j(\kappa)}_{\kappa}=\varphi. \end{align*}$$

Proof Regularity of $\kappa $ in $\mathop {\mathrm {Ult}}$ is needed so that $j(\varphi )\mathrm {|}^{j(\kappa )}_{\kappa }$ could possibly be defined in $\mathop {\mathrm {Ult}}$ . The proof proceeds by induction on $\xi <\kappa ^+$ . The case when $\xi <\kappa $ is easy, for then by Remark 2.6(1),

$$\begin{align*}j(\varphi(A_1,\ldots,A_n))=\varphi(j(A_1),\ldots,j(A_n)), \end{align*}$$

and thus $j(\varphi )\mathrm {|}^{j(\kappa )}_{\kappa }=\varphi $ by the definition of the restriction operation in this case. Successor steps above $\kappa $ are easily treated as well, for by Remark 2.6(2), in this case,

$$\begin{align*}j(\forall X\psi(X))=\forall X j(\psi(X)). \end{align*}$$

At limit steps $\xi \ge \kappa $ , if $\varphi =\bigwedge _{\zeta <\xi }\psi _{\zeta }$ is a $\Pi ^1_{\xi }$ -formula, let $\vec \psi =\langle \psi _{\zeta }\mid \zeta <\xi \rangle $ , and let $\vec \pi =\langle \pi ^{\kappa }_{\xi ,\alpha }\mid \alpha <\kappa \rangle $ . Then, by Remark 2.6(3), $j(\varphi )=\bigwedge _{\zeta <j(\xi )}j(\vec \psi )_{\zeta }$ , and therefore

$$\begin{align*}j(\varphi)\mathrm{|}^{j(\kappa)}_{\kappa}=\bigwedge_{\zeta\in f^{j(\kappa)}_{j(\xi)}(\kappa)}j(\vec\psi)_{j(\vec\pi)_{\kappa}^{-1}(\zeta)}\mathrm{|}^{j(\kappa)}_{\kappa}=\bigwedge_{\zeta\in\xi}(j(\psi_{\zeta}))\mathrm{|}^{j(\kappa)}_{\kappa}=\varphi,\end{align*}$$

using that $f^{j(\kappa )}_{j(\xi )}(\kappa )=j(f^{\kappa }_{\xi })(\kappa )=\xi $ by our choice of canonical functions at the level of $j(\kappa )$ in $\mathop {\mathrm {Ult}}$ , and by Observation 2.7.

The case when $\varphi $ is a $\Sigma ^1_{\xi }$ -formula is treated in exactly the same way.

A neat feature, which could also be seen as a possible motivation for our restriction operation, is now the following.

Lemma 2.9. Assume that $\kappa $ is regular in $\mathop {\mathrm {Ult}}$ , that $\varphi $ is either a $\Pi ^1_{\xi }$ - or $\Sigma ^1_{\xi }$ -formula over $V_{\kappa }$ for some $\xi <\kappa ^+$ , and that $\Phi \colon \kappa \to V_{\kappa }$ is a function with $\Phi (\alpha )=\varphi \mathrm {|}^{\kappa }_{\alpha }$ for every regular $\alpha <\kappa $ . Then, $\Phi $ represents $\varphi $ in $\mathop {\mathrm {Ult}}$ . That is, $j(\Phi )(\kappa )=[\Phi ]_U=\varphi $ .

Proof Note that $j(\Phi )(\kappa )=j(\varphi )\mathrm {|}^{j(\kappa )}_{\kappa }=\varphi $ by Lemma 2.8.

The following was essentially shown as [7, Lemma 3.6] using an elementary proof, and becomes almost trivial with a generic ultrapower argument.

Lemma 2.10. Suppose $\kappa $ is weakly Mahlo. For any $\xi <\kappa ^+$ , if $\varphi $ is a $\Pi ^1_{\xi }$ - or $\Sigma ^1_{\xi }$ -formula over $V_{\kappa }$ , then there is a club subset C of $\kappa $ such that for any regular $\alpha \in C$ , $\varphi \mathrm {|}^{\kappa }_{\alpha }$ is defined, and therefore a $\Pi ^1_{f^{\kappa }_{\xi }(\alpha )}$ - or $\Sigma ^1_{f^{\kappa }_{\xi }(\alpha )}$ -formula over $V_{\alpha }$ respectively by Observation 2.5.

Proof Assume for a contradiction that the conclusion of the lemma fails. This means that there is a stationary set T consisting of regular and uncountable cardinals $\alpha $ such that $\varphi \mathrm {|}^{\kappa }_{\alpha }$ is not defined. Assume that $T\in G$ . Then, $\kappa \in j(T)$ , and therefore $\kappa $ is regular in $\mathop {\mathrm {Ult}}$ ; however $j(\varphi )\mathrm {|}^{j(\kappa )}_{\kappa }$ is not defined in $\mathop {\mathrm {Ult}}$ . But, by Lemma 2.8, $j(\varphi )\mathrm {|}^{j(\kappa )}_{\kappa }=\varphi $ , which clearly yields a contradiction.

We will need the following property of restrictions of formulas, which is established using an argument similar to that of Lemma 2.8.

Lemma 2.11. Suppose that $\kappa $ is regular in $\mathop {\mathrm {Ult}}$ . If $\varphi $ is either a $\Pi ^1_{\xi }$ - or $\Sigma ^1_{\xi }$ -formula over $V_{\kappa }$ for some $\xi <\kappa ^+$ , and $\alpha <\kappa $ is regular such that $\varphi \mathrm {|}^{\kappa }_{\alpha }$ is defined, then

$$\begin{align*}j(\varphi)\mathrm{|}^{j(\kappa)}_{\alpha}=\varphi\mathrm{|}^{\kappa}_{\alpha}, \end{align*}$$

with the former being calculated in $\mathop {\mathrm {Ult}}$ , and the latter being calculated in V.

Proof By induction on $\xi <\kappa ^+$ . This is immediate in case $\xi <\kappa $ , for then by Remark 2.6(1), $j(\varphi (A_1,\ldots ,A_n))=\varphi (j(A_1),\ldots ,j(A_n))$ , and thus $j(\varphi )\mathrm {|}^{j(\kappa )}_{\alpha }=\varphi \mathrm {|}^{\kappa }_{\alpha }$ by the definition of the restriction operation in this case. It is also immediate for successor steps above $\kappa $ , for then by Remark 2.6(2), $j(\forall \vec X\psi )=\forall \vec X j(\psi )$ .

At limit steps $\xi \ge \kappa $ , if $\varphi =\bigwedge _{\zeta <\xi }\psi _{\zeta }$ is a $\Pi ^1_{\xi }$ -formula, let $\vec \psi =\langle \psi _{\zeta }\mid \zeta <\xi \rangle $ , and let $\vec \pi =\langle \pi ^{\kappa }_{\xi ,\alpha }\mid \alpha <\kappa \rangle $ . Then, by Remark 2.6(3), $j(\varphi )=\bigwedge _{\zeta <j(\xi )}j(\vec \psi )_{\zeta }$ , and therefore, assuming for now that $j(\varphi )\mathrm {|}^{j(\kappa )}_{\alpha }$ is defined,

$$\begin{align*}j(\varphi)\mathrm{|}^{j(\kappa)}_{\alpha}=\bigwedge_{\zeta\in j(f^{\kappa}_{\xi})(\alpha)}j(\vec\psi)_{j(\vec\pi)_{\alpha}^{-1}(\zeta)}\mathrm{|}^{j(\kappa)}_{\alpha}=\bigwedge_{\zeta\in j(f^{\kappa}_{\xi})(\alpha)}j(\psi_{j^{-1}(j(\vec\pi)_{\alpha}^{-1}(\zeta))})\mathrm{|}^{j(\kappa)}_{\alpha},\end{align*}$$

using that $j(\vec \pi )_{\alpha }^{-1}[j(f^{\kappa }_{\xi })(\alpha )]=j(F^{\kappa }_{\xi })(\alpha )\subseteq j(F^{\kappa }_{\xi })(\kappa )=j"\xi $ . By our inductive hypothesis, for each $\gamma \in \xi $ and every regular $\alpha <\kappa $ , $j(\psi _{\gamma })\mathrm {|}^{j(\kappa )}_{\alpha }=\psi _{\gamma }\mathrm {|}^{\kappa }_{\alpha }$ . Thus,

$$\begin{align*}j(\varphi)\mathrm{|}^{j(\kappa)}_{\alpha}=\bigwedge_{\zeta\in j(f^{\kappa}_{\xi})(\alpha)}\psi_{j^{-1}(j(\vec\pi)_{\alpha}^{-1}(\zeta))}\mathrm{|}^{\kappa}_{\alpha}. \end{align*}$$

Now,

$$\begin{align*}\varphi\mathrm{|}^{\kappa}_{\alpha}=\bigwedge_{\zeta\in f^{\kappa}_{\xi}(\alpha)}\psi_{(\pi^{\kappa}_{\xi,\alpha})^{-1}(\zeta)}\mathrm{|}^{\kappa}_{\alpha}. \end{align*}$$

Since $\alpha <\kappa $ we have $j(f^{\kappa }_{\xi })(\alpha )=f^{\kappa }_{\xi }(\alpha )$ , and furthermore

$$\begin{align*}(\pi^{\kappa}_{\xi,\alpha})^{-1}[f^{\kappa}_{\xi}(\alpha)]=F^{\kappa}_{\xi}(\alpha)=(j^{-1}\circ j(\vec\pi)_{\alpha}^{-1})[j(f^{\kappa}_{\xi})(\alpha)], \end{align*}$$

showing the above restrictions of $\varphi $ and of $j(\varphi )$ to be equal, ⁵ and thus in particular also showing that $j(\varphi )\mathrm {|}^{j(\kappa )}_{\alpha }$ is defined, as desired.

The case when $\varphi $ is a $\Sigma ^1_{\xi }$ -formula is treated in exactly the same way.

We can now easily deduce the following, which was originally shown as [7, Proposition 5.7].

Proposition 2.12. Suppose $\kappa $ is weakly Mahlo, and $\xi <\kappa ^+$ . For any formula $\varphi $ which is either $\Pi ^1_{\xi }$ or $\Sigma ^1_{\xi }$ over $V_{\kappa }$ , there is a club $D\subseteq \kappa $ such that for all regular uncountable $\alpha \in D$ , $\varphi \mathrm {|}^{\kappa }_{\alpha }$ is defined, and the set $D_{\alpha }$ of all ordinals $\beta <\alpha $ such that $(\varphi \mathrm {|}^{\kappa }_{\alpha })\mathrm {|}^{\alpha }_{\beta }$ is defined and $(\varphi \mathrm {|}^{\kappa }_{\alpha })\mathrm {|}^{\alpha }_{\beta }=\varphi \mathrm {|}^{\kappa }_{\beta }$ , is in the club filter on $\alpha $ .

Proof Assume for a contradiction that the conclusion of the proposition fails. By Lemma 2.10, this means that there is a stationary set T consisting of regular and uncountable cardinals $\alpha $ such that the set $D_{\alpha }$ has stationary complement $E_{\alpha }\subseteq \alpha $ . Using Lemma 2.10 once again, we may assume that $(\varphi \mathrm {|}^{\kappa }_{\alpha })\mathrm {|}^{\alpha }_{\beta }$ is defined for every $\alpha \in T$ and every $\beta \in E_{\alpha }$ . Let $\vec E$ denote the sequence $\langle E_{\alpha }\mid \alpha \in T\rangle $ . Assume that $T\in G$ . Then, $\kappa \in j(T)$ , and thus $j(\vec E)_{\kappa }$ is stationary in $\mathop {\mathrm {Ult}}$ . But,

$$\begin{align*}j(\vec E)_{\kappa}=\{\beta<\kappa\mid(j(\varphi)\mathrm{|}^{j(\kappa)}_{\kappa})\mathrm{|}^{\kappa}_{\beta}\ne j(\varphi)\mathrm{|}^{j(\kappa)}_{\beta}\}.\end{align*}$$

Note that by Lemma 2.9, $j(\varphi )\mathrm {|}^{j(\kappa )}_{\kappa }=\varphi $ . But then, by Lemmas 2.10 and 2.11, $j(\vec E)_{\kappa }$ is nonstationary in $\mathop {\mathrm {Ult}}$ , which gives our desired contradiction.

2.4 Higher indescribability

The notion of $\Pi ^1_{\xi }$ -indescribability of (subsets of) a cardinal $\kappa $ when $\xi <\kappa $ was introduced by Bagaria in [2], and was extended by the first author as follows.

Definition 2.13 [7, Definition 3.4].

Suppose $\kappa $ is a cardinal and $\xi <\kappa ^+$ . A set $S\subseteq \kappa $ is $\Pi ^1_{\xi }$ -indescribable if for every $\Pi ^1_{\xi }$ sentence $\varphi $ over $V_{\kappa }$ , if $V_{\kappa }\models \varphi $ then there is some $\alpha \in S$ such that $\varphi \mathrm {|}^{\kappa }_{\alpha }$ is defined, and $V_{\alpha }\models \varphi \mathrm {|}^{\kappa }_{\alpha }$ .

Note that the value of any particular $\varphi \mathrm {|}^{\kappa }_{\alpha }$ depends on our choice of bijections $b_{\kappa ,\xi }$ ; however using Lemma 2.8 and Proposition 2.2, it is easy to see that for any two choices of sequences $\langle b_{\kappa ,\xi }\mid \xi <\kappa ^+\rangle $ , the corresponding $\varphi \mathrm {|}^{\kappa }_{\alpha }$ ’s agree on a club, and thus in particular the above notion of higher indescribability is independent of that choice. ⁶

Note also that $S\subseteq \kappa $ is $\Pi ^1_0$ -indescribable if and only if S is a stationary subset of $\kappa $ . We will say that $S\subseteq \kappa $ is $\Pi ^1_{-1}$ -indescribable in case S is an unbounded subset of $\kappa $ . For $\xi \in \{-1\}\cup \kappa ^+$ , we let $\Pi ^1_{\xi }(\kappa )^+$ be the collection of all $\Pi ^1_{\xi }$ -indescribable subsets of $\kappa $ . It was shown by the first author in [7, Theorem 5.5] that if $\kappa $ is a cardinal, $\xi <\kappa ^+$ , and $\kappa $ is $\Pi ^1_{\xi }$ -indescribable, then $\Pi ^1_{\xi }(\kappa )$ is a nontrivial normal ideal on $\kappa $ .

3 Generalizations of a result of Baumgartner

A key result from Baumgartner’s [3] is the following theorem that indicates the strength of subtlety. Recall that $S\subseteq \kappa $ is subtle in case whenever $\vec S=\langle S_{\alpha }\mid \alpha \in S\rangle $ is an S-list and $C\subseteq \kappa $ is club, then there are $\alpha <\beta $ both in $S\cap C$ such that $S_{\alpha }=S_{\beta }\cap \alpha $ .

Theorem 3.1 (Baumgartner [3, Theorem 4.1]).

Suppose $S\subseteq \kappa $ is subtle and $\vec S=\langle S_{\alpha }\mid \alpha \in S\rangle $ is an S-list. Let

$$\begin{align*}A=\{\alpha\in S\mid(\exists X\subseteq S\cap\alpha)(\forall \eta<\omega\ X\in\Pi^1_{\eta}(\alpha)^+)\land(X\cup\{\alpha\}\text{ is hom. for }\vec{S})\}. \end{align*}$$

Then, $S\setminus A$ is not subtle.

In this section, we want to provide a strengthening of Baumgartner’s theorem with respect to higher indescribability, and then apply this to obtain a related result on iterations of the ineffability operator ${\mathcal I}$ .

3.1 Coding formulas

When $\kappa $ is inaccessible, we will need a sort of improved coding of $\Pi ^1_{\xi }$ - and $\Sigma ^1_{\xi }$ -formulas over $V_{\kappa }$ for $\xi <\kappa ^+$ , with the property that such formulas over $V_{\kappa }$ are coded as subsets of $\kappa $ .

Definition 3.2. Suppose $\kappa $ is an inaccessible cardinal, and fix a bijection

$$\begin{align*}b^{\kappa}\colon V_{\kappa}\to\kappa. \end{align*}$$

For ⁷ each $\kappa \le \xi <\kappa ^+$ , fix a well-ordering

$$\begin{align*}R^{\kappa}_{\xi}\subseteq\kappa\times\kappa \end{align*}$$

of $\kappa $ such that $\mathop {\mathrm {ot}}\nolimits (\kappa ,R^{\kappa }_{\xi })=\xi $ and let

$$\begin{align*}b_{\kappa,\xi}:\kappa\to\xi \end{align*}$$

be the bijection derived from $R^{\kappa }_{\xi }$ . We call the above the coding parameters at $\kappa $ . In what follows, by induction on formula complexity, we define a coding function $\mathrm {code}^{\kappa }$ such that whenever for some $\xi <\kappa ^+$ , $\varphi $ is a $\Pi ^1_{\xi }$ - or $\Sigma ^1_{\xi }$ -formula over $V_{\kappa }$ , $\mathrm {code}^{\kappa }(\varphi )$ is a subset of $\kappa $ .

If $\xi <\kappa $ , we let $\mathrm {code}^{\kappa }(\varphi )$ be a subset of $\kappa $ coding $\varphi $ in some reasonable way, making use of the bijection $b^{\kappa }$ to code the parameters of $\varphi $ . In particular, we require that its first slice of code, $(\mathrm {code}^{\kappa }(\varphi ))_0=\varnothing $ , ⁸ we use its slices with finite indices to code the second order parameters of $\varphi $ , we only use boundedly many nonempty slices to code $\varphi $ , and all such slices with infinite index are bounded subsets of $\kappa $ .

Let $\varphi $ be a $\Pi ^1_{\xi }$ - or $\Sigma ^1_{\xi }$ -formula over $V_{\kappa }$ for some $\kappa \le \xi <\kappa ^+$ , and assume that we have inductively defined $\mathrm { code}^{\kappa }(\psi )$ whenever $\psi $ is of lower complexity. We define $\mathrm {code}^{\kappa }(\varphi )$ to be a subset of $\kappa $ as follows.

• • Suppose $\xi =\zeta +1$ is a successor ordinal. If $\varphi =\forall X_{k_0}\ldots \forall X_{k_m}\psi $ is $\Pi ^1_{\zeta +1}$ with $m\in \omega $ and $\psi $ being $\Sigma ^1_{\zeta }$ , we define

  $$\begin{align*}(\mathrm{code}^{\kappa}(\varphi))_0=\{k_0,\ldots,k_m,\omega\} \end{align*}$$
  and if $\varphi =\exists X_{k_0}\ldots \exists X_{k_m}\psi $ is $\Sigma ^1_{\zeta +1}$ with $m\in \omega $ and $\psi $ being $\Pi ^1_{\zeta }$ , we define
  $$\begin{align*}(\mathrm{code}^{\kappa}(\varphi))_0=\{k_0,\ldots,k_m,\omega+1\}. \end{align*}$$
  In either case, we let
  $$\begin{align*}(\mathrm{code}^{\kappa}(\varphi))_1=\mathrm{code}^{\kappa}(\psi), \end{align*}$$
  and for $1<\nu <\kappa $ , we let
  $$\begin{align*}(\mathrm{code}^{\kappa}(\varphi))_{\nu}=\varnothing. \end{align*}$$

• • Suppose $\xi $ is a limit ordinal. If $\varphi =\bigwedge _{\zeta <\xi }\psi _{\zeta }$ is $\Pi ^1_{\xi }$ , we let

  $$\begin{align*}(\mathrm{code}^{\kappa}(\varphi))_0=\{0\}, \end{align*}$$
  and if $\varphi =\bigvee _{\zeta <\xi }\psi _{\zeta }$ is $\Sigma ^1_{\xi }$ , we let
  $$\begin{align*}(\mathrm{code}^{\kappa}(\varphi))_0=\{1\}. \end{align*}$$
  In either case, we let
  $$\begin{align*}(\mathrm{code}^{\kappa}(\varphi))_1=\Gamma[R^{\kappa}_{\xi}], \end{align*}$$
  where $\Gamma $ denotes the Gödel pairing function, and for all $\nu <\kappa $ we let
  $$\begin{align*}(\mathrm{code}^{\kappa}(\varphi))_{2+\nu}=\mathrm{code}^{\kappa}(\psi_{b_{\kappa,\xi}(\nu)}). \end{align*}$$

Fix a sequence of coding parameters at $\alpha $ for every inaccessible $\alpha \le \kappa $ , such that the sequence $\langle b^{\alpha }\mid \alpha \le \kappa \rangle $ is $\subseteq $ -increasing. Note that, using the j-images of our coding parameters to code in $\mathop {\mathrm {Ult}}$ , by elementarity, for any relevant formula $\varphi $ over $V_{\kappa }$ , we have

$$\begin{align*}j(\mathrm{code}^{\kappa}(\varphi))=\mathrm{code}^{j(\kappa)}(j(\varphi)). \end{align*}$$

Lemma 3.3. Suppose $\kappa $ is Mahlo. If $\varphi $ is a $\Pi ^1_{\xi }$ - or $\Sigma ^1_{\xi }$ -formula over $V_{\kappa }$ for some $\xi <\kappa ^+$ , $H\subseteq \kappa $ is stationary and consists only of regular cardinals, and for each $\alpha \in H$ , there is a $\Pi ^1_{f^{\kappa }_{\xi }(\alpha )}$ - or $\Sigma ^1_{f^{\kappa }_{\xi }(\alpha )}$ -formula $\varphi ^{\alpha }$ over $V_{\alpha }$ respectively, such that

$$\begin{align*}\mathrm{code}^{\alpha}(\varphi^{\alpha})=\mathrm{code}^{\kappa}(\varphi)\cap\alpha, \end{align*}$$

then there is a club $C\subseteq \kappa $ such that for each $\alpha \in H\cap C$ , $\varphi \mathrm {|}^{\kappa }_{\alpha }$ is defined, and

$$\begin{align*}\mathrm{code}^{\kappa}(\varphi)\cap\alpha=\mathrm{code}^{\alpha}(\varphi\mathrm{|}^{\kappa}_{\alpha}). \end{align*}$$

Proof Suppose for a contradiction that the conclusion of the lemma fails. Using Lemma 2.10, this means that there is a stationary set $T\subseteq H$ such that for every $\alpha \in T$ ,

$$\begin{align*}\mathrm{code}^{\kappa}(\varphi)\cap\alpha\ne\mathrm{code}^{\alpha}(\varphi\mathrm{|}^{\kappa}_{\alpha}). \end{align*}$$

Assume $T\in G$ . Then, $\kappa \in j(T)$ , and hence in $\mathop {\mathrm {Ult}}$ , $\kappa $ is regular and

$$\begin{align*}\mathrm{code}^{j(\kappa)}(j(\varphi))\cap\kappa\ne\mathrm{ code}^{\kappa}(j(\varphi)\mathrm{|}^{j(\kappa)}_{\kappa}). \end{align*}$$

But by Lemma 2.8, this means that in $\mathop {\mathrm {Ult}}$ ,

(1) $$ \begin{align} \mathrm{code}^{j(\kappa)}(j(\varphi))\cap\kappa\ne\mathrm{code}^{\kappa}(\varphi). \end{align} $$

Let us show that (1) is false, thus yielding our desired contradiction. First, let us consider the case in which $\xi <\kappa $ . We have $j(\varphi (A_1,\ldots ,A_n))=\varphi (j(A_1),\ldots ,j(A_n))$ in this case, and by our choice of bijections we see that in $\mathop {\mathrm {Ult}}$ , $b^{j(\kappa )}\supseteq b^{\kappa }$ and hence first order parameters are coded in the same way. Furthermore, by our choice of reasonable coding, we observe that $\mathrm {code}^{j(\kappa )}(j(\varphi ))\cap \kappa $ and $\mathrm {code}^{\kappa }(\varphi )$ have the same slices and hence $\mathrm { code}^{j(\kappa )}(j(\varphi ))\cap \kappa = \mathrm {code}^{\kappa }(\varphi )$ .

Let us inductively look at the cases when $\xi \ge \kappa $ . The successor ordinal case is immediate, comparing all slices of $\mathrm {code}^{j(\kappa )}(j(\varphi ))\cap \kappa $ and of $\mathrm {code}^{\kappa }(\varphi )$ .

Assume now that $\xi $ is a limit ordinal. We will again be comparing the slices of $\mathrm {code}^{j(\kappa )}(j(\varphi ))\cap \kappa $ and of $\mathrm { code}^{\kappa }(\varphi )$ . The slices with index $0$ clearly agree. Our assumption, which we haven’t used yet, yields a formula $\psi $ of complexity $\xi $ such that in $\mathop {\mathrm {Ult}}$ ,

$$\begin{align*}\mathrm{code}^{\kappa}(\psi)=\mathrm{code}^{j(\kappa)}(j(\varphi))\cap\kappa.\end{align*}$$

The slices with index $1$ thus agree between $\mathrm { code}^{\kappa }(\psi )$ and $\mathrm {code}^{\kappa }(\varphi )$ , for they are coding the same well-ordering, and hence they also agree for our desired formulas. The remaining slices agree inductively, contradicting (1) as desired.

3.2 Generalizing Baumgartner’s lemma to higher indescribability

In this section, we provide the promised strengthening of Theorem 3.1.

Theorem 3.4. Suppose $S\subseteq \kappa $ is subtle and $\vec {S}=\langle S_{\alpha }\mid \alpha \in S\rangle $ is an S-list. Let

$$\begin{align*}A=\{\alpha\in S\mid(\exists X\subseteq S\cap\alpha)(\forall \eta<\alpha^+\ X\in\Pi^1_{\eta}(\alpha)^+)\land(X\cup\{\alpha\}\text{ is hom. for }\vec{S})\}. \end{align*}$$

Then, $S\setminus A$ is not subtle.

Proof Suppose $S\subseteq \kappa $ is subtle, $\vec {S}$ is an S-list, and suppose for a contradiction that $S\setminus A$ is subtle. Using that the set of inaccessible cardinals below $\kappa $ is in the subtle filter on $\kappa $ , we may assume that all elements of S are inaccessible. We will also assume our canonical functions at $\kappa $ to be based on the bijections $b_{\kappa ,\xi }$ used to define the coding at $\kappa $ in the above.

Suppose $\beta \in S\setminus A$ . Let $B_{\beta }=\{\alpha \in S\cap \beta \mid S_{\alpha }=S_{\beta }\cap \alpha \}$ . Since $B_{\beta }\cup \{\beta \}$ is homogeneous for $\vec {S}$ , it follows that for some limit ordinal $\xi _{\beta }$ with $\beta \le \xi _{\beta }<\beta ^+$ , the set $B_{\beta }$ is not $\Pi ^1_{\xi _{\beta }}$ -indescribable in $\beta $ . Let $\varphi ^{\beta }$ thus be a $\Pi ^1_{\xi _{\beta }}$ sentence over $V_{\beta }$ such that $V_{\beta }\models \varphi ^{\beta }$ , and for all $\alpha \in B_{\beta }$ , we have that $V_{\alpha }\not \models \varphi ^{\beta }\mathrm {|}^{\beta }_{\alpha }$ whenever $\varphi ^{\beta }\mathrm {|}^{\beta }_{\alpha }$ is defined.

For each $\beta \in S\setminus A$ , let $E_{\beta }$ code the pair

$$\begin{align*}\langle S_{\beta},\mathrm{code}^{\beta}(\varphi^{\beta})\rangle \end{align*}$$

as a subset of $\beta $ in a natural way. This defines an $(S\setminus A)$ -list $\vec {E}=\langle E_{\beta }\mid \beta \in S\setminus A\rangle $ . By Theorem 3.1, there is a Mahlo cardinal $\beta \in S\setminus A$ and a stationary set $H\subseteq (S\setminus A)\cap \beta $ such that $H\cup \{\beta \}$ is homogeneous for $\vec {E}$ . Let $\varphi =\varphi ^{\beta }$ , and let $\xi =\xi _{\beta }$ .

Recall that for each $\alpha \in S\setminus A$ , we have $(\mathrm {code}^{\alpha }(\varphi ^{\alpha }))_1=\Gamma [R^{\alpha }_{\xi _{\alpha }}]$ , and thus the homogeneity of $H\cup \{\beta \}$ implies that for all $\alpha \in H$ , we have

$$\begin{align*}R^{\beta}_{\xi}\cap(\alpha\times\alpha)=R^{\alpha}_{\xi_{\alpha}}, \end{align*}$$

and therefore

$$\begin{align*}\xi_{\alpha}=\mathop{\mathrm{ot}}\nolimits(\alpha,R^{\alpha}_{\xi_{\alpha}})=\mathop{\mathrm{ot}}\nolimits(\alpha,R^{\beta}_{\xi}\cap(\alpha\times\alpha))=\mathop{\mathrm{ ot}}\nolimits(b_{\beta,\xi}[\alpha])=f^{\beta}_{\xi}(\alpha). \end{align*}$$

Thus, for each $\alpha \in H$ , we have a $\Pi ^1_{f^{\beta }_{\xi }(\alpha )}$ -formula $\varphi ^{\alpha }$ over $V_{\alpha }$ such that

$$\begin{align*}\mathrm{code}^{\alpha}(\varphi^{\alpha})=\mathrm{code}^{\beta}(\varphi)\cap\alpha. \end{align*}$$

Therefore, by Lemma 3.3, there is a club $C\subseteq \beta $ such that $\varphi \mathrm {|}^{\beta }_{\alpha }$ is defined, and

$$\begin{align*}\mathrm{code}^{\beta}(\varphi)\cap\alpha=\mathrm{code}^{\alpha}(\varphi\mathrm{|}^{\beta}_{\alpha}) \end{align*}$$

for all $\alpha \in H\cap C\ne \varnothing $ . Fix some $\alpha \in H\cap C$ . We have

$$\begin{align*}\mathrm{code}^{\alpha}(\varphi^{\alpha})=\mathrm{code}^{\beta}(\varphi)\cap\alpha=\mathrm{code}^{\alpha}(\varphi\mathrm{|}^{\beta}_{\alpha}), \end{align*}$$

and hence $\varphi ^{\alpha }=\varphi \mathrm {|}^{\beta }_{\alpha }$ . However, since $S_{\beta }\cap \alpha =S_{\alpha }$ as well by the homogeneity of H, we have $\alpha \in B_{\beta }$ , and hence $V_{\alpha }\not \models \varphi \mathrm {|}^{\beta }_{\alpha }$ , contradicting that $V_{\alpha }\models \varphi ^{\alpha }$ .

The following is now immediate from Theorem 3.4.

Corollary 3.5. Suppose $\kappa $ is subtle. Then, the set

$$\begin{align*}\{\alpha<\kappa\mid(\forall\eta<\alpha^+)\ \alpha \text{ is } \Pi^1_{\eta}\text{-indescribable}\} \end{align*}$$

is in the subtle filter on $\kappa $ .⊣

3.3 Pushing Baumgartner’s lemma up the ineffability hierarchy

In this section, we provide our promised application of Theorem 3.4 on the ineffability hierarchy, showing that iterated applications of the ineffability operator yield strong generalizations of the consequences of subtlety described in Theorem 3.4. We will also apply this result in order to obtain further results on the ineffability hierarchy later on in our paper. We will first need an easy auxiliary lemma, which is the analogue of [8, Theorem 2.1] for the ineffability operator. This result is essentially due to Baumgartner (two particular instances are mentioned as [3, Theorems 2.3 and 2.4]), and can be seen to follow from a combination of several results from [12], and in particular its [12, Proposition 10.2]. We would rather like to provide a self-contained proof, which is a minor adaption of the proof of [3, Theorem 2.2].

Lemma 3.6. If I is an ideal on $\kappa $ , then ${\mathcal I}(I)$ is a normal ideal on $\kappa $ .

Proof The only nontrivial property is normality. Note first that ${\mathcal I}(I)\supseteq {\mathcal I}([\kappa ]^{<\kappa })\supseteq {\mathop {\mathrm {NS}}}_{\kappa }$ , where the latter is due to an easy argument that may be found within the proof of [3, Theorem 2.3].

Assume now that $A\in {\mathcal I}(I)^+$ . Let C be the club set of all ordinals below $\kappa $ that are closed under the Gödel pairing function $\Gamma $ . By the above, it follows that $A\cap C\in {\mathcal I}(I)^+$ , and we may thus assume that every element of A is closed under Gödel pairing.

Assume that $f\colon A\to \kappa $ is regressive, and let $A_{\alpha }=f^{-1}(\alpha )$ for every $\alpha <\kappa $ . Assume for a contradiction that $A_{\alpha }\in {\mathcal I}(I)$ for every $\alpha <\kappa $ . Thus, for every $\alpha <\kappa $ , we may fix an $A_{\alpha }$ -list $\vec A_{\alpha }=\langle a^{\alpha }_{\beta }\mid \beta \in A_{\alpha }\rangle $ which has no homogeneous set in $I^+$ .

Let $\vec A=\langle a_{\beta }\mid \beta \in A\rangle $ be an A-list defined by letting, for every $\beta \in A$ , $a_{\beta }$ code both $\{f(\beta )\}$ and $a^{f(\beta )}_{\beta }$ , using Gödel pairing. Since $A\in {\mathcal I}(I)^+$ , there is $H\in I^+$ that is homogeneous for $\vec A$ . It follows that H is homogeneous for f, and we let $\alpha $ be the value of f on H. It then follows that H is homogeneous for $\vec A_{\alpha }$ . This yields our desired contradiction.

We need another auxiliary observation, which has already been used by Baumgartner in [3], but for which we couldn’t find a proof of in the set-theoretic literature. For the convenience of our readers, we would therefore like to provide the easy argument.

Observation 3.7. For any cardinal $\kappa $ , the subtle ideal on $\kappa $ is contained in the weakly ineffable ideal on $\kappa $ .

Proof Assume that $A\subseteq \kappa $ is not an element of the weakly ineffable ideal ${\mathcal I}([\kappa ]^{<\kappa })$ on $\kappa $ , that $\vec a$ is an A-list, and let C be a club subset of $\kappa $ . By Lemma 3.6, $A\cap C\in {\mathcal I}([\kappa ]^{<\kappa })^+$ . It follows that we can find $H\subseteq A\cap C$ that is homogeneous for H. This shows that A is subtle.

Theorem 3.8. Suppose $\gamma <\kappa ^+$ , $S\in {\mathcal I}^{\gamma +1}([\kappa ]^{<\kappa })^+$ , and $\vec {S}=\langle S_{\alpha }\mid \alpha \in S\rangle $ is an S-list. Let A be the set of all ordinals $\alpha \in S$ such that

$$\begin{align*}\exists X\subseteq S\cap\alpha\left[(\forall \xi<\alpha^+\ X\in {\mathcal I}^{f^{\kappa}_{\gamma}(\alpha)}(\Pi^1_{\xi}(\alpha))^+) \land (X\cup\{\alpha\}\text{ is hom. for }\vec{S})\right]. \end{align*}$$

Then, $S\setminus A\in {\mathcal I}^{\gamma +1}([\kappa ]^{<\kappa })$ .

Proof We proceed by induction on $\gamma <\kappa ^+$ . When $\gamma =0$ , the result follows directly from Theorem 3.4, because the subtle ideal is contained in the weakly ineffable ideal ${\mathcal I}([\kappa ]^{<\kappa })$ , and $f^{\kappa }_0(\alpha )=0$ for all $\alpha <\kappa $ .

Suppose $\gamma =\delta +1<\kappa ^+$ is a successor ordinal, and suppose for a contradiction that $S\setminus A\in {\mathcal I}^{\delta +2}([\kappa ]^{<\kappa })^+$ . Let $C=\{\alpha <\kappa \mid f^{\kappa }_{\delta +1}(\alpha )=f^{\kappa }_{\delta }(\alpha )+1\}$ be the club subset of $\kappa $ obtained from Lemma 2.3. Then, the set

$$\begin{align*}E=\{\alpha\in S\setminus A\mid \alpha\text{ is inaccessible}\}\cap C \end{align*}$$

is in ${\mathcal I}^{\delta +2}([\kappa ]^{<\kappa })^+$ . For each $\alpha \in E$ , let $B_{\alpha }=\{\beta \in S\cap \alpha \mid S_{\beta }=S_{\alpha }\cap \beta \}$ . Since $B_{\alpha }\cup \{\alpha \}$ is homogeneous for $\vec {S}$ and $\alpha \in S\setminus A$ , there is an ordinal $\xi _{\alpha }<\alpha ^+$ such that $B_{\alpha }\in {\mathcal I}^{f^{\kappa }_{\delta }(\alpha )+1}(\Pi ^1_{\xi _{\alpha }}(\alpha ))$ , and hence we may fix a $B_{\alpha }$ -list $\vec {B}^{\alpha }=\langle b^{\alpha }_{\beta }\mid \beta \in B_{\alpha }\rangle $ such that $\vec {B}^{\alpha }$ has no homogeneous set in ${\mathcal I}^{f^{\kappa }_{\delta }(\alpha )}(\Pi ^1_{\xi _{\alpha }}(\alpha ))^+$ .

For $\alpha \in E$ , let $E_{\alpha }$ code the triple $\langle S_{\alpha },B_{\alpha },\vec {B}^{\alpha }\rangle $ as a subset of $\alpha $ in a natural way. This defines an E-list $\vec {E}=\langle E_{\alpha }\mid \alpha \in E\rangle $ . Since $E\in {\mathcal I}^{\delta +2}([\kappa ]^{<\kappa })^+$ , we may fix $H\in \mathcal P(E)\cap {\mathcal I}^{\delta +1}([\kappa ]^{<\kappa })^+$ which is homogeneous for $\vec {E}$ . It follows that H is homogeneous for $\langle S_{\alpha }\mid \alpha \in E\rangle $ , $\langle B_{\alpha }\mid \alpha \in E\rangle $ , and $\langle \vec {B}^{\alpha }\mid \alpha \in E\rangle $ . We let $D=\bigcup _{\alpha \in H} S_{\alpha }$ , $B=\bigcup _{\alpha \in H}B_{\alpha }$ , and $\vec {B}=\bigcup _{\alpha \in H}\vec {B}^{\alpha }=\langle b_{\alpha }\mid \alpha \in B\rangle $ . Since $B=\{\alpha <\kappa \mid S_{\alpha }=D\cap \alpha \}$ , it follows that $H\subseteq B$ .

Let $A_0$ be the set of all ordinals $\alpha \in H$ such that

$$\begin{align*}\exists X\subseteq H\cap\alpha\left[(\forall\xi<\alpha^+\ X\in {\mathcal I}^{f^{\kappa}_{\delta}(\alpha)}(\Pi^1_{\xi}(\alpha))^+) \land (X\cup\{\alpha\}\text{ is hom. for }\vec{B})\right].\end{align*}$$

By our inductive hypothesis, $H\setminus A_0\in {\mathcal I}^{\delta +1}([\kappa ]^{<\kappa })$ , and hence $A_0\in {\mathcal I}^{\delta +1}([\kappa ]^{<\kappa })^+$ . Thus, we may fix an $\alpha \in A_0$ . Since $\alpha \in H$ , it follows by homogeneity that $\vec {B}\upharpoonright (H\cap \alpha )=\vec {B}^{\alpha }\upharpoonright H$ . But by the definition of $A_0$ , and since $\xi _{\alpha }<\alpha ^+$ , there is some $X\in \mathcal P(H\cap \alpha )\cap {\mathcal I}^{f^{\kappa }_{\delta }(\alpha )}(\Pi ^1_{\xi _{\alpha }}(\alpha ))^+$ which is homogeneous for $\vec {B}^{\alpha }$ , which is a contradiction.

Now let us suppose $\gamma <\kappa ^+$ is a limit ordinal, and suppose again for a contradiction that $S\setminus A\in {\mathcal I}^{\gamma +1}([\kappa ]^{<\kappa })^+$ . Since by Lemma 2.3, the set

$$\begin{align*}C=\{\alpha<\kappa\mid f^{\kappa}_{\gamma}(\alpha)\text{ is a limit ordinal and } f^{\kappa}_{\gamma}(\alpha)=\bigcup_{\delta\in F^{\kappa}_{\gamma}(\alpha)} f^{\kappa}_{\delta}(\alpha) \} \end{align*}$$

is in the club filter on $\kappa $ , it follows that the set

$$\begin{align*}E=\{\alpha\in S\setminus A\mid \alpha \text{ is inaccessible}\}\cap C \end{align*}$$

is in ${\mathcal I}^{\gamma +1}([\kappa ]^{<\kappa })^+$ .

For each $\alpha \in E$ , let $B_{\alpha }=\{\beta \in S\cap \alpha \mid S_{\beta }=S_{\alpha }\cap \beta \}$ . Since $B_{\alpha }\cup \{\alpha \}$ is homogeneous for $\vec {S}$ , and $\alpha \in S\setminus A$ , there is some $\xi _{\alpha }<\alpha ^+$ such that $B_{\alpha }\in {\mathcal I}^{f^{\kappa }_{\gamma }(\alpha )}(\Pi ^1_{\xi _{\alpha }}(\alpha ))$ . Since $\alpha \in C$ , we have

$$\begin{align*}B_{\alpha}\in{\mathcal I}^{f^{\kappa}_{\gamma}(\alpha)}(\Pi^1_{\xi_{\alpha}}(\alpha))=\bigcup_{\delta\in F^{\kappa}_{\gamma}(\alpha)}{\mathcal I}^{f^{\kappa}_{\delta}(\alpha)}(\Pi^1_{\xi_{\alpha}}(\alpha))=\bigcup_{\beta<\alpha}{\mathcal I}^{f^{\kappa}_{b_{\kappa,\gamma}(\beta)}(\alpha)}(\Pi^1_{\xi_{\alpha}}(\alpha)). \end{align*}$$

Using that $f^{\kappa }_{\gamma }(\alpha )$ is a limit ordinal once again, we may choose an ordinal $g(\alpha )<\alpha $ such that

$$\begin{align*}B_{\alpha}\in{\mathcal I}^{f^{\kappa}_{b_{\kappa,\gamma}(g(\alpha))}(\alpha)+1}(\Pi^1_{\xi_{\alpha}}(\alpha)).\end{align*}$$

This defines a regressive function $g:E\to \kappa $ , and by the normality of ${\mathcal I}^{\gamma +1}([\kappa ]^{<\kappa })^+$ that follows from Lemma 3.6, there is an $E^*\in \mathcal P(E)\cap {\mathcal I}^{\gamma +1}([\kappa ]^{<\kappa })^+$ and some $\beta _0<\kappa $ such that $g(\alpha )=\beta _0$ for all $\alpha \in E^*$ . Let $\nu =b_{\kappa ,\gamma }(\beta _0)$ and notice that for all $\alpha \in E^*$ ,

$$\begin{align*}B_{\alpha}\in{\mathcal I}^{f^{\kappa}_{\nu}(\alpha)+1}(\Pi^1_{\xi_{\alpha}}(\alpha)). \end{align*}$$

For each $\alpha \in E^*$ , we fix a $B_{\alpha }$ -list $\vec {B}^{\alpha }=\langle b^{\alpha }_{\beta }\mid \beta \in B_{\alpha }\rangle $ such that $\vec {B}^{\alpha }$ has no homogeneous set in ${\mathcal I}^{f^{\kappa }_{\nu }(\alpha )}(\Pi ^1_{\xi _{\alpha }}(\alpha ))^+$ .

Now we define an $E^*$ -list by letting $E^*_{\alpha }$ code the triple $\langle S_{\alpha },B_{\alpha },\vec {B}^{\alpha }\rangle $ as a subset of $\alpha $ in a natural way, for all $\alpha \in E^*$ . This defines $\vec {E}^*=\langle E^*_{\alpha }\mid \alpha \in E^*\rangle $ . Since $E^*\in {\mathcal I}^{\gamma +1}([\kappa ]^{<\kappa })^+$ , we may fix an $H\in \mathcal P(E^*)\cap {\mathcal I}^{\gamma }([\kappa ]^{<\kappa })^+$ which is homogeneous for $\vec {E}^*$ . Then, H is homogeneous for $\langle S_{\alpha }\mid \alpha \in S\rangle $ , $\langle B_{\alpha }\mid \alpha \in E^*\rangle $ and $\langle \vec {B}^{\alpha }\mid \alpha \in E^*\rangle $ . We let $D=\bigcup _{\alpha \in H}S_{\alpha }$ , $B=\bigcup _{\alpha \in H}B_{\alpha }$ and $\vec {B}=\bigcup _{\alpha \in H}\vec {B}^{\alpha }=\langle b_{\alpha }\mid \alpha \in B\rangle $ . Since $B=\{\alpha <\kappa \mid S_{\alpha }=D\cap \alpha \}$ , it follows that $H\subseteq B$ .

Now since $\nu <\gamma $ , we have $H\in {\mathcal I}^{\gamma }([\kappa ]^{<\kappa })^+\subseteq {\mathcal I}^{\nu +1}([\kappa ]^{<\kappa })^+$ , and we may apply the inductive hypothesis to the H-list $\vec {B}\upharpoonright H$ . Let $A_0$ be the set of all ordinals $\alpha \in H$ such that

$$\begin{align*}\exists X\subseteq H\cap\alpha\left[(\forall\xi<\alpha^+\ X\in{\mathcal I}^{f^{\kappa}_{\nu}(\alpha)}(\Pi^1_{\xi}(\alpha))^+)\land (X\cup\{\alpha\}\text{ is hom. for }\vec{B}\upharpoonright H)\right]. \end{align*}$$

It follows that $H\setminus A_0\in {\mathcal I}^{\nu +1}([\kappa ]^{<\kappa })$ , which implies that $A_0\in {\mathcal I}^{\nu +1}([\kappa ]^{<\kappa })^+$ . Fix $\alpha \in A_0$ . Since $\alpha \in H$ , homogeneity implies that $\vec {B}\upharpoonright (H\cap \alpha )=\vec {B}^{\alpha }\upharpoonright H$ . But, by the definition of $A_0$ , and the fact that $\xi _{\alpha }<\alpha ^+$ , it follows that there is some $X\in \mathcal P(H\cap \alpha )\cap {\mathcal I}^{f^{\kappa }_{\nu }}(\Pi ^1_{\xi _{\alpha }}(\alpha ))^+$ which is homogeneous for $\vec {B}^{\alpha }$ , a contradiction.

The following is now immediate from Theorem 3.8.

Corollary 3.9. Suppose $\kappa \in {\mathcal I}^{\gamma +1}([\kappa ]^{<\kappa })^+$ where $\gamma <\kappa ^+$ . Then the set

$$\begin{align*}\{\alpha<\kappa\mid (\forall\xi<\alpha^+)\ \alpha\in{\mathcal I}^{f^{\kappa}_{\gamma}(\alpha)}(\Pi^1_{\xi}(\alpha))^+\} \end{align*}$$

is in the filter ${\mathcal I}^{\gamma +1}([\kappa ]^{<\kappa })^*$ .⊣

3.4 A version of Baumgartner’s lemma for the strongly Ramsey ideal

Recall that in Section 1, we introduced the strongly Ramsey subset operator $\mathcal {S}$ . Next, we show that Baumgartner’s lemma can, in a sense, be generalized to the strongly Ramsey ideal ${\mathcal S}([\kappa ]^{<\kappa })^+$ .

Theorem 3.10. Suppose $S\in {\mathcal S}([\kappa ]^{<\kappa })^+$ and $f:[S]^{<\omega }\to \kappa $ is a regressive function. Let

$$\begin{align*}A=\{\alpha\in S\mid(\exists X\subseteq S\cap \alpha)(\forall\eta<\alpha^+ X\in\Pi^1_{\eta}(\alpha)^+)\land( X \text{ is hom. for } f)\}. \end{align*}$$

Then, $S\setminus A\in {\mathcal S}([\kappa ]^{<\kappa })$ .

Proof Suppose $S\setminus A\in {\mathcal S}([\kappa ]^{<\kappa })^+$ . Let M be a $\kappa $ -model with $S\setminus A,f\in M$ and let $U\subseteq [\kappa ]^{\kappa }$ be a $\kappa $ -amenable M-normal M-ultrafilter such that $S\setminus A\in U$ . Let $j:M\to N$ be the usual elementary embedding obtained from U such that N is transitive.

Since U is $\kappa $ -amenable, it follows that for every $B\subseteq \kappa ^n\times \kappa $ in M, the set $\{\vec {\alpha }\in \kappa ^n\mid B_{\vec {\alpha }}\in U\}$ is in M where $B_{\vec {\alpha }}=\{\beta <\kappa \mid \vec {\alpha }\mathbin {{}^{\smallfrown }}\beta \in B\}$ . Thus, we can define the product ultrafilters $U^n$ on $P(\kappa ^n)^M$ by induction as follows. For $B\subseteq \kappa ^n\times \kappa $ , we let $B\in U^{n+1}=U^n\times U$ if and only if $B\in M$ and $\{\vec {\alpha }\in \kappa ^n\mid B_{\vec {\alpha }}\in U\}\in U^n$ . For each $n\in \omega \setminus \{0\}$ , we let $j_{U^n}:M\to N_{U^n}$ be the ultrapower of M by $U^n$ and note that, it follows from [10, Proposition 2.32] that $N_{U^n}$ is well-founded. Furthermore, by [10, Lemma 2.31], we have $j_{U^{n+1}}=j_{j_{U^n}(U)}\circ j_{U^n}$ where $j_{j_{U^n}(U)}$ is the ultrapower of $N_{U^n}$ by $j_{U^n}(U)$ (for more details on product ultrafilters one may consult [10, Chapter 2] or [11]).

For each $n\in \omega \setminus \{0\}$ , let $f_n=f\upharpoonright [S]^n$ . Since f is regressive, it follows by elementarity that the ordinal $\gamma _1=j_{U^1}(f)(\{\kappa \})$ is less than $\kappa $ . Furthermore, for each $n<\omega $ the ordinal $\gamma _{n+2}=j_{U^{n+2}}(f)(\{\kappa ,j_U(\kappa ),j_{U^2}(\kappa ),\ldots ,j_{U^{n+1}}(\kappa )\})$ is less than $\kappa $ . Fix $n\in \omega \setminus \{0\}$ . Let $A_n=\{\vec {\alpha }\in [S]^n\mid f_n(\vec {\alpha })=\gamma _n\}$ . By [10, Lemma 2.34], there is a $B_n\in U$ such that for all $\beta _1<\cdots <\beta _n$ in $B_n$ we have $(\beta _1,\ldots ,\beta _n)\in A_n$ , that is, $f_n(\beta _1,\ldots ,\beta _n)=\gamma _n$ . Hence $B_n$ is homogeneous for $f_n$ .

Clearly $B=\bigcap _{n<\omega }B_n$ is homogeneous for f, and since M is a $\kappa $ -model and U is M-normal, it follows that $B\in U$ and hence $\kappa \in j(B)$ . Now we have $B\in N$ and furthermore, N thinks that B is homogeneous for f. But, since $\kappa \in j(S\setminus A)$ , it follows that N thinks that there are no subsets of S which are both $\Pi ^1_{\eta }$ -indescribable in $\kappa $ for all $\eta <(\kappa ^+)^N$ and homogeneous for f. Hence, in N, there must be some $\eta <(\kappa ^+)^N$ such that B is not $\Pi ^1_{\eta }$ -indescribable in $\kappa $ . Working in N, fix a $\Pi ^1_{\eta }$ -sentence $\varphi $ over $V_{\kappa }$ that is true in $V_{\kappa }$ such that for all $\alpha \in B$ we have $V_{\alpha }\models \lnot \varphi \mathrm {|}^{\kappa }_{\alpha }$ . Since M and N are both $\kappa $ -models and since $P(\kappa )^M=P(\kappa )^N$ , it follows that $(V_{\kappa }\models \varphi )^M$ and

$$\begin{align*}((\forall\alpha\in B)\ V_{\alpha}\models\lnot\varphi\mathrm{|}^{\kappa}_{\alpha})^M. \end{align*}$$

Hence by elementarity,

$$\begin{align*}((\forall\alpha\in j(B))\ V_{\alpha}\models\lnot j(\varphi)\mathrm{|}^{j(\kappa)}_{\alpha})^N. \end{align*}$$

But this is a contradiction because $\kappa \in j(B)$ and $(V_{\kappa }\models \varphi )^N$ where $j(\varphi )\mathrm {|}^{j(\kappa )}_{\kappa }=\varphi $ (see the proof of Lemma 2.8).

4 Indescribability from homogeneity

Extending [3, Lemma 7.1] and [6, Lemma 5.1], we show that for all $\xi <\kappa ^+$ , $S\in {\mathcal I}(\Pi ^1_{\xi }(\kappa ))^+$ implies $S\in \Pi ^1_{\xi +2}(\kappa )^+$ . Let us note that the following lemma has precursors in the work of Welch et al. (see [15, Corollary 3.24] and [5]).

Lemma 4.1. Suppose $S\subseteq \kappa $ , $\xi <\kappa ^+$ , and for every S-list $\vec {S}$ , there is a set $H\in \mathcal P(S)\cap \bigcap _{\zeta \in \{-1\}\cup \xi }\Pi ^1_{\zeta }(\kappa )^+$ that is homogeneous for $\vec {S}$ . Then, S is a $\Pi ^1_{\xi +1}$ -indescribable subset of $\kappa $ .

Proof The case in which $\xi <\kappa $ is handled by [6, Lemma 2.20]. The case in which $\kappa <\xi <\kappa ^+$ and $\xi $ is a successor ordinal is similar (see the corresponding case in [6, Lemma 2.20]), and is thus left to the reader.

Suppose $\kappa \leq \xi <\kappa ^+$ , $\xi $ is a limit ordinal, and every S-list has a homogeneous set $H\in \mathcal P(S)\cap \bigcap _{\zeta <\xi }\Pi ^1_{\zeta }(\kappa )^+$ . Suppose for a contradiction that S is not $\Pi ^1_{\xi +1}$ -indescribable. Let

$$\begin{align*}\varphi=\forall X\left(\bigvee_{\zeta<\xi}\psi_{\zeta}\right) \end{align*}$$

be $\Pi ^1_{\xi +1}$ over $V_{\kappa }$ , such that $V_{\kappa }\models \varphi $ , and such that for all $\alpha \in S$ , we have $V_{\alpha }\not \models \varphi \mathrm {|}^{\kappa }_{\alpha }$ whenever $\varphi \mathrm {|}^{\kappa }_{\alpha }$ is defined. Fix a bijection $b:V_{\kappa }\to \kappa $ , let

$$\begin{align*}C=\{\alpha<\kappa\mid b\upharpoonright\alpha:V_{\alpha}\to\alpha\text{ is a bijection and }\varphi\mathrm{|}^{\kappa}_{\alpha}\text{ is defined}\}, \end{align*}$$

and note that C is a club subset of $\kappa $ by Lemma 2.10. For each $\alpha \in S\cap C$ , the sentence $\varphi \mathrm {|}^{\kappa }_{\alpha }$ thus is $\Pi ^1_{f^{\kappa }_{\xi +1}(\alpha )}$ over $V_{\alpha }$ , and hence $V_{\alpha }\models \lnot \varphi \mathrm {|}^{\kappa }_{\alpha }$ , where

$$\begin{align*}\lnot\varphi\mathrm{|}^{\kappa}_{\alpha}=\exists X\left(\bigwedge_{\zeta\in f^{\kappa}_{\xi}(\alpha)}\lnot\psi_{(\pi^{\kappa}_{\xi,\alpha})^{-1}(\zeta)}\mathrm{|}^{\kappa}_{\alpha}\right). \end{align*}$$

For each $\alpha \in S\cap C$ , let $T_{\alpha }\subseteq V_{\alpha }$ be such that

(2) $$ \begin{align} V_{\alpha}\models\bigwedge_{\zeta\in f^{\kappa}_{\xi}(\alpha)}(\lnot\psi_{(\pi^{\kappa}_{\xi,\alpha})^{-1}(\zeta)}\mathrm{|}^{\kappa}_{\alpha})(T_{\alpha}). \end{align} $$

For $\alpha \in S\setminus C$ let $T_{\alpha }\subseteq \alpha $ be arbitrary. Now we define an S-list $\vec {S}=\langle S_{\alpha }\mid \alpha \in S\rangle $ where $S_{\alpha }=b[T_{\alpha }]$ for all $\alpha \in S$ . Let $H\in \mathcal P(S)\cap \bigcap _{\zeta <\xi }\Pi ^1_{\zeta }(\kappa )^+$ be homogeneous for $\vec {S}$ . Notice that $H\cap C\in \bigcap _{\zeta <\xi }\Pi ^1_{\zeta }(\kappa )^+$ , let $R=\bigcup _{\alpha \in H\cap C}S_{\alpha }$ , and let $T=b^{-1}[R]$ . Since $V_{\kappa }\models \varphi $ , we have $V_{\kappa }\models \psi _{\zeta }(T)$ for some fixed $\zeta <\xi $ . Since the set

$$\begin{align*}H\cap C\cap\{\alpha<\kappa\mid \zeta\in F^{\kappa}_{\xi}(\alpha)\} \end{align*}$$

is $\Pi ^1_{\zeta }$ -indescribable in $\kappa $ , it follows that for some $\alpha \in H\cap C$ with $\zeta \in F^{\kappa }_{\xi }(\alpha )$ we have $V_{\alpha }\models (\psi _{\zeta }\mathrm {|}^{\kappa }_{\alpha })(T\cap V_{\alpha })$ . By homogeneity, we have $R\cap \alpha =S_{\alpha }$ , and hence $T\cap V_{\alpha }=T_{\alpha }$ . This implies that $V_{\alpha } \models (\psi _{\zeta }\mathrm {|}^{\kappa }_{\alpha })(T_{\alpha })$ , but this contradicts (2).

It was shown in [7, Proposition 3.8] that measurable cardinals are $\Pi ^1_{\xi }$ -indescribable for every $\xi <\kappa ^+$ . We want to make use of Lemma 4.1 in order to provide a better upper bound. Recall that a cardinal $\kappa $ is completely ineffable if there is a collection $\mathcal S$ of stationary subsets of $\kappa $ that is closed under the taking of supersets (such collections are called a stationary class), such that whenever $S\in \mathcal S$ and $\vec S$ is an S-list, then there is a set $H\in \mathcal S$ that is homogeneous for $\vec S$ .

Proposition 4.2. If $\kappa $ is completely ineffable, then $\kappa $ is $\Pi ^1_{\xi }$ -indescribable for every $\xi <\kappa ^+$ .

Proof Let $\mathcal T$ be the union of all stationary classes witnessing that $\kappa $ is completely ineffable. It is easy to see that $\mathcal T$ itself is a stationary class witnessing that $\kappa $ is completely ineffable, and also that $I=\mathcal P(\kappa )\setminus \mathcal T$ is an ideal on $\kappa $ —in fact, I is what is called the completely ineffable ideal on $\kappa $ , as defined in [14]. Note that by the very definition of ${\mathcal I}$ , we see that ${\mathcal I}(I)=I$ . Using Lemma 4.1, and recalling that $\Pi ^1_0(\kappa )\subseteq I$ is the nonstationary ideal on $\kappa $ , a straightforward induction now yields $\kappa $ to be $\Pi ^1_{\xi }$ -indescribable for every $\xi <\kappa ^+$ .

5 Some properties of the ineffability and the Ramsey operator

In this section, we will provide two lemmas about the ineffability and the Ramsey operator which will be required later on, but which should also be of independent interest. For the Ramsey operator, when $\gamma $ and $\xi $ are both less than $\kappa $ , these are due to the first author in [6, Lemmas 3.1 and 3.2].

Lemma 5.1. Let ${\mathcal O}\in \{{\mathcal I},{\mathcal R}\}$ . Suppose $\kappa $ is a cardinal, $\gamma <\kappa ^+$ , $\xi \in \{-1\}\cup \kappa ^+$ , $S\in {\mathcal O}^{\gamma }(\Pi ^1_{\xi }(\kappa ))^+$ , and for each $\alpha \in S$ , let $S_{\alpha }\in {\mathcal O}^{f^{\kappa }_{\gamma }(\alpha )}(\Pi ^1_{f^{\kappa }_{\xi }(\alpha )}(\alpha ))^+$ . Then $\bigcup _{\alpha \in S}S_{\alpha }\in {\mathcal O}^{\gamma }(\Pi ^1_{\xi }(\kappa ))^+$ .

Proof Let us assume that ${\mathcal O}={\mathcal I}$ ; when ${\mathcal O}={\mathcal R}$ the proof is essentially the same, only one must replace lists by regressive functions. We proceed by induction on $\gamma $ . Suppose $\gamma =0$ , fix $\xi <\kappa ^+$ , $S\in \Pi ^1_{\xi }(\kappa )^+$ and let $S_{\alpha }\in \Pi ^1_{f^{\kappa }_{\xi }(\alpha )}(\alpha )^+$ for all $\alpha \in S$ . Fix a $\Pi ^1_{\xi }$ sentence $\varphi $ over $V_{\kappa }$ such that $V_{\kappa }\models \varphi $ . By Lemma 2.10, the set

$$\begin{align*}C=\{\alpha<\kappa\mid\varphi\mathrm{|}^{\kappa}_{\alpha}\text{ is defined, and hence } \Pi^1_{f^{\kappa}_{\xi}(\alpha)} \text{ over } V_{\alpha}\} \end{align*}$$

is in the club filter on $\kappa $ . By Proposition 2.12, there is a club subset D of $\kappa $ such that for all regular uncountable $\alpha \in D$ , the set $D_{\alpha }$ of all ordinals $\beta <\alpha $ for which $(\varphi \mathrm {|}^{\kappa }_{\alpha })\mathrm {|}^{\alpha }_{\beta }$ is defined and $(\varphi \mathrm {|}^{\kappa }_{\alpha })\mathrm {|}^{\alpha }_{\beta }=\varphi \mathrm {|}^{\kappa }_{\beta }$ is in the club filter on $\alpha $ . Since $S\cap C\cap D\in \Pi ^1_{\xi }(\kappa )^+$ , we may fix an $\alpha \in S\cap C\cap D$ such that $V_{\alpha }\models \varphi \mathrm {|}^{\kappa }_{\alpha }$ . Now, since $S_{\alpha }\cap D_{\alpha }\in \Pi ^1_{f^{\kappa }_{\xi }(\alpha )}(\alpha )^+$ and $\varphi \mathrm {|}^{\kappa }_{\alpha }$ is $\Pi ^1_{f^{\kappa }_{\xi }(\alpha )}$ over $V_{\alpha }$ , we may fix $\beta \in S_{\alpha }\cap D_{\alpha }$ such that $V_{\beta }\models (\varphi \mathrm {|}^{\kappa }_{\alpha })\mathrm {|}^{\alpha }_{\beta }$ . Since $\beta \in D_{\alpha }$ implies $(\varphi \mathrm {|}^{\kappa }_{\alpha })\mathrm {|}^{\alpha }_{\beta }=\varphi \mathrm {|}^{\kappa }_{\beta }$ , we have $V_{\beta }\models \varphi \mathrm {|}^{\kappa }_{\beta }$ . Thus, $\bigcup _{\alpha \in S}S_{\alpha }$ is a $\Pi ^1_{\xi }$ -indescribable subset of $\kappa $ .

Suppose $\gamma <\kappa ^+$ is a limit ordinal. Fix $\xi <\kappa ^+$ , $S\in {\mathcal I}^{\gamma }(\Pi ^1_{\xi }(\kappa ))^+$ and let $S_{\alpha }\in {\mathcal I}^{f^{\kappa }_{\gamma }(\alpha )}(\Pi ^1_{f^{\kappa }_{\xi }(\alpha )}(\alpha ))^+$ for all $\alpha \in S$ . It suffices to show that $\bigcup _{\alpha \in S}S_{\alpha }\in {\mathcal I}^{\delta }(\Pi ^1_{\xi }(\kappa ))^+$ for all $\delta <\gamma $ . Fix $\delta <\gamma $ . Since $\delta <\gamma $ in any generic ultrapower $\mathop {\mathrm {Ult}}$ obtained by forcing with $P(\kappa )/{\mathop {\mathrm {NS}}}_{\kappa }$ , the set

$$\begin{align*}C=\{\alpha<\kappa\mid f^{\kappa}_{\delta}(\alpha)<f^{\kappa}_{\gamma}(\alpha)\}\end{align*}$$

is in the club filter on $\kappa $ . Thus, $S\cap C\in {\mathcal I}^{\delta }(\Pi ^1_{\xi }(\kappa ))^+$ , and for each $\alpha \in S\cap C$ , we have $S_{\alpha }\in {\mathcal I}^{f^{\kappa }_{\delta }(\alpha )}(\Pi ^1_{f^{\kappa }_{\xi }(\alpha )})^+$ . By our inductive hypothesis, we have $\bigcup _{\alpha \in S\cap C}S_{\alpha }\in {\mathcal I}^{\delta }(\Pi ^1_{\xi }(\kappa ))^+$ . Since $\bigcup _{\alpha \in S\cap C}S_{\alpha }\subseteq \bigcup _{\alpha \in S}S_{\alpha }$ , we thus see that $\bigcup _{\alpha \in S}S_{\alpha }\in {\mathcal I}^{\gamma }(\Pi ^1_{\xi }(\kappa ))^+$ .

Suppose $\gamma =\delta +1$ is a successor ordinal. Fix $\xi <\kappa ^+$ , $S\in {\mathcal I}^{\delta +1}(\Pi ^1_{\xi }(\kappa ))^+$ and let $S_{\alpha }\in {\mathcal I}^{f^{\kappa }_{\delta +1}(\alpha )}(\Pi ^1_{f^{\kappa }_{\xi }(\alpha )})^+$ for each $\alpha \in S$ . Let $T=\bigcup _{\alpha \in S}S_{\alpha }$ . Fix a T-list $\vec {T}=\langle T_{\alpha }\mid \alpha \in T\rangle $ . We must show that there is a homogeneous set for $\vec {T}$ in ${\mathcal I}^{\delta }(\Pi ^1_{\xi }(\kappa ))^+$ . By Lemma 2.3, the set

$$\begin{align*}C=\{\alpha<\kappa\mid f^{\kappa}_{\delta+1}(\alpha)=f^{\kappa}_{\delta}(\alpha)+1\}\end{align*}$$

is in the club filter on $\kappa $ . Thus $S\cap C\in {\mathcal I}^{\delta +1}(\Pi ^1_{\xi }(\kappa ))^+$ . For each $\alpha \in S\cap C$ , the $S_{\alpha }$ -list $\vec {T}\upharpoonright S_{\alpha }$ has a homogeneous set $H_{\alpha }\in \mathcal P(S_{\alpha })\cap \Pi ^1_{f^{\kappa }_{\delta }(\alpha )}(\alpha )^+$ . Let $H\in {\mathcal I}^{\delta }(\Pi ^1_{\xi }(\kappa ))^+$ be homogeneous for the $(S\cap C)$ -list $\langle H_{\alpha }\mid \alpha \in S\cap C\rangle $ . By our inductive hypothesis, $\bigcup _{\alpha \in H}H_{\alpha }\in {\mathcal I}^{\delta }(\Pi ^1_{\xi }(\kappa ))^+$ , and it is easy to see that this set is homogeneous for $\vec {T}$ .

Lemma 5.2. Let ${\mathcal O}\in \{{\mathcal I},{\mathcal R}\}$ . Suppose $\kappa $ is a cardinal, $\gamma <\kappa ^+$ , and $\xi \in \{-1\}\cup \kappa ^+$ . If $\kappa \in {\mathcal O}^{\gamma }(\Pi ^1_{\xi }(\kappa ))^+$ , then the set

$$\begin{align*}S_{\kappa}=\{\alpha<\kappa\mid\alpha\in{\mathcal O}^{f^{\kappa}_{\gamma}(\alpha)}(\Pi^1_{f^{\kappa}_{\xi}(\alpha)}(\alpha))\}\end{align*}$$

is in ${\mathcal O}^{\gamma }(\Pi ^1_{\xi }(\kappa ))^+$ .

Proof Assume for a contradiction that the statement of the lemma does not hold true, and let $\kappa $ be the least counterexample: the least cardinal for which there are $\gamma <\kappa ^+$ and $\xi \in \{-1\}\cup \kappa ^+$ such that $\kappa \in {\mathcal O}^{\gamma }(\Pi ^1_{\xi }(\kappa ))^+$ and $S:=S_{\kappa }\in {\mathcal O}^{\gamma }(\Pi ^1_{\xi }(\kappa ))$ . Then, $\kappa \setminus S\in {\mathcal O}^{\gamma }(\Pi ^1_{\xi }(\kappa ))^+$ . For each $\alpha \in \kappa \setminus S$ , we have $\alpha \in {\mathcal O}^{f^{\kappa }_{\gamma }(\alpha )}(\Pi ^1_{f^{\kappa }_{\xi }(\alpha )}(\alpha ))^+$ , and by the minimality of $\kappa $ , the set $S_{\alpha }=S\cap \alpha $ is in ${\mathcal O}^{f^{\kappa }_{\gamma }(\alpha )}(\Pi ^1_{f^{\kappa }_{\xi }(\alpha )}(\alpha ))^+$ . Thus, by Lemma 5.1, the set $S=\bigcup _{\alpha \in \kappa \setminus S}S_{\alpha }$ is in ${\mathcal O}^{\gamma }(\Pi ^1_{\xi }(\kappa ))^+$ , a contradiction.

Next, we provide a result for the strongly Ramsey ideal which is analogous to the base case of Lemma 5.2. This result follows from more general results in [13, Lemma 14.2] (with the core argument being [13, Lemma 9.15]); however we would like to provide a proof for the particular case of strongly Ramsey cardinals, also in order to allow for the discussion of possible generalizations that follows in the remark below.

Lemma 5.3 (Holy-Lücke).

For every cardinal $\kappa $ , if $\kappa \in {\mathcal S}([\kappa ]^{<\kappa })^+$ , then the set

$$\begin{align*}T=\{\alpha<\kappa\mid\alpha\in{\mathcal S}([\alpha]^{<\alpha})\}\end{align*}$$

is in ${\mathcal S}([\kappa ]^{<\kappa })^+$ .

Proof Suppose the result is false and let $\kappa $ be the least counterexample. Then $\kappa $ is strongly Ramsey and $T\in {\mathcal S}([\kappa ]^{<\kappa })$ . This implies $\kappa \setminus T\in {\mathcal S}([\kappa ]^{<\kappa })^*$ and hence there is an $A_T\subseteq \kappa $ such that whenever M is a $\kappa $ -model with $A_T,\kappa \setminus T\in M$ and whenever $U\subseteq [\kappa ]^{\kappa }$ is a $\kappa $ -amenable M-normal M-ultrafilter on $\kappa $ , it must follow that $\kappa \setminus T\in U$ . Let $j:M\to N$ be the ultrapower embedding obtained from U, and notice that $\kappa \in j(\kappa \setminus T)$ and hence $\kappa $ is strongly Ramsey in N.

By our assumption on $\kappa $ , it follows that for all $\alpha <\kappa $ , if $\alpha $ is strongly Ramsey then the set

$$\begin{align*}T\cap\alpha=\{\beta<\alpha\mid\beta\in{\mathcal S}([\beta]^{<\beta})\}\end{align*}$$

is in ${\mathcal S}([\alpha ]^{<\alpha })^+$ . Since M is a $\kappa $ -model, this statement also holds in M. So, since $\kappa $ is strongly Ramsey in N, it follows by elementarity that in N, the set $j(T)\cap \kappa =T$ is in $({\mathcal S}([\kappa ]^{<\kappa })^+)^N$ . Working in N, we let $\bar M$ be a $\kappa $ -model with $A_T,T\in \bar M$ and we let $\bar U$ be a $\kappa $ -amenable $\bar M$ -normal $\bar M$ -ultrafilter on $\kappa $ with $\bar U\subseteq ([\kappa ]^{\kappa })^N$ and $T\in \bar U$ . Since N is a $\kappa $ -model, it follows that in V the set $\bar M$ is a $\kappa $ -model with $A_T,T\in \bar M$ , $\bar U$ is a $\kappa $ -amenable $\bar M$ -normal $\bar M$ -ultrafilter on $\kappa $ with $\bar U\subseteq [\kappa ]^{\kappa }$ , and $T\in U$ . This contradicts the definition of $A_T$ .

Remark 5.4. Let us note that we do not know whether a version of Lemma 5.3 holds for the ideal ${\mathcal S}^2([\kappa ]^{<\kappa })$ . Suppose $\kappa \in {\mathcal S}^2([\kappa ]^{<\kappa })^+$ and let $T=\{\alpha <\kappa \mid \alpha \in {\mathcal S}^2([\alpha ]^{<\alpha })\}$ . Does it follow that $T\in {\mathcal S}^2([\kappa ]^{<\kappa })^+$ ? If we try to generalize the proof of Lemma 5.3 to this situation, we would like to show that if a $\kappa $ -model N thinks that $\bar M$ is a $\kappa $ -model and $\bar U$ is a $\kappa $ -amenable $\bar M$ -normal $\bar M$ -ultrafilter with $\bar U\subseteq ({\mathcal S}([\kappa ]^{<\kappa })^+)^N$ , then it is the case that in V we have $\bar U\subseteq ({\mathcal S}([\kappa ]^{<\kappa })^+)^V$ . However, we do not see how to prove this. One would want to show that $({\mathcal S}([\kappa ]^{<\kappa })^+)^N\subseteq ({\mathcal S}([\kappa ]^{<\kappa })^+)^V$ . But this seems to be problematic because $P(\kappa )^N\subsetneq P(\kappa )^V$ .

6 Expressibility results

First, let us recall an expressibility result for higher indescribability due to the first author, which extends results of Bagaria from [2].

Theorem 6.1 [7, Theorem 5.8].

Suppose $\kappa>\omega $ is regular and $\xi <\kappa ^+$ . Then, there is a $\Pi ^1_{\xi +1}$ formula $\Phi $ over $V_{\kappa }$ and a club $C\subseteq \kappa $ such that for all $S\subseteq \kappa $ we have

$$\begin{align*}S \text{ is a } \Pi^1_{\xi}\text{-indescribable subset of } \kappa \text{ if and only if } V_{\kappa}\models\Phi(S) \end{align*}$$

and for all regular $\alpha \in C$ , we have

$$\begin{align*}S\cap\alpha \text{ is a } \Pi^1_{f^{\kappa}_{\xi}(\alpha)}\text{-indescribable subset of } \alpha \text{ if and only if } V_{\alpha}\models\Phi(S)\mathrm{|}^{\kappa}_{\alpha}. \end{align*}$$

Note that, within our usual generic ultrapower setup, using Lemma 2.8, the existence of a club C as for the second statement of Theorem 6.1 above is equivalent to its first statement holding in the generic ultrapower $\mathop {\mathrm {Ult}}$ . This could be used to extract a fairly simple proof of the second statement from the original proof of the first statement that is provided in [7]. Since doing this in detail would involve going through quite a lot of material from [7] however, we will leave this task to the interested reader.

We will next need an easy lemma, whose proof, via a standard closing-off argument, is left to the reader as well.

Lemma 6.2. Suppose $\kappa $ is a regular cardinal and $\gamma ,\gamma '<\kappa ^+$ are ordinals such that $\gamma \leq \gamma '$ . If $f:\gamma \to \gamma '$ is any function then the set

$$\begin{align*}\{\alpha<\kappa\mid f[F^{\kappa}_{\gamma}(\alpha)]\subseteq F^{\kappa}_{\gamma'}(\alpha)\} \end{align*}$$

is in the club filter on $\kappa $ .

Building on Theorem 6.1 and [6, Lemma 5.1], we obtain the following.

Lemma 6.3. Suppose $\kappa $ is a regular cardinal, and $\gamma <\kappa ^+$ , and $\xi \in \{-1\}\cup \kappa ^+$ . Let $\mathcal {O}\in \{\mathcal {I},\mathcal {R}\}$ be either the ineffability operator or the Ramsey operator. Then, there is a $\Pi ^1_{\xi +1+2\gamma }$ formula $\Theta ^{\kappa }_{\gamma ,\xi }$ over $V_{\kappa }$ and a club subset $C^{\kappa }_{\gamma ,\xi }$ of $\kappa $ such that for all $S\subseteq \kappa $ we have

$$\begin{align*}S\in{\mathcal O}^{\gamma}(\Pi^1_{\xi}(\kappa))^+\text{ if and only if }V_{\kappa}\models\Theta^{\kappa}_{\gamma,\xi}(S) \end{align*}$$

and for all regular cardinals $\alpha \in C^{\kappa }_{\gamma ,\xi }$ we have

$$\begin{align*}S\cap\alpha\in {\mathcal O}^{f^{\kappa}_{\gamma}(\alpha)}(\Pi^1_{f^{\kappa}_{\xi}(\alpha)}(\alpha))^+\text{ if and only if }V_{\alpha}\models\Theta^{\kappa}_{\gamma,\xi}(S)\mathrm{|}^{\kappa}_{\alpha}. \end{align*}$$

Proof For the final statement, we will make use of our usual generic ultrapower setup once again: Using Lemma 2.8, it easily follows that the existence of a club as for the second statement of Lemma 6.3 above is equivalent to its first statement holding in any generic ultrapower $\mathop {\mathrm {Ult}}$ obtained by forcing with $P(\kappa )/{\mathop {\mathrm {NS}}}_{\kappa }$ . ⁹ We thus proceed by induction on $\gamma <\kappa ^+$ to verify the first statement in V and in $\mathop {\mathrm {Ult}}$ simultaneously. Let us consider the case in which $\mathcal {O}=\mathcal {I}$ ; the case in which $\mathcal {O}=\mathcal {R}$ is similar. If $\gamma =0$ , then for all $\xi <\kappa ^+$ , we have ${\mathcal I}^{\gamma }(\Pi ^1_{\xi }(\kappa ))^+=\Pi ^1_{\xi }(\kappa )^+$ , and the result follows directly from Theorem 6.1 and the comments made afterwards (regarding the case of the generic ultrapower $\mathop {\mathrm {Ult}}$ ).

Suppose $\gamma =\delta +1$ , and that the result holds for $\delta $ . Fix $\xi <\kappa ^+$ . Then, there is a $\Pi ^1_{\xi +1+2\delta }$ -formula $\Theta ^{\kappa }_{\delta ,\xi }$ over $V_{\kappa }$ such that both in V and in $\mathop {\mathrm {Ult}}$ , for all $S\subseteq \kappa $ , we have

(3) $$ \begin{align} S\in{\mathcal I}^{\delta}(\Pi^1_{\xi}(\kappa))^+\text{ if and only if }V_{\kappa}\models\Theta^{\kappa}_{\delta,\xi}(S). \end{align} $$

We simply define $\Theta ^{\kappa }_{\gamma ,\xi }$ to be the $\Pi ^1_{\xi +1+2\gamma }$ -formula over $V_{\kappa }$ which asserts that every X-list has a homogeneous set Y such that $\Theta ^{\kappa }_{\delta ,\xi }(Y)$ holds. It is clear that this formula is as desired both in V and in $\mathop {\mathrm {Ult}}$ .

Suppose now that $\gamma $ is a limit ordinal, and that the result holds for all ordinals $\delta <\gamma $ . By definition, we have $X\in {\mathcal I}^{\gamma }(\Pi ^1_{\xi }(\kappa ))^+$ if and only if $X\in {\mathcal I}^{\delta }(\Pi ^1_{\xi }(\kappa ))^+$ for all $\delta <\gamma $ . Note that $\xi +1+2\gamma =\xi +\gamma $ in this case. We define a $\Pi ^1_{\xi +\gamma }$ -formula

$$\begin{align*}\Theta^{\kappa}_{\gamma,\xi}=\bigwedge_{\zeta<\xi+\gamma}\psi^{\kappa}_{\zeta} \end{align*}$$

as follows. For each $\zeta <\xi +\gamma $ , if it exists, define $\delta _{\zeta }$ to be the greatest ordinal $\delta <\gamma $ such that $\xi +1+2\delta \leq \zeta $ and let $\psi ^{\kappa }_{\zeta }=\Theta ^{\kappa }_{\delta _{\zeta },\xi }$ . Otherwise, let $\psi ^{\kappa }_{\zeta }$ be the formula “ $0=0$ .” Since the sequence $\vec {\delta }=\langle \delta _{\zeta }\mid \zeta <\xi +\gamma \rangle $ is cofinal in $\gamma $ , it follows that both in V and in $\mathop {\mathrm {Ult}}$ , for all $S\subseteq \kappa $ ,

$$\begin{align*}S\in{\mathcal I}^{\gamma}(\Pi^1_{\xi}(\kappa))^+\text{ if and only if }V_{\kappa}\models\Theta^{\kappa}_{\gamma,\xi}(S).\\[-35pt] \end{align*}$$

7 A framework for large cardinal operators

In this section, we review a framework for large cardinal operators that was introduced by the second author [12], which in particular fits the ineffability operator $\mathcal I$ , the Ramsey operator $\mathcal R$ , and the strongly Ramsey subset operator $\mathcal S$ (the latter was denoted as $\mathbf T_{\mathrm {cl}}$ in [12]). This framework builds on statements about the existence of certain ultrafilters for small models of set theory, and is itself based on a framework for the characterization of large cardinal ideals that was introduced in [13]. In the present paper, we apply this framework from [12], verifying a number of results on the relationship between higher indescribability and large cardinal operators in a uniform way. In particular, we thus obtain a number of new results on the relationship between higher indescribability and the operators ${\mathcal I}$ and ${\mathcal R}$ , and also ${\mathcal S}$ . For readers only interested in these examples, our framework is still useful, for it provides uniform arguments that work for each of these operators. We will also mention (see Remark 7.13) two additional operators, introduced by the second author [12], that fit into this framework: the $\mathbf T_{\omega }^{\kappa }$ -Ramsey subset operator $\mathbf T$ that is connected to the notion of $\mathbf T_{\omega }^{\kappa }$ -Ramsey cardinals introduced in [13], and the $\mathbf {wf}^{\kappa }_{\omega }$ -Ramsey subset operator $\mathbf {wf}$ that is connected to the notion of weakly Ramsey cardinals from [11], to which our results thus apply.

Let us assume throughout this section that $\kappa $ denotes an inaccessible cardinal, and that I denotes an ideal on $\kappa $ . Recall that an M-ultrafilter U on $\kappa $ is $\kappa $ -amenable for M if whenever $\mathcal A\in M$ is a $\kappa $ -sized collection of subsets of $\kappa $ in M, then $\mathcal A\cap U\in M$ . We next provide the definition of the model version ${\mathcal I}_{mod}$ of the ineffability operator, as introduced in [12].

Definition 7.1.

• • For any $y\subseteq \kappa $ , we first define the local instance of ${\mathcal I}_{mod}$ at y, by letting $x\in {\mathcal I}_{mod}^y(I)^+$ if there is a transitive weak $\kappa $ -model M with $y\in M$ , and an M-ultrafilter U on $\kappa $ with $x\in U$ , such that every diagonal intersection of U is in $I^+$ —we abbreviate this latter property of U and of I by stating that $\Delta U\in I^+$ . ¹⁰

• • We let ${\mathcal I}_{mod}(I)^+=\bigcap _{y\subseteq \kappa }{\mathcal I}_{mod}^y(I)^+$ .

Proposition 7.2 ([12], Proposition 2.5).

Let $I\supseteq {\mathop {\mathrm {NS}}}_{\kappa }$ be an ideal on $\kappa $ . Then, ${\mathcal I}_{mod}(I)=\mathcal I(I)$ .

We also provide the model version of the Ramsey operator from [12].

Definition 7.3.

• • For any $y\subseteq \kappa $ , we first define the local instance of ${\mathcal R}_{mod}$ at y, by letting $x\in {\mathcal R}_{mod}^y(I)^+$ if there is a transitive weak $\kappa $ -model M with $y\in M$ , and an M-normal M-ultrafilter U on $\kappa $ with $x\in U$ that is $\kappa $ -amenable for M, such that every countable intersection of elements of U is in $I^+$ .

• • We let ${\mathcal R}_{mod}(I)^+=\bigcap _{y\subseteq \kappa }{\mathcal R}_{mod}^y(I)^+$ .

The Ramsey operator and its model version were shown to be equivalent in [15]. See also [12].

Theorem 7.4 (Sharpe and Welch [15]).

For any ideal I,

$$\begin{align*}{\mathcal R}_{mod}(I)=\mathcal R(I).\end{align*}$$

Taking the above characterizations of the ineffability and of the Ramsey operator as an inspiration, a framework for large cardinal operators was developed in [12], which we would now like to review.

Definition 7.5. Let $\Psi (M,U)$ and $\Omega (U,I)$ be parameter-free first order formulae such that $\mathrm {ZFC}$ proves that for any ideal I on a regular uncountable cardinal $\kappa $ , any transitive weak $\kappa $ -model M and any M-ultrafilter U on $\kappa $ ,

• • $\Omega (U,I)$ implies that $U\subseteq I^+$ , and

• • for any ideal J on $\kappa $ , $\left [I\supseteq J\,\land \,\Omega (U,I)\right ]\to \Omega (U,J)$ .

Let us say that a pair of formulas $\langle \Psi ,\Omega \rangle $ satisfying the above is regular.

We define an ideal operator $\mathfrak O\Psi \Omega $ as follows. For any ideal I on $\kappa $ and $y\subseteq \kappa $ , we first define a local instance by letting

• • $x\in \mathfrak O\Psi \Omega ^y(I)^+$ if there exists a transitive weak $\kappa $ -model M with $y\in M$ and an M-ultrafilter U on $\kappa $ with $x\in U$ such that $\Psi (M,U)$ and $\Omega (U,I)$ hold, and we let

• • $\mathfrak O\Psi \Omega (I)^+=\bigcap _{y\subseteq \kappa }\mathfrak O\Psi \Omega ^y(I)^+$ .

Let us remark that, since we assume $\kappa $ to be inaccessible, we could additionally require that $M\supseteq V_{\kappa }$ in the above, for given any $y\subseteq \kappa $ , we can easily find $y'\subseteq \kappa $ such that $y'\in M$ implies both that $y\in M$ and that $V_{\kappa }\subseteq M$ .

Let us check how the examples we saw so far fit into these schemes:

• • If $\Psi (M,U)$ is trivial, and $\Omega (U,I)$ denotes the property that $\Delta U{\in }I^+$ , then $\mathfrak O\Psi \Omega $ is the model version ${\mathcal I}_{mod}$ of the ineffability operator.

• • If $\Psi (M,U)$ denotes the property that U is M-normal and $\kappa $ -amenable for M, and $\Omega (U,I)$ denotes the property that every countable intersection of elements of U is in $I^+$ , then $\mathfrak O\Psi \Omega $ is (the model version ${\mathcal R}_{mod}$ of) the Ramsey operator.

• • If $\Psi (M,U)$ denotes the property that M is closed under ${<}\kappa $ -sequences, U is M-normal, and U is $\kappa $ -amenable for M, and $\Omega (U,I)$ denotes the property $U\subseteq I^+$ , then $\mathfrak O\Psi \Omega $ is the strongly Ramsey subset operator ${\mathcal S}$ .

Proposition 7.6 [12, Proposition 10.2].

Assume that $\langle \Psi ,\Omega \rangle $ is regular, and that $I\supseteq J$ are ideals on $\kappa $ . Then, the following hold.

• • $\mathfrak O\Psi \Omega (I)\supseteq I$ is an ideal on $\kappa $ .

• • $\mathfrak O\Psi \Omega (I)\supseteq \mathfrak O\Psi \Omega (J)$ .

• • If for any transitive weak $\kappa $ -model M and any M-ultrafilter U on $\kappa $ , the conjunction $\Psi (M,U)\,\land \,\Omega (U,I)$ implies that U is M-normal, then $\mathfrak O\Psi \Omega (I)$ is normal.

• • In particular, if $I\supseteq {\mathop {\mathrm {NS}}}_{\kappa }$ , then $\Delta U\in I^+$ implies that U is M-normal.

• • If $\langle \Psi ',\Omega '\rangle $ is regular as well, and $\Psi '(M,U)\land \Omega '(U,I)$ implies $\Psi (M,U)\land \Omega (U,I)$ for any transitive weak $\kappa $ -model M and any M-ultrafilter U on $\kappa $ , then $\mathfrak O\Psi '\Omega '(I)\supseteq \mathfrak O\Psi \Omega (I)$ .

A crucial property of ideal operators is ineffability, as introduced in [12].

Definition 7.7. Let $\langle \Psi ,\Omega \rangle $ be a pair of formulas, and let $\mathcal O$ be an ideal operator.

• • The pair $\langle \Psi ,\Omega \rangle $ is ineffable in case $\mathrm {ZFC}$ proves that for any ideal I on a regular uncountable cardinal $\kappa $ , any transitive weak $\kappa $ -model M and any M-ultrafilter U on $\kappa $ , $\Psi (M,U)\,\land \,\Omega (U,I)$ implies that for every $A\in U$ , every A-list $\vec a\in M$ has a homogeneous set in $I^+$ .

• • The operator $\mathcal O$ is ineffable in case $\mathrm {ZFC}$ proves that for any ideal I on a regular uncountable cardinal $\kappa $ , whenever $A\in \mathcal O(I)^+$ and $\vec a$ is an A-list, then $\vec a$ has a homogeneous set in $I^+$ .

Note that by the above, the ineffability operator ${\mathcal I}$ is ineffable. But also, if $\mathcal O$ can be characterized to be of the form $\mathcal O=\mathfrak O\Psi \Omega $ for some ineffable pair of formulas $\langle \Psi ,\Omega \rangle $ , then $\mathcal O$ is ineffable.

Observation 7.8 [12, Observation 10.4].

Let $\langle \Psi ,\Omega \rangle $ be regular, and let $\mathcal O$ be the operator $\mathfrak O\Psi \Omega $ . Then:

• • If $\Psi (M,U) \ \mathrm {ZFC}$ -provably implies that U is $\kappa $ -amenable for M and contains all club subsets of $\kappa $ in M as elements, then $\mathcal O$ is ineffable.

• • If $\Omega (U,I) \ \mathrm {ZFC}$ -provably implies that $\Delta U\in I^+$ , then $\mathcal O$ is ineffable.

• • If $\mathcal O$ is ineffable, then for any ideal I on a regular uncountable cardinal $\kappa $ , $\mathcal O(I)\supseteq \mathcal I(I)\supseteq {\mathop {\mathrm {NS}}}_{\kappa }$ .

• • If $\mathrm {ZFC}$ proves that for any ideal I on a regular and uncountable cardinal $\kappa $ , $\mathcal O(I)\supseteq \mathcal I(I)$ , then $\mathcal O$ is ineffable.

In particular, the above implies that the operators $\mathcal R$ and ${\mathcal S}$ are ineffable.

Given an ordinal $\beta <\kappa ^+$ , let us use the notation

$$\begin{align*}\Pi^1_{<\beta}(\kappa)=\bigcup_{\xi\in\{-1\}\cup\beta}\Pi^1_{\xi}(\kappa).\end{align*}$$

The next corollary is immediate from Lemma 4.1 together with a straightforward induction on $\gamma $ .

Corollary 7.9. Assume that $\mathcal O$ is ineffable, $\beta <\kappa ^+$ is an ordinal, and $I\supseteq \Pi ^1_{<\beta }(\kappa )$ . Then,

$$\begin{align*}\mathcal O^{\gamma}(I)\supseteq\Pi^1_{<(\beta+2\gamma)}(\kappa). \end{align*}$$

We will now review material from [12, Section 13] on coding weak $\kappa $ -models M and M-ultrafilters U on $\kappa $ as subsets of $V_{\kappa }$ . These definitions are tailored so that any transitive weak $\kappa $ -model that can be coded will have to be a superset of $V_{\kappa }$ , with elements x of $V_{\kappa }$ being coded as ordered pairs of the form $\langle 0,x\rangle $ , and we code $\kappa $ by $0$ .

Definition 7.10. We say that $\mathcal M\subseteq V_{\kappa }$ is a code for a transitive weak $\kappa $ -model if $\mathcal M\subseteq V_{\kappa }$ with the following properties:

• • ${\mathcal M}$ is a binary relation on $V_{\kappa }$ , such that $\mathop {\mathrm {dom}}(\mathcal M)=V_{\kappa }$ ,

• • for all $x,y\in V_{\kappa }$ , $\langle 0,x\rangle {\mathcal M}\langle 0,y\rangle $ if and only if $x\in y$ ,

• • for all x, $x\,\mathcal M\,0\iff \exists y\in \kappa \ x=\langle 0,y\rangle $ ,

• • $\mathcal M$ is well-founded and extensional, and

• • $\langle V_{\kappa },\mathcal M\rangle \models \mathrm {ZFC}^-$ .

Note that the weak $\kappa $ -model that is coded here is the model M such that $\langle M,\in \rangle $ is the transitive collapse of $\langle V_{\kappa },\mathcal M\rangle $ . On the other hand, any transitive weak $\kappa $ -model $M\supseteq V_{\kappa }$ has a code as described above, using a suitable bijection between M and $V_{\kappa }$ . Let $\pi _{\mathcal M}$ denote the transitive collapsing map of $\langle V_{\kappa },\mathcal M\rangle $ . If $X=\pi _{\mathcal M}(x)$ , we say that x is the code of X (within $\mathcal M$ ).

Using standard arguments (see [12, Lemma 12.2]), it is easy to see that the property that $\mathcal M$ is a code for a transitive weak $\kappa $ -model is a $\Delta ^1_1$ -property over $\langle V_{\kappa },\mathcal M\rangle $ . Note also that we can easily shift between subsets X of $V_{\kappa }$ in M and their codes within $\mathcal M$ using the fact that for $X\subseteq V_{\kappa }$ in M and $x\in V_{\kappa }$ , the property $\pi _{\mathcal M}^{-1}(X)=x$ is equivalent to the first-order sentence $\forall y\ \left [\langle 0,y\rangle \mathcal M x\longleftrightarrow y\in X\right ]$ in $\langle V_{\kappa },\in ,\mathcal M,X\rangle $ .

Next, we want to define what it means to code an M-ultrafilter on $\kappa $ , which is easily seen to be a $\Delta ^1_1$ -property over $\langle V_{\kappa },\mathcal M,\mathcal U\rangle $ .

Definition 7.11. Given a code $\mathcal M$ for a transitive weak $\kappa $ -model M, we say that $\mathcal U\subseteq V_{\kappa }$ is a code for an M-ultrafilter on $\kappa $ if $\langle V_{\kappa },\mathcal M,\mathcal U\rangle $ thinks that $\mathcal U$ is an ultrafilter on $0$ (note that our setup is so that $0$ codes $\kappa $ ).

For our desired applications, we will need our operators to satisfy some properties of simple definability that were introduced in [12].

Definition 7.12. Let $\langle \Psi ,\Omega \rangle $ be a pair of formulas, and let $\mathcal O$ be an ideal operator.

• • $\langle \Psi ,\Omega \rangle $ is simple in case $\mathrm {ZFC}$ proves the following:

  1. (a) whenever M is a transitive weak $\kappa $ -model, and U is an M-ultrafilter on $\kappa $ , then $\Psi (M,U)$ translates to a $\Delta ^1_1$ -property of any pair of codes $\langle \mathcal M,\mathcal U\rangle $ for $\langle M,U\rangle $ over $V_{\kappa }$ , and

  2. (b) whenever the property $X\in I^+$ is definable over $V_{\kappa }$ by a $\Pi ^1_{\beta }$ -formula $\varphi (X)$ for some $0<\beta <\kappa $ , then $\Omega (U,I)$ translates to a $\Pi ^1_{\beta }$ -property of any code $\mathcal U$ of U over $V_{\kappa }$ .

• • $\langle \Psi ,\Omega \rangle $ is always simple in case $\mathrm {ZFC}$ additionally proves that if in (b), the property $X\in I^+$ is first order definable over $V_{\kappa }$ , then $\Omega (U,I)$ translates to a $\Delta ^1_1$ -property of any code $\mathcal U$ of U over $V_{\kappa }$ .

• • $\mathcal O$ is simple or always simple in case $\mathrm {ZFC}$ proves that $\mathcal O$ can be characterized in the form $\mathcal O=\mathfrak O\Psi \Omega $ for some pair of formulas $\langle \Psi ,\Omega \rangle $ that is simple or always simple respectively.

Definition 7.12(a) is immediate if $\Psi $ can be expressed as a first order property of the structure $\langle M,\in ,U\rangle $ . For example, this is the case when $\Psi (M,U)$ denotes the statement that U is $\kappa $ -amenable for M.

The property that U is countably complete translates to the following first order statement about $\mathcal {U}$ over $V_{\kappa }$ : for any countable sequence $\langle u_i\mid i<\omega \rangle $ of elements of $\mathcal U$ , ¹¹ there is x such that $x\mathcal M u_i$ for every $i<\omega $ .

The statement that M is closed under ${<}\kappa $ -sequences translates to the following first order statement about $\mathcal M$ over $V_{\kappa }$ : $\forall p\,\exists t\,\forall x\ \left (x\,\mathcal M\,t\ \iff \ x\in p\right )$ .

For other examples, see [12].

Let us now look at some examples in which Definition 7.12(b) holds.

• • If $\Omega (U,I)$ denotes the statement that $U\subseteq I^+$ , then this translates to the statement that $\forall x\,\mathcal M\,\mathcal U\,\forall X\ [\pi _{\mathcal M}^{-1}(X)=x\to \varphi (X)]$ , where $\varphi $ is a formula defining $I^+$ over $V_{\kappa }$ .

• • If $\Omega (U,I)$ denotes the property that countable intersections from U are in $I^+$ , then this translates to the statement that for any countable sequence $\langle u_{\beta }\mid \beta <\omega \rangle $ of $\mathcal M$ -elements of $\mathcal U$ ,

  $$\begin{align*}\varphi(\{\alpha<\kappa\mid\forall\beta<\omega\ \langle 0,\alpha\rangle\,\mathcal M\,u_{\beta}\}). \end{align*}$$

• • If $\Omega (U,I)$ denotes the property that $\Delta U\in I^+$ , then this translates to the statement that for any $\kappa $ -enumeration $\langle u_{\beta }\mid \beta <\kappa \rangle $ of the $\mathcal M$ -elements of $\mathcal U$ ,

  $$\begin{align*}\varphi(\{\alpha<\kappa\mid\forall\beta<\alpha\ \langle 0,\alpha\rangle\,\mathcal M\,u_{\beta}\}). \end{align*}$$

If the property $X\in I^+$ is first order definable, observe that we obtain a $\Delta ^1_1$ -statement in the first two cases above, for we can equivalently rephrase the above to use existential rather than universal second order quantifiers. However this does not work in the third case (see the remarks made in Footnote 10). In particular, this means that the Ramsey operator and the strongly Ramsey subset operator are always simple, while (the model version of) the ineffability operator is simple.

Remark 7.13. Further examples of operators that are both ineffable and always simple have been introduced in [12, Sections 12 and 13], including in particular the $\mathbf T_{\omega }^{\kappa }$ -Ramsey subset operator $\mathbf T$ , and the $\mathbf {wf}^{\kappa }_{\omega }$ -Ramsey subset operator $\mathbf {wf}$ . All of our results on ineffable always simple operators that follow will thus apply to these operators as well.

As a first application, we want to show that Lemma 6.3 can be extended to work for our framework, and we want to generalize it even further by considering ideals other than the indescribability ideals (which are particular instances of the below by Theorem 6.1). Note that the lemma below does not include the case of applying the ineffability operator to the bounded ideal, which however is already handled as a special case of Lemma 6.3.

Lemma 7.14. Suppose $\kappa $ is a regular cardinal, $\gamma ,\xi <\kappa ^+$ are ordinals with $\xi>0$ , I is an ideal on $\kappa $ such that $I^+$ is $\Pi ^1_{\xi }$ -definable over $V_{\kappa }$ , I is represented by $\langle I_{\alpha }\mid \alpha <\kappa \rangle $ in $\mathop {\mathrm {Ult}}$ , ¹² and that $\mathcal O=\mathfrak O\Psi \Omega $ is simple. Then, there is a $\Pi ^1_{\xi +2\gamma }$ -formula $\Theta ^{\kappa }_{\gamma ,\xi }(X)$ over $V_{\kappa }$ and a club subset $C^{\kappa }_{\gamma ,\xi }$ of $\kappa $ such that for all $S\subseteq \kappa $ , we have

$$\begin{align*}S\in{\mathcal O}^{\gamma}(I)^+\text{ if and only if }V_{\kappa}\models\Theta^{\kappa}_{\gamma,\xi}(S) \end{align*}$$

and for all regular cardinals $\alpha \in C^{\kappa }_{\gamma ,\xi }$ , we have

$$\begin{align*}S\cap\alpha\in {\mathcal O}^{f^{\kappa}_{\gamma}(\alpha)}(I_{\alpha})^+\text{ if and only if } V_{\alpha}\models\Theta^{\kappa}_{\gamma,\xi}(S)\mathrm{|}^{\kappa}_{\alpha}. \end{align*}$$

If $\mathcal O$ is always simple, then the above also holds for $\xi =0$ .

Proof The second statement is handled as usual, namely it is equivalent to the first statement holding in all generic ultrapowers $\mathop {\mathrm {Ult}}$ , ¹³ obtained by forcing with $P(\kappa )/{\mathop {\mathrm {NS}}}_{\kappa }$ . Thus it suffices to verify the first statement both in V and in $\mathop {\mathrm {Ult}}$ . We do so by induction on $\gamma <\kappa ^+$ . The case when $\gamma =0$ is immediate from our assumption. The case when $\gamma $ is a limit ordinal is handled as in the proof of Lemma 6.3.

Suppose $\gamma =\delta +1$ and the result holds for $\delta $ . Then, there is a $\Pi ^1_{\xi +2\delta }$ -formula $\Theta ^{\kappa }_{\delta ,\xi }(X)$ over $V_{\kappa }$ such that both in V and in $\mathop {\mathrm {Ult}}$ , for all $S\subseteq \kappa $ , we have

(4) $$ \begin{align} S\in\mathcal O^{\delta}(I)^+\text{ if and only if }V_{\kappa}\models\Theta^{\kappa}_{\delta,\xi}(S). \end{align} $$

We simply define $\Theta ^{\kappa }_{\gamma ,\xi }(X)$ to be the $\Pi ^1_{\xi +2\gamma }$ -formula over $V_{\kappa }$ which asserts that for every (code $\mathcal M$ for a) transitive weak $\kappa $ -model M there is (a code $\mathcal U$ for) an M-ultrafilter U on $\kappa $ such that $\Psi (M,U)$ and $\Omega (U,\mathcal O^{\delta }(I))$ hold. Since $\mathcal O$ is simple, it follows that this formula is as desired, both in V and in $\mathop {\mathrm {Ult}}$ . Clearly, if $\mathcal O$ is always simple, this works also in case $\xi =0$ .

8 Pre-operators

Our ideal operators are defined via local instances that are parametrized by certain objects. Given a cardinal $\kappa $ , we refer to the collection of all such objects on $\kappa $ as the object type at $\kappa $ of such an operator $\mathcal O$ , and denote this by $\mathcal T(\mathcal O,\kappa )$ . The object type $\mathcal T(\mathcal I,\kappa )$ of the ineffability operator at $\kappa $ is the collection of all $\kappa $ -lists, the object type $\mathcal T(\mathcal R,\kappa )$ of the Ramsey operator at $\kappa $ is the collection of all regressive functions $c\colon [\kappa ]^{<\omega }\to \kappa $ , and the object type of our model based operators at $\kappa $ is simply the powerset of $\kappa $ .

Each object type $\mathcal T$ at $\kappa $ comes with an associated restriction operator, which, given some $y\in \mathcal T$ and some $\alpha <\kappa $ , outputs its natural restriction $y\operatorname {\mathrm {\!\upharpoonright \!}}\alpha $ . The following definition should not bear any surprises.

Definition 8.1. Suppose $\kappa $ is a cardinal and $\alpha <\kappa $ .

• • If $\mathcal T=\mathcal P(\kappa )$ and $y\in \mathcal T$ , then $y\operatorname {\mathrm {\!\upharpoonright \!}}\alpha =y\cap \alpha $ .

• • If $\mathcal T$ is the collection of all $\kappa $ -lists and $y\in \mathcal T$ , then $y\operatorname {\mathrm {\!\upharpoonright \!}}\alpha $ is the restriction of y to the domain $\alpha $ , i.e., the initial segment of length $\alpha $ of the $\kappa $ -sequence y.

• • If $\mathcal T$ is the collection of all functions $c\colon [\kappa ]^{<\omega }\to 2$ and $y\in \mathcal T$ , then $y\operatorname {\mathrm {\!\upharpoonright \!}}\alpha $ is the restriction of y to the domain $[\alpha ]^{<\omega }$ .

Each ideal operator $\mathcal O$ with local instances has an associated pre-operator.

Definition 8.2. Given an ideal operator $\mathcal O$ together with local instances $\mathcal O^y$ at $\kappa $ for $y\in \mathcal T(\mathcal O,\kappa )$ , we define its associated pre-operator $\mathcal O_0$ as follows. Given an ideal I on $\kappa $ such that $I^+$ is definable by a $\Pi ^1_{\xi }$ -formula over $V_{\kappa }$ for some $\xi <\kappa ^+$ , and such that I (in the sense of $\mathop {\mathrm {Ult}}$ ) is represented by $\langle I_{\alpha }\mid \alpha <\kappa \rangle $ in $\mathop {\mathrm {Ult}}$ ,

$$\begin{align*}\mathcal {O}_0(I)^+=\{x\!\subseteq\!\kappa\mid\forall y\in\mathcal T(\mathcal O,\kappa)\,\forall C\!\subseteq\!\kappa\,\textrm{club }\exists\alpha\!\in\!x\ x\cap C\cap\alpha\in\mathcal O^{y\operatorname{\mathrm{\!\upharpoonright\!}}\alpha}(I_{\alpha})^+\}, \end{align*}$$

where $\alpha $ is understood to range over regular uncountable cardinals.

${\mathcal I}_0$ is the subtle operator, and ${\mathcal R}_0$ is the pre-Ramsey operator.

Remark 8.3. Notice that by Theorem 3.4, ${\mathcal I}_0([\kappa ]^{<\kappa })={\mathcal I}_0(\Pi ^1_{\xi }(\kappa ))$ is the subtle ideal on $\kappa $ for any $\xi <\kappa ^+$ , that is the collection of all subsets of $\kappa $ which are not subtle. ${\mathcal R}_0([\kappa ]^{<\kappa })$ is the pre-Ramsey ideal on $\kappa $ . Since we do not know whether an analogue of Theorem 3.4 holds for the pre-Ramsey operator (see Question 10.8), we do not know whether ${\mathcal R}_0(\Pi ^1_{\xi }(\kappa ))={\mathcal R}_0([\kappa ]^{<\kappa })$ for all (or any) $\xi <\kappa ^+$ .

The second author has shown in [12] that the subtle and the pre-Ramsey operators are equivalent to their respective model versions (for ideals containing the nonstationary ideal in case of the subtle operator).

Theorem 8.4 [12, Theorems 7.3 and 9.1].

Whenever $I\supseteq {\mathop {\mathrm {NS}}}_{\kappa }$ ,

$$\begin{align*}{\mathcal I}_0(I)=({\mathcal I}_{mod})_0(I), \end{align*}$$

and for arbitrary ideals I on $\kappa $ ,

$$\begin{align*}{\mathcal R}_0(I)=({\mathcal R}_{mod})_0(I). \end{align*}$$

9 Generating ideals

In this section, we analyze the interplay between our generalized operators, and ideals of higher indescribability. Such an analysis for the Ramsey operator ${\mathcal R}$ and the ideals $\Pi ^1_{\xi }(\kappa )$ for $\xi <\kappa $ has been performed by the first author in his [6]. Let us start this section by citing a classic result of Baumgartner, that we will extend afterwards. Given ideals I and J on $\kappa $ , $\overline {I\cup J}$ denotes the collection of all sets $X\cup Y$ for which $X\in I$ and $Y\in J$ .

Theorem 9.1 [3, Section 7].

For all cardinals $\kappa $ and all $\xi \in \{-1\}\cup \omega $ , $\kappa \in {\mathcal I}(\Pi ^1_{\xi }(\kappa ))^+$ if and only if:

1. (1) $\kappa \in {\mathcal I}_0(\Pi ^1_{\xi }(\kappa ))^+\cap \Pi ^1_{\xi +2}(\kappa )^+$ and

2. (2) the ideal $\overline {{\mathcal I}_0(\Pi ^1_{\xi }(\kappa ))\cup \Pi ^1_{\xi +2}(\kappa )}$ is nontrivial and equals ${\mathcal I}(\Pi ^1_{\xi }(\kappa ))$ .

Moreover, (2) is necessary in the above characterization of $\kappa \in {\mathcal I}(\Pi ^1_{\xi }(\kappa ))^+$ , for the least $\Pi ^1_{\xi +2}$ -indescribable cardinal $\kappa $ such that $\kappa \in {\mathcal I}_0(\Pi ^1_{\xi }(\kappa ))^+$ is strictly below the least cardinal $\kappa $ for which $\kappa \in {\mathcal I}(\Pi ^1_{\xi }(\kappa ))^+$ (if such exists).

We will frequently use the following.

Remark 9.2. Suppose $I_0$ , $I_1$ , and J are ideals on $\kappa $ . In order to prove that $J=\overline {I_0\cup I_1}$ , part of what we must show is that $J\supseteq \overline {I_0\cup I_1}$ , or in other words $J^+\subseteq \overline {I_0\cup I_1}^+$ . Notice that we may obtain a chain of equivalences directly from the definitions involved:

$$ \begin{align*} J^+\subseteq\overline{I_0\cup I_1}^+ &\iff \overline{I_0\cup I_1}\subseteq J \\ &\iff I_0\cup I_1\subseteq J\\ &\iff J^+\subseteq I_0^+\cap I_1^+. \end{align*} $$

In the following, we extend Baumgartner’s result to simple ineffable operators, and to ideals of higher indescribability. For readers who are only interested in the operators ${\mathcal I}$ and ${\mathcal R}$ , it should be possible to read this section without having read Section 7 in full detail. In this case, it is only relevant to know that both ${\mathcal I}$ and ${\mathcal R}$ are ineffable and simple, that ${\mathcal R}$ is always simple, and that ${\mathcal I}$ and ${\mathcal R}$ are monotonic in the sense that if $\mathcal O\in \{{\mathcal I},{\mathcal R}\}$ , and $I\subseteq J$ are both ideals on a cardinal $\kappa $ , then ${\mathcal O}(I)\subseteq {\mathcal O}(J)$ . It should then be easy to read the present section, perhaps checking some relevant bits of Section 7 when needed. Let us remind our readers of the following, which provides a fairly large class of ideals that Theorem 9.4 applies to. It is immediate from Corollary 7.9 and from Lemma 7.14.

Observation 9.3. Assume that $\mathcal O$ is ineffable and simple, and that $\Pi ^1_{<\xi }(\kappa )\subseteq I$ .

• • If $\gamma <\kappa ^+$ , $0<\xi <\kappa ^+$ , and $I^+$ is $\Pi ^1_{\xi }$ -definable over $V_{\kappa }$ , ¹⁴ then $\Pi ^1_{<(\xi +2\gamma )}(\kappa )\subseteq \mathcal O^{\gamma }(I)$ , and the latter ideal is $\Pi ^1_{\xi +2\gamma }$ -definable over $V_{\kappa }$ .

• • If $\xi =0$ , the same holds true if either $\mathcal O$ is always simple, or if $\gamma \ge \omega $ , and $I^+$ is $\Pi ^1_n$ -definable for some $n<\omega $ . ¹⁵ ⊣

The conclusion of Theorem 9.4 in case ${\mathcal O}={\mathcal R}$ and $I=\Pi ^1_{\xi }(\kappa )$ for $\xi <\omega $ is due to Feng in [8, Theorem 4.8], and has been extended to $\xi <\kappa $ by the first author in [6, Corollary 6.2].

Theorem 9.4. Assume that I is an ideal on $\kappa $ , $\mathcal O=\mathfrak O\Psi \Omega $ is ineffable and simple, $\gamma ,\xi <\kappa ^+$ , $\xi>0$ , $\Pi ^1_{<\xi }(\kappa )\subseteq I$ , $I^+$ is $\Pi ^1_{\xi }$ -definable over $V_{\kappa }$ , and $\kappa \in {\mathcal O}^{\gamma +1}(I)^+$ . Then,

$$\begin{align*}{\mathcal O}^{\gamma+1}(I)=\overline{{\mathcal O}_0({\mathcal O}^{\gamma}(I))\cup\Pi^1_{\xi+2\gamma+1}(\kappa)}. \end{align*}$$

If either $\mathcal O=\mathcal I$ , $\mathcal O$ is ineffable and always simple, or $\gamma \ge \omega $ in the above, then the conclusion also holds in case $\xi =0$ .

Proof Let us first treat the cases when $\mathcal O$ is either simple or always simple. Let $J=\overline {{\mathcal O}_0({\mathcal O}^{\gamma }(I))\cup \Pi ^1_{\xi +2\gamma +1}(\kappa )}$ , and assume that I, in the sense of $\mathop {\mathrm {Ult}}$ , is represented by $\langle I_{\alpha }\mid \alpha <\kappa \rangle $ in $\mathop {\mathrm {Ult}}$ .

Suppose $S\in J^+$ , and for the sake of a contradiction, suppose $S\in {\mathcal O}^{\gamma +1}(I)$ . Let $y\subseteq \kappa $ be such that whenever M is a transitive weak $\kappa $ -model with $y\in M$ and U is an M-ultrafilter on $\kappa $ such that $\Psi (M,U)$ and $\Omega (U,\mathcal O^{\gamma }(I))$ hold, then $S\not \in U$ . Using that $\mathcal O$ is simple, or always simple in case $\xi =0$ , and using Lemma 7.14, this property of y and of S can be expressed by a natural $\Pi ^1_{\xi +2\gamma +1}$ -formula $\varphi (y,S)$ over $V_{\kappa }$ , and there is a club $C\subseteq \kappa $ such that for all regular $\alpha \in C$ ,

$$\begin{align*}V_{\alpha}\models\varphi(y,S)\mathrm{|}^{\kappa}_{\alpha}\end{align*}$$

if and only if whenever M is a transitive weak $\alpha $ -model with $y\cap \alpha \in M$ and U is an M-ultrafilter on $\alpha $ such that $\Psi (M,U)$ and $\Omega (U,\mathcal O^{f^{\kappa }_{\gamma }(\alpha )}(I_{\alpha }))$ hold, then $S\cap \alpha \not \in U$ .

Since $V_{\kappa }\models \varphi (y,S)$ , the set

$$\begin{align*}D=\{\alpha<\kappa\mid V_{\alpha}\models\varphi(y,S)\mathrm{|}^{\kappa}_{\alpha}\} \end{align*}$$

is in the filter $\Pi ^1_{\xi +2\gamma +1}(\kappa )^*$ . Since $S\notin J$ , S is not the union of a set in ${\mathcal O}_0({\mathcal O}^{\gamma }(I))$ and a set in $\Pi ^1_{\xi +2\gamma +1}(\kappa )$ . Since $S=(S\cap C\cap D)\cup (S\setminus (C\cap D))$ and $S\setminus (C\cap D)\in \Pi ^1_{\xi +2\gamma +1}(\kappa )$ , we see that $S\cap C\cap D\in {\mathcal O}_0({\mathcal O}^{\gamma }(I))^+$ . Thus, by definition of ${\mathcal O}_0$ , there is some ordinal $\alpha \in S\cap C\cap D$ for which there exists a transitive weak $\alpha $ -model M with $y\cap \alpha \in M$ and an M-ultrafilter U on $\alpha $ such that $\Psi (M,U)$ and $\Omega (U,{\mathcal O}^{f^{\kappa }_{\gamma }(\alpha )}(I_{\alpha }))$ hold, and such that $S\cap C\cap D\cap \alpha \in U$ . However, since $\alpha \in C\cap D$ we have $V_{\alpha }\models \varphi (y,S)\mathrm {|}^{\kappa }_{\alpha }$ , contradicting the above.

Now suppose $S\in {\mathcal O}^{\gamma +1}(I)^+$ . Since $\mathcal O$ is ineffable on I, and by our assumption that $\Pi ^1_{<\xi }(\kappa )\subseteq I$ , this implies that $S\in \Pi ^1_{\xi +2\gamma +1}(\kappa )^+$ by Corollary 7.9. Let us show that $S\in {\mathcal O}_0({\mathcal O}^{\gamma }(I))^+$ . Suppose $y\subseteq \kappa $ and fix a club subset C of $\kappa $ . By the third item in Observation 7.8, it follows that $S\cap C\in {\mathcal O}^{\gamma +1}(I)^+$ , and thus there is a weak $\kappa $ -model $M\supseteq V_{\kappa }$ with $y\in M$ , and an M-ultrafilter U on $\kappa $ such that $\Psi (M,U)$ and $\Omega (U,O^{\gamma }(I))$ hold, and such that $S\cap C\in U$ . By our assumptions and using Lemma 7.14, this property of $S\cap C$ , y, and the codes $\mathcal M$ and $\mathcal U$ of M and U respectively is expressible by a $\Pi ^1_{\xi +2\gamma }$ -formula $\varphi (S\cap C,y,\mathcal M,\mathcal U)$ over $V_{\kappa }$ , which additionally states that $\mathcal M$ is a code for a weak $\kappa $ -model M, and that $\mathcal U$ is a code for an M-ultrafilter on $\kappa $ . Moreover using Lemma 7.14, there is a club subset D of $\kappa $ such that for $\alpha \in D$ , $\varphi (S\cap C,y,\mathcal M,\mathcal U)\mathrm {|}^{\kappa }_{\alpha }$ expresses the corresponding property over $V_{\alpha }$ , namely that $\mathcal M\cap V_{\alpha }$ is a code for a weak $\alpha $ -model $\bar M$ , that $\mathcal U\cap V_{\alpha }$ is a code for an $\bar M$ -ultrafilter $\bar U$ on $\alpha $ , that $y\cap \alpha \in \bar M$ , $\Psi (\bar M,\bar U)$ and $\Omega (\bar U,{\mathcal O}^{f^{\kappa }_{\gamma }(\alpha )}(I_{\alpha }))$ hold, and that $S\cap C\cap \alpha \in \bar U$ . Since $S\cap C\cap D$ is $\Pi ^1_{\xi +2\gamma +1}$ -indescribable, there is some $\alpha \in S\cap C\cap D$ such that $V_{\alpha }\models \varphi (S\cap C,y,\mathcal M,\mathcal U)\mathrm {|}^{\kappa }_{\alpha }$ . Thus, $S\in {\mathcal O}_0({\mathcal O}^{\gamma }(I))^+$ .

When ${\mathcal O}={\mathcal I}$ (and $\xi =0$ ), note that the case when $\gamma =0$ is handled by Theorem 9.1. For $\gamma>1$ , note that if $\gamma =1+\bar \gamma $ , we have ${\mathcal I}^{\gamma }(I)={\mathcal I}_{mod}^{\bar \gamma }({\mathcal I}(I))$ , and that ${\mathcal I}(I)$ has the properties that $\Pi ^1_1(\kappa )\subseteq I$ and that ${\mathcal I}(I)^+$ is $\Pi ^1_2$ -definable over $V_{\kappa }$ . We can now apply the main case of the theorem using the operator $\mathcal O={\mathcal I}_{mod}$ .

Similarly, if $\gamma =\omega +\delta \ge \omega $ (and $\xi =0$ ), the desired conclusion of the theorem can be rewritten as

$$\begin{align*}{\mathcal O}^{\delta+1}({\mathcal O}^{\omega}(I))=\overline{{\mathcal O}_0({\mathcal O}^{\delta}({\mathcal O}^{\omega}(I)))\cup\Pi^1_{\xi+2\gamma+1}(\kappa)},\end{align*}$$

and we can deduce this conclusion from applying the theorem to the ideal ${\mathcal O}^{\omega }(I)$ , using Observation 9.3.

9.1 On finite iterates of operators

By Remark 8.3, we have ${\mathcal I}_0(\Pi ^1_{\xi }(\kappa ))={\mathcal I}_0([\kappa ]^{<\kappa })$ for any $\xi <\kappa ^+$ , and hence we easily obtain the following corollary of Theorem 9.4.

Corollary 9.5. Suppose $\kappa \in {\mathcal I}(\Pi ^1_{\xi }(\kappa ))^+$ where $\xi \in \{-1\}\cup \kappa ^+$ . Then

$$\begin{align*}{\mathcal I}(\Pi^1_{\xi}(\kappa))=\overline{{\mathcal I}_0([\kappa]^{<\kappa})\cup\Pi^1_{\xi+2}(\kappa)}.\end{align*}$$

We can obtain a variant of Theorem 9.4 for finite iterates of operators as follows.

Corollary 9.6. Assume that I is an ideal on $\kappa $ , $\mathcal O$ is ineffable and simple, $\gamma <\omega $ , $0<\xi <\kappa ^+$ , $\Pi ^1_{<\xi }(\kappa )\subseteq I$ , and $I^+$ is $\Pi ^1_{\xi }$ -definable over $V_{\kappa }$ . Then,

$$\begin{align*}{\mathcal O}^{\gamma+1}(I)=\overline{{\mathcal O}_0({\mathcal O}^{\gamma}(I))\cup{\mathcal O}^{\gamma}(\Pi^1_{\xi+1}(\kappa))}. \end{align*}$$

If either $\mathcal O=\mathcal I$ , $\mathcal O$ is ineffable and always simple, or $\gamma \ge \omega $ in the above, then the above conclusion also holds in case $\xi =0$ .

Proof On the one hand, by our assumptions and Corollary 7.9, ${\mathcal O}(I)\supseteq \Pi ^1_{\xi +1}(\kappa )$ , and therefore, ${\mathcal O}^{\gamma +1}(I)\supseteq {\mathcal O}^{\gamma }(\Pi ^1_{\xi +1}(\kappa ))$ by the monotonicity of ${\mathcal O}$ (see Proposition 7.6). On the other hand, ${\mathcal O}^{\gamma }(\Pi ^1_{\xi +1}(\kappa ))\supseteq \Pi ^1_{\xi +2\gamma +1}(\kappa )$ . Thus, the result follows immediately from Theorem 9.4.

There is also a sort of analogue of the above for infinite iterates of operators. This has been worked out for the Ramsey operator in [6, Theorem 7.8], and can analogously be performed for our generalized operators. We will leave all details to the interested reader.

As an easy corollary of Corollary 7.9, again using the monotonicity of our operators, we obtain the following generalization of [6, Corollary 6.8]:

Corollary 9.7. If ${\mathcal O}$ is ineffable, $\xi \in \{-1\}\cup \kappa ^+$ , and $n<\omega $ , then

$$\begin{align*}{\mathcal O}^{\omega}(\Pi^1_{\xi}(\kappa))={\mathcal O}^{\omega}(\Pi^1_{\xi+n}(\kappa)). \end{align*}$$

The next corollary is a starting point in relating assumptions of the form $\kappa \in {\mathcal O}^{\gamma }(\Pi ^1_{\xi }(\kappa ))$ , for different $\gamma $ and $\xi $ below $\kappa ^+$ , with respect to consistency strength.

Corollary 9.8. Assume that $\mathcal O$ is ineffable and simple. Suppose $\gamma <\omega $ , $\xi <\kappa ^+$ , and $\kappa \in {\mathcal O}^{\gamma }(\Pi ^1_{\xi }(\kappa ))^+$ . If $S\in {\mathcal O}^{\delta }(\Pi ^1_{\zeta }(\kappa ))^+$ where $\zeta +1+2\delta \leq \xi +2\gamma $ , then

$$\begin{align*}T=\{\alpha<\kappa\textrm{ regular}\mid S\cap\alpha\in{\mathcal O}^{f^{\kappa}_{\delta}(\alpha)}(\Pi^1_{f^{\kappa}_{\zeta}(\alpha)}(\alpha))^+\}\in{\mathcal O}^{\gamma}(\Pi^1_{\xi}(\kappa))^*.\end{align*}$$

If either $\mathcal O=\mathcal I$ , or $\mathcal O$ is ineffable and always simple, then the above conclusion also holds in case $\xi =-1$ .

Proof The fact that $S\in {\mathcal O}^{\delta }(\Pi ^1_{\zeta }(\kappa ))^+$ is expressible by a $\Pi ^1_{\zeta +1+2\delta }$ -formula $\Theta $ over $V_{\kappa }$ by Lemma 7.14, or by Lemma 6.3 in case ${\mathcal O}={\mathcal I}$ . Let C be the corresponding club obtained from the relevant lemma. Since $\zeta +1+2\delta \leq \xi +2\gamma $ , we have $\Pi ^1_{\zeta +1+2\delta }(\kappa )\subseteq \Pi ^1_{\xi +2\gamma }(\kappa )\subseteq {\mathcal O}^{\gamma }(\Pi ^1_{\xi }(\kappa ))$ by Corollary 7.9. The set

$$\begin{align*}\{\alpha\in C\textrm{ regular}\mid V_{\alpha}\models\Theta(S)\mathrm{|}^{\kappa}_{\alpha}\}\end{align*}$$

is contained in T and is in $\Pi ^1_{\zeta +1+2\delta }(\kappa )^*\subseteq {\mathcal O}^{\gamma }(\Pi ^1_{\xi }(\kappa ))^*$ . Therefore, $T\in {\mathcal O}^{\gamma }(\Pi ^1_{\xi }(\kappa ))^*$ , as desired.

We do not know whether the next result on the proper containment of certain ideals generated by applications of ${\mathcal I}$ and ${\mathcal R}$ generalizes to our framework of operators, for we do not know whether Lemma 5.2 does.

Corollary 9.9. Let ${\mathcal O}\in \{{\mathcal I},{\mathcal R}\}$ . Suppose $\gamma <\omega $ , $\xi \in \{-1\}\cup \kappa ^+$ , and $\kappa \in {\mathcal O}^{\gamma }(\Pi ^1_{\xi +1}(\kappa ))^+$ . Then,

$$\begin{align*}{\mathcal O}^{\gamma}(\Pi^1_{\xi}(\kappa))\subsetneq{\mathcal O}^{\gamma}(\Pi^1_{\xi+1}(\kappa)).\end{align*}$$

Proof Clearly ${\mathcal O}^{\gamma }(\Pi ^1_{\xi }(\kappa ))\subseteq {\mathcal O}^{\gamma }(\Pi ^1_{\xi +1}(\kappa ))$ , so we just need to show that the containment is proper. Let $S=\{\alpha <\kappa \mid \alpha \in {\mathcal O}^{f^{\kappa }_{\gamma }(\alpha )}(\Pi ^1_{f^{\kappa }_{\xi }(\alpha )}(\alpha ))\}$ . Then $S\in {\mathcal O}^{\gamma }(\Pi ^1_{\xi }(\kappa ))^+$ by Lemma 5.2, and Corollary 9.8 implies that $S\in {\mathcal O}^{\gamma }(\Pi ^1_{\xi +1}(\kappa ))$ .

We can show yet another form of proper containment of ideals when ${\mathcal O}={\mathcal I}$ . An analogous result for the operator ${\mathcal R}$ was claimed by the first author in [6] (see our Question 10.6).

Corollary 9.10. Suppose $\gamma <\kappa ^+$ , $\xi \in \{-1\}\cup \kappa ^+$ , and $\kappa \in {\mathcal I}^{\gamma }(\Pi ^1_{\xi +2}(\kappa ))^+$ . Then,

$$\begin{align*}{\mathcal I}^{\gamma}(\Pi^1_{\xi+2}(\kappa))\subsetneq{\mathcal I}^{\gamma+1}(\Pi^1_{\xi}(\kappa)). \end{align*}$$

Proof The ¹⁶ inclusion itself is immediate, since $\Pi ^1_{\xi +2}(\kappa )\subseteq {\mathcal I}(\Pi ^1_{\xi }(\kappa ))$ by the ineffability of ${\mathcal I}$ , and it only remains to verify its properness. Since $\kappa \in {\mathcal I}^{\gamma }(\Pi ^1_{\xi +2}(\kappa ))^+$ , it follows by Lemma 5.2 that the set

$$\begin{align*}S=\{\alpha<\kappa\mid\alpha\in{\mathcal I}^{f^{\kappa}_{\gamma}(\alpha)}(\Pi^1_{f^{\kappa}_{\xi+2}(\alpha)}(\alpha))\}\end{align*}$$

is in ${\mathcal I}^{\gamma }(\Pi ^1_{\xi +2}(\kappa ))^+$ . From Corollary 3.9, it follows that the set

$$\begin{align*}C=\{\alpha<\kappa\mid(\forall\eta<\alpha^+)\ \alpha\in{\mathcal I}^{f^{\kappa}_{\gamma}(\alpha)}(\Pi^1_{\eta}(\alpha))^+\}\end{align*}$$

is in the filter ${\mathcal I}^{\gamma +1}([\kappa ]^{<\kappa })^*$ . Since $C\subseteq \kappa \setminus S$ , we see that $\kappa \setminus S\in {\mathcal I}^{\gamma +1}([\kappa ]^{<\kappa })^*\subseteq {\mathcal I}^{\gamma +1}(\Pi ^1_{\xi }(\kappa ))^*$ . Hence, this implies that $S\in {\mathcal I}^{\gamma +1}(\Pi ^1_{\xi }(\kappa ))\setminus {\mathcal I}^{\gamma }(\Pi ^1_{\xi +2}(\kappa ))$ .

The next corollary extends Baumgartner’s observation that the use of ideals is necessary in Theorem 9.1.

Corollary 9.11. Assume that $\mathcal O=\mathfrak O\Psi \Omega $ is ineffable and simple, $\gamma <\omega $ , $\xi <\kappa $ , and $I=\Pi ^1_{<\xi }(\kappa )$ . Then, $\kappa \in {\mathcal O}^{\gamma +1}(I)^+$ if and only if:

1. (1) $\kappa \in {\mathcal O}_0({\mathcal O}^{\gamma }(I))^+\cap \Pi ^1_{\xi +2\gamma +1}(\kappa )^+$ and

2. (2) the ideal $\overline {{\mathcal O}_0({\mathcal O}^{\gamma }(I))\cup \Pi ^1_{\xi +2\gamma +1}(\kappa )}$ is nontrivial and equals ${\mathcal O}^{\gamma +1}(I)$ .

If either $\mathcal O=\mathcal I$ , or $\mathcal O$ is ineffable and always simple in the above, then the conclusion also holds in case $\xi =0$ .

Moreover, (2) is necessary in the above characterization, that is, the least $\Pi ^1_{\xi +2\gamma +1}$ -indescribable cardinal $\kappa $ that satisfies $\kappa \in {\mathcal O}_0({\mathcal O}^{\gamma }(I))^+$ is strictly below the least cardinal $\kappa $ that satisfies $\kappa \in {\mathcal O}^{\gamma +1}(I)^+$ .

Proof Note that $\kappa $ being $\Pi ^1_{\xi +2\gamma +1}$ -indescribable and $\kappa \in {\mathcal O}_0({\mathcal O}^{\gamma }(I))^+$ are $\Pi ^1_{\xi +2\gamma +2}$ -properties over $V_{\kappa }$ , and $\kappa \in {\mathcal O}^{\gamma +1}(I)^+$ implies that $\kappa $ is $\Pi ^1_{<(\xi +1+2\gamma +2)}$ -indescribable by Corollary 7.9, and hence $\kappa $ is $\Pi ^1_{\xi +2\gamma +2}$ -indescribable using that $\gamma $ is finite. Now since $\xi <\kappa $ , this yields some $\xi <\alpha <\kappa $ such that $\alpha \in {\mathcal O}_0({\mathcal O}^{\gamma }(\Pi ^1_{\xi }(\alpha )))^+$ and $\alpha \in \Pi ^1_{\xi +2\gamma +1}(\alpha )^+$ .

In the above, one could obtain analogous results when $\kappa \le \xi <\kappa ^+$ ; however the statement that is reflected down from $\kappa $ to $\alpha $ will be changed for $\xi $ will be reflected down to $f^{\kappa }_{\xi }(\alpha )$ . This still yields a satisfactory analogue of Corollary 9.11 when $\kappa \leq \xi <\kappa ^+$ and $\xi $ is definable from $\kappa $ (for example, if $\xi =\kappa $ , or $\xi =\kappa +\kappa $ , etc.). We will leave the easy and straightforward details to our interested readers.

9.2 On infinite iterates of operators

We would like to use Theorem 9.4 to prove an analogue of Corollary 9.8 for infinite $\gamma $ , which would, in a sense, say that the strength of the hypothesis “ $\kappa \in {\mathcal O}^{\gamma }(\Pi ^1_{\xi }(\kappa ))^+$ ” increases as $\xi $ increases. However, there is an added complication, as illustrated in Corollary 9.7, which is that if $\xi _0<\xi _1<\kappa ^+$ , it may be that $\kappa \in {\mathcal O}^{\gamma }(\Pi ^1_{\xi _0}(\kappa ))^+$ is equivalent to $\kappa \in {\mathcal O}^{\gamma }(\Pi ^1_{\xi _1}(\kappa ))^+$ , if $\gamma $ is large enough. In the next theorem, we determine the least $\gamma $ for which this occurs when ${\mathcal O}\in \{{\mathcal I},{\mathcal R}\}$ . Let us note that we do not know how to verify this leastness for operators other than ${\mathcal I}$ and ${\mathcal R}$ . Even though the other statements of the theorem below in fact hold for simple ineffable operators, we therefore only state the below result for these two operators.

Theorem 9.12. Suppose $\kappa $ is a cardinal, $\xi _0<\xi _1$ are in $\{-1\}\cup \kappa ^+$ , and ${\mathcal O}\in \{{\mathcal I},{\mathcal R}\}$ . Then, the ideal chains

$$\begin{align*}\langle{\mathcal O}^{\gamma}(\Pi^1_{\xi_0}(\kappa))\mid\gamma<\kappa^+\rangle\textrm{ and }\langle{\mathcal O}^{\gamma}(\Pi^1_{\xi_1}(\kappa))\mid\gamma<\kappa^+\rangle\end{align*}$$

are eventually equal. Moreover, letting $\delta =\mathop {\mathrm {ot}}\nolimits (\xi _1\setminus \xi _0)\cdot \omega $ , if the ideal ${\mathcal O}^{\delta }(\Pi ^1_{\xi _1}(\kappa ))$ is nontrivial, then $\delta $ is least ordinal such that

$$\begin{align*}{\mathcal O}^{\delta}(\Pi^1_{\xi_0}(\kappa))={\mathcal O}^{\delta}(\Pi^1_{\xi_1}(\kappa)).\end{align*}$$

Proof First, let us show that ${\mathcal O}^{\delta }(\Pi ^1_{\xi _0}(\kappa ))={\mathcal O}^{\delta }(\Pi ^1_{\xi _1}(\kappa ))$ , where $\delta =\mathop {\mathrm {ot}}\nolimits (\xi _1\setminus \xi _0)\cdot \omega $ . Since $\xi _0<\xi _1$ , it is clear that ${\mathcal O}^{\delta }(\Pi ^1_{\xi _0}(\kappa ))\subseteq {\mathcal O}^{\delta }(\Pi ^1_{\xi _1}(\kappa ))$ . Let us show that ${\mathcal O}^{\delta }(\Pi ^1_{\xi _0}(\kappa ))\supseteq {\mathcal O}^{\delta }(\Pi ^1_{\xi _1}(\kappa ))$ . If $\sigma =\mathop {\mathrm {ot}}\nolimits (\xi _1\setminus \xi _0)=n$ is finite, then $\delta =n\cdot \omega =\omega $ and the result follows from Corollary 9.7. Suppose $\sigma \geq \omega $ . Then $\delta =\sigma \cdot \omega $ is a limit of limit ordinals. Thus, it will suffice to show that ${\mathcal O}^{\eta }(\Pi ^1_{\xi _1}(\kappa ))\subseteq {\mathcal O}^{\delta }(\Pi ^1_{\xi _0}(\kappa ))$ for all limit ordinals $\eta <\delta $ . Fix a limit ordinal $\eta <\delta $ . By Corollary 7.9, we have

(5) $$ \begin{align} \Pi^1_{\xi_1}(\kappa)=\Pi^1_{\xi_0+\sigma}(\kappa)\subseteq{\mathcal O}^{\sigma+1}(\Pi^1_{\xi_0}(\kappa)). \end{align} $$

Applying the operator ${\mathcal O} \ \eta $ -many times to (5) yields

$$\begin{align*}{\mathcal O}^{\eta}(\Pi^1_{\xi_1}(\kappa))\subseteq{\mathcal O}^{\sigma+1+\eta}(\Pi^1_{\xi_0}(\kappa))\subseteq{\mathcal O}^{\delta}(\Pi^1_{\xi_0}(\kappa)),\end{align*}$$

where the final subset relation follows since $\sigma +1+\eta <\delta $ .

Next, let us show that if $\eta <\delta $ , then ${\mathcal O}^{\eta }(\Pi ^1_{\xi _0}(\kappa ))\subsetneq {\mathcal O}^{\eta }(\Pi ^1_{\xi _1}(\kappa ))$ . If $\sigma =\mathop {\mathrm { ot}}\nolimits (\xi _1\setminus \xi _0)$ is finite, in which case $\delta =\omega $ , then the result follows from Corollary 9.9. On the other hand, if $\sigma $ is infinite, then $\delta =\sigma \cdot \omega $ is a limit of limit ordinals. Let $\nu $ be a limit ordinal with $\eta \le \nu <\delta $ . It suffices to show that ${\mathcal O}^{\nu +1}(\Pi ^1_{\xi _0}(\kappa ))\subsetneq {\mathcal O}^{\nu +1}(\Pi ^1_{\xi _1}(\kappa ))$ , for this contradicts ${\mathcal O}^{\eta }(\Pi ^1_{\xi _0}(\kappa ))={\mathcal O}^{\eta }(\Pi ^1_{\xi _1}(\kappa ))$ . Let

$$\begin{align*}S=\Big\{\alpha<\kappa\mid\alpha\in{\mathcal O}^{f^{\kappa}_{\nu+1}(\alpha)}(\Pi^1_{f^{\kappa}_{\xi_0}(\alpha)}(\alpha))\Big\}.\end{align*}$$

Since $\kappa \in {\mathcal O}^{\delta }(\Pi ^1_{\xi _1}(\kappa ))^+$ , it follows from Lemma 5.2 that $S\notin {\mathcal O}^{\nu +1}(\Pi ^1_{\xi _0}(\kappa ))$ . Furthermore, the fact that $S\notin {\mathcal O}^{\nu +1}(\Pi ^1_{\xi _0}(\kappa ))$ is expressible by a $\Pi ^1_{\xi _0+\nu +2}$ -sentence $\Theta $ over $V_{\kappa }$ , by Lemma 6.3. Let C be the corresponding club subset of $\kappa $ obtained from that lemma. It follows that the set

$$\begin{align*}D=\{\alpha\in C\mid V_{\alpha}\models\Theta(S)\mathrm{|}^{\kappa}_{\alpha}\}\end{align*}$$

is in the filter $\Pi ^1_{\xi _0+\nu +2}(\kappa )^*$ and is contained in $\kappa \setminus S$ . Hence, $S\in \Pi ^1_{\xi _0+\nu +2}(\kappa )$ . By Corollary 7.9, since $\xi _1=\xi _0+\sigma $ , it follows that $\Pi ^1_{\xi _0+\sigma +\nu +1}(\kappa )\subseteq {\mathcal O}^{\nu +1}(\Pi ^1_{\xi _1}(\kappa ))$ . Since $\nu <\delta =\sigma \cdot \omega $ , it follows that $\xi _0+\nu +2<\xi _0+\sigma +\nu +1$ and thus $\Pi ^1_{\xi _0+\nu +2}(\kappa )\subseteq {\mathcal O}^{\nu +1}(\Pi ^1_{\xi _1}(\kappa ))$ . Together with the above, this implies that $S\in {\mathcal O}^{\nu +1}(\Pi ^1_{\xi _1}(\kappa ))$ .

Next, extending Corollary 9.8 to infinite iterates of operators, we show that for ineffable and simple operators ${\mathcal O}$ , $\gamma <\kappa ^+$ , and $\xi _0<\xi _1$ in $\kappa ^+\setminus \omega $ , the hypothesis $\kappa \in {\mathcal O}^{\gamma }(\Pi ^1_{\xi _1}(\kappa ))^+$ implies that there are many $\alpha <\kappa $ which satisfy $\alpha \in {\mathcal O}^{f^{\kappa }_{\gamma }(\alpha )}(\Pi ^1_{f^{\kappa }_{\xi _0}(\alpha )}(\alpha ))^+$ , assuming $\xi _0$ and $\xi _1$ are far enough apart. Thus, the hypotheses of the form $\kappa \in {\mathcal O}^{\gamma }(\Pi ^1_{\xi }(\kappa ))^+$ for (certain) $\xi <\kappa ^+$ provide a strictly increasing hierarchy of length $\kappa ^+$ .

Theorem 9.13. Suppose ${\mathcal O}$ is ineffable and simple, $\kappa $ is a cardinal, $\omega \le \xi _0<\xi _1$ are in $\{-1\}\cup \kappa ^+$ , and $\gamma <\mathop {\mathrm { ot}}\nolimits (\xi _1\setminus \xi _0)\cdot \omega $ . If $\kappa \in {\mathcal O}^{\gamma }(\Pi ^1_{\xi _1}(\kappa ))^+$ , then the set

$$\begin{align*}\{\alpha<\kappa\mid\alpha\in{\mathcal O}^{f^{\kappa}_{\gamma}(\alpha)}(\Pi^1_{f^{\kappa}_{\xi_0}(\alpha)}(\alpha))^+\}\end{align*}$$

is in ${\mathcal O}^{\gamma }(\Pi ^1_{\xi _1}(\kappa ))^*$ .

Proof Since $\kappa \in {\mathcal O}^{\gamma }(\Pi ^1_{\xi _1}(\kappa ))^+$ and $\xi _0<\xi _1$ , we have $\kappa \in {\mathcal O}^{\gamma }(\Pi ^1_{\xi _0}(\kappa ))^+$ , which is expressible by a $\Pi ^1_{\xi _0+1+2\gamma }$ -formula $\Theta $ over $V_{\kappa }$ by Lemma 7.14. Let C be the corresponding club subset of $\kappa $ obtained from that lemma. Since $\gamma <\mathop {\mathrm {ot}}\nolimits (\xi _1\setminus \xi _0)\cdot \omega $ , it follows that $\xi _0+1+2\gamma <\xi _1+1+2\gamma $ . Now by Corollary 7.9, we see that $\Pi ^1_{<(\xi _1+1+2\gamma )}\subseteq {\mathcal O}^{\gamma }(\Pi ^1_{\xi _1}(\kappa ))$ , and thus the set

$$\begin{align*}D=\{\alpha\in C\mid V_{\alpha}\models\Theta(\kappa)\mathrm{|}^{\kappa}_{\alpha}\}\subseteq\{\alpha<\kappa\mid\alpha\in{\mathcal O}^{f^{\kappa}_{\gamma}(\alpha)}(\Pi^1_{f^{\kappa}_{\xi_0}(\alpha)}(\alpha))^+\}\end{align*}$$

is in ${\mathcal O}^{\gamma }(\Pi ^1_{\xi _1}(\kappa ))^*$ .

10 On some results of the first author

In [6, Theorem 4.1], the first author claimed the following: If $S\in \mathcal R([\kappa ]^{<\kappa })^+$ , then

$$\begin{align*}T=\{\alpha<\kappa\mid\forall\xi<\alpha\ S\cap\alpha\in\Pi^1_{\xi}(\alpha)^+\}\in\mathcal R([\kappa]^{<\kappa})^*.\end{align*}$$

The proof that is provided however is slightly flawed, and in fact only yields a somewhat weaker result, namely a weak form of the analogue of Theorem 3.8 for the Ramsey operator $\mathcal R$ rather than the ineffability operator $\mathcal I$ , in the special case when $\gamma =0$ (see below). We first want to provide a counterexample for the above statement that is claimed in [6], and then follow it with a corrected version of that theorem. We then shortly discuss the consequences that this has on other results of [6]. For the very start, we need an auxiliary result.

Lemma 10.1. If $\kappa $ is a measurable cardinal, then

$$\begin{align*}\{\alpha<\kappa\mid\alpha\textrm{ is not Ramsey}\}\not\in\mathcal R([\kappa]^{<\kappa})^*.\end{align*}$$

Proof Using Theorem 7.4, $\mathcal R([\kappa ]^{<\kappa })^*={\mathcal R}_{mod}([\kappa ]^{<\kappa })^*$ . Let $U^*$ be a measurable ultrafilter on $\kappa $ and let $A\subseteq \kappa $ be arbitrary. Let $M^*\prec H((2^{\kappa })^+)$ have size $\kappa $ with $A,U^*\in M^*$ and such that $\kappa +1\subseteq M^*$ , let M be the transitive collapse of $M^*$ , and let U be the image of $U^*$ under the collapsing map. Then, U is M-normal, $\kappa $ -amenable for M and countably complete, and since $\kappa $ is Ramsey in the ultrapower of V by $U^*$ , it is also Ramsey in the ultrapower of M by U, and hence $\{\alpha <\kappa \mid \alpha $ is Ramsey $\}\in U$ . This shows that $\{\alpha <\kappa \mid \alpha $ is not Ramsey $\}\not \in {\mathcal R}_{mod}([\kappa ]^{<\kappa })^*$ .

Counterexample 10.2. Assume that $\kappa $ is Ramsey, such that

$$\begin{align*}S=\{\alpha<\kappa\mid\alpha\textrm{ is not Ramsey}\}\not\in\mathcal R([\kappa]^{<\kappa})^*.\end{align*}$$

Then,

$$\begin{align*}T=\{\alpha<\kappa\mid\forall\xi<\alpha\ S\cap\alpha\in\Pi^1_{\xi}(\alpha)^+\}\subseteq \{\alpha<\kappa\mid S\cap\alpha\in\Pi^1_2(\alpha)^+\}\end{align*}$$

$$\begin{align*}=\{\alpha<\kappa\mid\{\beta<\alpha\mid\beta\textrm{ is not Ramsey}\}\in\Pi^1_2(\alpha)^+\}.\end{align*}$$

But since being Ramsey is a $\Pi ^1_2$ -property, if $\alpha $ is a Ramsey cardinal, then every set in $\Pi ^1_2(\alpha )^+$ contains a Ramsey cardinal. Hence the latter set, and thus also T, is contained in S. This shows that $T\not \in \mathcal R([\kappa ]^{<\kappa })^*$ .

The following seems to be exactly the statement that is shown to hold true by the proof of [6, Theorem 4.1].

Theorem 10.3. If $\kappa $ is a cardinal, $S\in \mathcal R([\kappa ]^{<\kappa })^+$ , and

$$\begin{align*}T=\{\alpha<\kappa\mid\forall\xi<\alpha\ S\cap\alpha\in\Pi^1_{\xi}(\alpha)^+\},\end{align*}$$

then $S\setminus T\in \mathcal R([\kappa ]^{<\kappa })$ .

As for the case when $\gamma =0$ in the proof of Theorem 3.8 however, this result now follows directly from Theorem 3.4 because the subtle ideal is contained in the Ramsey ideal ${\mathcal R}([\kappa ]^{<\kappa })$ . Note that Theorem 3.4 in fact yields the stronger statement that if

$$\begin{align*}T^*=\{\alpha<\kappa\mid\forall\xi<\alpha^+\ S\cap\alpha\in\Pi^1_{\xi}(\alpha)^+\},\end{align*}$$

then $S\setminus T^*\in \mathcal R([\kappa ]^{<\kappa })$ .

The next result that is claimed in [6] is its [6, Theorem 4.2], which suffers the same kind of problem as does its [6, Theorem 4.1] (and it is now seen to be wrong for it includes the base case when $\alpha =0$ , which is [6, Theorem 4.1]). However, if its statement is modified according to the modification of [6, Theorem 4.1] that we provided in Theorem 10.3, it is not clear as to whether Cody’s argument can be adapted to work. Let us thus state what might be a good candidate for a corrected version of [6, Theorem 4.2] as an open question. ¹⁷

Question 10.4. Assume that $\kappa $ is a cardinal, that $\gamma <\kappa $ is an ordinal, that $S\in \mathcal R^{\gamma +1}([\kappa ]^{<\kappa })^+$ , and that $T=\{\alpha <\kappa \mid \forall \xi <\alpha ^+\ S\cap \alpha \in \mathcal R^{\gamma }(\Pi ^1_{\xi }(\alpha ))^+\}$ . Does it follow that $S\setminus T\in \mathcal R^{\gamma +1}([\kappa ]^{<\kappa })$ ? ¹⁸

[6, Theorem 4.2] is then used to deduce [6, Corollary 4.3], which we would like to pose as yet another open question, since it now seems unclear how to prove the below when $\gamma>0$ (its instance for $\gamma =0$ however follows directly from Theorem 10.3).

Question 10.5. Assume that $\kappa \in \mathcal R^{\gamma +1}([\kappa ]^{<\kappa })^+$ . Does it follow that

$$\begin{align*}\{\alpha<\kappa\mid\forall\xi<\alpha\ \alpha\in\mathcal R^{\gamma}(\Pi^1_{\xi}(\alpha))^+\}\in\mathcal R^{\gamma+1}([\kappa]^{<\kappa})^*?\end{align*}$$

In the remainder of [6], the results from its Section 4 are only used in a few places. The first result that becomes unclear is [6, Theorem 6.7(2)] (except for the case when $m=1$ , for the proof of which the case when $\gamma =0$ in Question 10.5 suffices), which we thus state as an open question.

Question 10.6. Suppose $1<m<\omega $ and $\xi <\kappa $ . If $\kappa \in {\mathcal R}^m(\Pi ^1_{\xi }(\kappa ))^+$ , does it follow that the inclusion

$$\begin{align*}{\mathcal R}^{m-1}(\Pi^1_{\xi+2}(\kappa))\subseteq{\mathcal R}^m(\Pi^1_{\xi}(\kappa))\end{align*}$$

is a proper inclusion?

The only other result from [6] that becomes unclear is (the properness of the containments in) [6, Theorem 7.9], which is essentially a version of [6, Theorem 6.7(2)] (and thus would yield a version of Question 10.6) for infinite m.

In order to answer Question 10.4 in the affirmative, it seems one would need to proceed by induction on $\gamma $ to prove a statement similar to that of Theorem 3.8, but with the ineffable operator ${\mathcal I}$ replaced with the Ramsey operator ${\mathcal R}$ and the S-list $\vec {S}$ replaced with a regressive function. This suggests the following.

Question 10.7. Suppose $\gamma <\kappa ^+$ , $S\in {\mathcal R}^{\gamma +1}([\kappa ]^{<\kappa })^+$ , and $f:[S]^{<\omega }\to \kappa $ is a regressive function. Let A be the set of all ordinals $\alpha \in S$ such that

$$\begin{align*}\exists X\subseteq S\cap\alpha\left[(\forall \xi<\alpha^+\ X\in {\mathcal R}^{f^{\kappa}_{\gamma}(\alpha)}(\Pi^1_{\xi}(\alpha))^+) \land (X\cup\{\alpha\}\text{ is hom. for }\vec{S})\right].\end{align*}$$

Does it follow that $S\setminus A\in {\mathcal R}^{\gamma +1}([\kappa ]^{<\kappa })$ ?

Notice that in order to address Question 10.7, one might attempt an argument similar to that of Theorem 3.10, using Ramseyness embeddings instead of strong Ramseyness embeddings. However, the elementary embedding characterization of Ramseyness involves weak $\kappa $ -models which are not in general closed under $\omega $ -sequences, and therefore, in the context of the proof of Theorem 3.10, if one only assumes that M is a weak $\kappa $ -model, there is no reason to expect that the sequence $\langle B_n\mid n<\omega \rangle $ is in M, and hence $B=\bigcap _{n<\omega }B_n$ may not be in M.

Feng [8] showed that the Ramsey operator can be characterized using $(\omega ,S)$ -sequences. Recall that for any set S of ordinals, an $(\omega ,S)$ -sequence is a sequence $\vec S$ of the form $\vec S=\langle S_{\alpha _1\ldots \alpha _n}\mid 1\le n<\omega ,\,\alpha _1<\cdots <\alpha _n,\,\alpha _1,\ldots ,\alpha _n\in S\rangle $ , where each $S_{\alpha _1\ldots \alpha _n}\subseteq \alpha _1$ . We say that $H\subseteq S$ is homogeneous for $\vec S$ if for all $n>0$ , and all $\alpha _1<\cdots <\alpha _n$ and $\beta _1<\cdots <\beta _n$ from H, if $\alpha _1\le \beta _1$ , then $S_{\alpha _1\ldots \alpha _n}=S_{\beta _1\ldots \beta _n}\cap \alpha _1$ . Feng proved that for any ideal I on a regular cardinal $\kappa $ we have $S\in {\mathcal R}(I)^+$ if and only if every $(\omega ,S)$ -list has a homogeneous set $H\in P(S)\cap I^+$ . Thus, in Question 10.7 one may replace the regressive function with an $(\omega ,S)$ -list if desired.

It seems that in order to handle the base case ( $\gamma =0$ ) of Question 10.7, one would want to address the following question about the pre-Ramsey ideal; this is in analogy to the fact that the base case of Theorem 3.8 follows from the corresponding result, namely Theorem 3.1, about the subtle ideal. It is straightforward to check that the pre-Ramsey ideal can be characterized in terms of $(\omega ,S)$ -sequences, so let us formulate the question as follows.

Question 10.8. If $S\in \mathcal R_0([\kappa ]^{<\kappa })^+$ , $\vec a$ is an $(\omega ,S)$ -sequence, and

$$\begin{align*}A=\{\alpha\!\in\!S\mid\exists X\!\subseteq\!S\cap\alpha\,\forall\xi<\alpha^+\ X\!\in\!\Pi^1_{\xi}(\alpha)^+\,\land\,X\!\cup\!\{\alpha\}\textrm{ is homogeneous for }\vec a\},\end{align*}$$

does it follow that $S\setminus A\in \mathcal R_0([\kappa ]^{<\kappa })$ ?

Furthermore, note that in Theorem 3.1, we are only stating a particular instance of Baumgartner’s original result, for it is not only about subtlety, but about n-subtlety for any particular $n<\omega $ ; here n-subtlety is a property which resembles subtlety but is formulated in terms of $(n,S)$ -sequences (see [3]). Since pre-Ramseyness is, in a certain sense, simultaneous n-subtlety for all $n<\omega $ , one could hope for Baumgartner’s argument to somehow be adaptable to the context of our Question 10.8, and thus answer it positively. However, our attempts to do so have as yet been unsuccessful.

Let us close by posing the simplest version of Question 10.7 which remains open.

Question 10.9. Is the hypothesis “ $\exists \kappa \ \kappa \in {\mathcal R}^2([\kappa ]^{<\kappa })^+$ ” stronger in consistency strength than “ $\exists \kappa \ \kappa \in {\mathcal R}(\Pi ^1_1(\kappa ))^+$ ?”