A partition is finitary if all its members are finite. For a set A, $\mathscr {B}(A)$ denotes the set of all finitary partitions of A. It is shown consistent with $\mathsf {ZF}$ (without the axiom of choice) that there exist an infinite set A and a surjection from A onto $\mathscr {B}(A)$. On the other hand, we prove in $\mathsf {ZF}$ some theorems concerning $\mathscr {B}(A)$ for infinite sets A, among which are the following:

1. (1) If there is a finitary partition of A without singleton blocks, then there are no surjections from A onto $\mathscr {B}(A)$ and no finite-to-one functions from $\mathscr {B}(A)$ to A.

2. (2) For all $n\in \omega $, $|A^n|<|\mathscr {B}(A)|$.

3. (3) $|\mathscr {B}(A)|\neq |\mathrm {seq}(A)|$, where $\mathrm {seq}(A)$ is the set of all finite sequences of elements of A.

On the basis of Poincaré and Weyl’s view of predicativity as invariance, we develop an extensive framework for predicative, type-free first-order set theory in which $\Gamma _0$ and much bigger ordinals can be defined as von Neumann ordinals. This refutes the accepted view of $\Gamma _0$ as the “limit of predicativity”.

The structure of Borel reducibility for equivalence relations on $\omega _1$ is determined.

An infinite graph is said to be highly connected if the induced subgraph on the complement of any set of vertices of smaller size is connected. We continue the study of weaker versions of Ramsey’s theorem on uncountable cardinals asserting that if we color edges of the complete graph, we can find a large highly connected monochromatic subgraph. In particular, several questions of Bergfalk, Hrušák, and Shelah (2021, Acta Mathematica Hungarica 163, 309–322) are answered by showing that assuming the consistency of suitable large cardinals, the following are relatively consistent with ZFC:

• • $\kappa \to _{hc} (\kappa )^2_\omega $ for every regular cardinal $\kappa \geq \aleph _2$,

• • $\neg \mathsf {CH}+ \aleph _2 \to _{hc} (\aleph _1)^2_\omega $.

Building on a work of Lambie-Hanson (2023, Fundamenta Mathematicae. 260(2):181–197), we also show that

• • $\aleph _2 \to _{hc} [\aleph _2]^2_{\omega ,2}$ is consistent with $\neg \mathsf {CH}$.

To prove these results, we use the existence of ideals with strong combinatorial properties after collapsing suitable large cardinals.

We write $\mathcal {S}_n(A)$ for the set of permutations of a set A with n non-fixed points and $\mathrm {{seq}}^{1-1}_n(A)$ for the set of one-to-one sequences of elements of A with length n where n is a natural number greater than $1$. With the Axiom of Choice, $|\mathcal {S}_n(A)|$ and $|\mathrm {{seq}}^{1-1}_n(A)|$ are equal for all infinite sets A. Among our results, we show, in ZF, that $|\mathcal {S}_n(A)|\leq |\mathrm {{seq}}^{1-1}_n(A)|$ for any infinite set A if ${\mathrm {AC}}_{\leq n}$ is assumed and this assumption cannot be removed. In the other direction, we show that $|\mathrm {{seq}}^{1-1}_n(A)|\leq |\mathcal {S}_{n+1}(A)|$ for any infinite set A and the subscript $n+1$ cannot be reduced to n. Moreover, we also show that “$|\mathcal {S}_n(A)|\leq |\mathcal {S}_{n+1}(A)|$ for any infinite set A” is not provable in ZF.

It is proved in $\mathsf {ZF}$ (without the axiom of choice) that, for all infinite sets M, there are no surjections from $\omega \times M$ onto $\operatorname {\mathrm {\mathscr {P}}}(M)$.

In this paper, we give an axiomatization of the ordinal number system, in the style of Dedekind’s axiomatization of the natural number system. The latter is based on a structure $(N,0,s)$ consisting of a set N, a distinguished element $0\in N$ and a function $s\colon N\to N$ . The structure in our axiomatization is a triple $(O,L,s)$ , where O is a class, L is a class function defined on all s-closed ‘subsets’ of O, and s is a class function $s\colon O\to O$ . In fact, we develop the theory relative to a Grothendieck-style universe (minus the power set axiom), as a way of bringing the natural and the ordinal cases under one framework. We also establish a universal property for the ordinal number system, analogous to the well-known universal property for the natural number system.

In this paper, we use algebra-valued models to study cardinal numbers in a class of non-classical set theories. The algebra-valued models of these non-classical set theories validate the Axiom of Choice, if the ground model validates it. Though the models are non-classical, the foundations of cardinal numbers in these models are similar to those in classical set theory. For example, we show that mathematical induction, Cantor’s theorem, and the Schröder–Bernstein theorem hold in these models. We also study a few basic properties of cardinal arithmetic. In addition, the generalized continuum hypothesis is proved to be independent of these non-classical set theories.

We show that assuming modest large cardinals, there is a definable class of ordinals, closed and unbounded beneath every uncountable cardinal, so that for any closed and unbounded subclasses $P, Q, {\langle L[P],\in ,P \rangle }$ and ${\langle L[Q],\in ,Q \rangle }$ possess the same reals, satisfy the Generalised Continuum Hypothesis, and moreover are elementarily equivalent. Examples of such P are Card, the class of uncountable cardinals, I the uniform indiscernibles, or for any n the class $C^{n}{=_{{\operatorname {df}}}}\{ \lambda \, | \, V_{\lambda } \prec _{{\Sigma }_{n}}V\}$ ; moreover the theory of such models is invariant under ZFC-preserving extensions. They also all have a rich structure satisfying many of the usual combinatorial principles and a definable wellorder of the reals. The inner model constructed using definability in the language augmented by the Härtig quantifier is thus also characterized.

Extending Aanderaa’s classical result that $\pi ^{1}_{1} < \sigma ^{1}_{1}$ , we determine the order between any two patterns of iterated $\Sigma ^{1}_{1}$ - and $\Pi ^{1}_{1}$ -reflection on ordinals. We show that this order of linear reflection is a prewellordering of length $\omega ^{\omega }$ . This requires considering the relationship between linear and some non-linear reflection patterns, such as $\sigma \wedge \pi $ , the pattern of simultaneous $\Sigma ^{1}_{1}$ - and $\Pi ^{1}_{1}$ -reflection. The proofs involve linking the lengths of $\alpha $ -recursive wellorderings to various forms of stability and reflection properties satisfied by ordinals $\alpha $ within standard and non-standard models of set theory.

We consider a seemingly weaker form of $\Delta ^{1}_{1}$ Turing determinacy.

Let $2 \leqslant \rho < \omega _{1}^{\mathsf {CK}}$ , $\textrm{Weak-Turing-Det}_{\rho }(\Delta ^{1}_{1})$ is the statement:

Every $\Delta ^{1}_{1}$ set of reals cofinal in the Turing degrees contains two Turing distinct, $\Delta ^{0}_{\rho }$ -equivalent reals.

We show in $\mathsf {ZF}^-$ :

$\textrm{Weak-Turing-Det}_{\rho }(\Delta ^{1}_{1})$ implies: for every $\nu < \omega _{1}^{\mathsf {CK}}$ there is a transitive model ${M \models \mathsf {ZF}^{-} + \textrm{``}\aleph _{\nu } \textrm{ exists''.}}$

As a corollary:

• If every cofinal $\Delta ^{1}_{1}$ set of Turing degrees contains both a degree and its jump, then for every $\nu < \omega_1^{\mathsf{CK}}$ , there is atransitive model: $M \models \mathsf{ZF}^{-} + \textrm{``}\aleph_\nu \textrm{ exists''.}$

• • With a simple proof, this improves upon a well-known result of Harvey Friedman on the strength of Borel determinacy (though not assessed level-by-level).

• • Invoking Tony Martin’s proof of Borel determinacy, $\textrm{Weak-Turing-Det}_{\rho }(\Delta ^{1}_{1})$ implies $\Delta ^{1}_{1}$ determinacy.

• • We show further that, assuming $\Delta ^{1}_{1}$ Turing determinacy, or Borel Turing determinacy, as needed:

  • – Every cofinal $\Sigma ^{1}_{1}$ set of Turing degrees contains a “hyp-Turing cone”: ${\{x \in \mathcal {D} \mid d_{0} \leqslant _{T} x \leqslant _{h} d_{0} \}}$ .

  • – For a sequence $(A_{k})_{k < \omega }$ of analytic sets of Turing degrees, cofinal in $\mathcal {D}$ , $\bigcap _{k} A_{k}$ is cofinal in $\mathcal {D}$ .

Skolem (1956) studied the germs at infinity of the smallest class of real valued functions on the positive real line containing the constant $1$ , the identity function ${\mathbf {x}}$ , and such that whenever f and g are in the set, $f+g,fg$ and $f^g$ are in the set. This set of germs is well ordered and Skolem conjectured that its order type is epsilon-zero. Van den Dries and Levitz (1984) computed the order type of the fragment below $2^{2^{\mathbf {x}}}$ . Here we prove that the set of asymptotic classes within any Archimedean class of Skolem functions has order type $\omega $ . As a consequence we obtain, for each positive integer n, an upper bound for the fragment below $2^{n^{\mathbf {x}}}$ . We deduce an epsilon-zero upper bound for the fragment below $2^{{\mathbf {x}}^{\mathbf {x}}}$ , improving the previous epsilon-omega bound by Levitz (1978). A novel feature of our approach is the use of Conway’s surreal number for asymptotic calculations.

This paper investigates the principles that one must add to Boolean algebra to capture reasoning not only about intersection, union, and complementation of sets, but also about the relative size of sets. We completely axiomatize such reasoning under the Cantorian definition of relative size in terms of injections.

In a recent paper by M. Rathjen and the present author it has been shown that the statement “every normal function has a derivative” is equivalent to $\Pi ^1_1$ -bar induction. The equivalence was proved over $\mathbf {ACA_0}$ , for a suitable representation of normal functions in terms of dilators. In the present paper, we show that the statement “every normal function has at least one fixed point” is equivalent to $\Pi ^1_1$ -induction along the natural numbers.

We investigate the effects of various forcings on several forms of the Halpern– Läuchli theorem. For inaccessible κ, we show they are preserved by forcings of size less than κ. Combining this with work of Zhang in [17] yields that the polarized partition relations associated with finite products of the κ-rationals are preserved by all forcings of size less than κ over models satisfying the Halpern– Läuchli theorem at κ. We also show that the Halpern–Läuchli theorem is preserved by <κ-closed forcings assuming κ is measurable, following some observed reflection properties.

We prove that WRP and saturation of the ideal NSω1 together imply $\left\{ {a \in [\lambda ]^{\omega _1 } :{\text{cof}}\left( {{\text{sup}}\left( a \right)} \right) = \omega _1 } \right\}$, for every cardinal λ with cof(λ) ≥ω2 .

We obtain a formula for the reflexivity index of a finite lattice of sets and of various types of infinite lattices of sets.

We extend the scope of B. Shapirovskii's results on the order of $\pi $-bases in compact spaces and answer some questions of V. Tkachuk.

Given a left Noetherian ring $R$, we give a necessary and sufficient condition in order that a complex of $R$-modules be DG-injective. Using this result we prove that if $(K_i)_{i\in I}$ is a family of DG-injective complexes of left $R$-modules and $K$ is the $\aleph_1$-product of $(K_i)_{i\in I}$ (i.e. $K\subset\prod_{i\in I}K_i$ is such that, for each $n$, $K^n\subset\prod_{i\in I}K_i^n$ consists of all $(x_i)_{i\in I}$ such that $\{i\mid x_i\neq0\}$ is at most countable), then $K$ is DG-injective.

We answer three problems by J. D. Monk on cardinal invariants of Boolean algebras. Two of these are whether taking the algebraic density πA resp. the topological density cL4 of a Boolean algebra A commutes with formation of ultraproducts; the third one compares the number of endomorphisms and of ideals of a Boolean algebra.