The following is part of a series of papers on theories for (iterated) admissible sets (cf. [10], [11], [12], [14], [15]). Although these theories are weak subsystems of Zermelo-Fraenkel set theory, they allow one to formalize and prove a fair amount of definability theory and generalized recursion theory. Using this machinery it is in general not very hard to establish the connections between theories for admissible sets and (for example) systems of second order arithmetic. A proof-theoretic analysis of theories for admissible sets therefore provides quite a uniform and powerful framework for the proof-theoretic treatment of many systems of set theory, second order arithmetic and constructive mathematics (see [12] and [15]). The strongest result in this direction so far is the pair of proof-theoretic equivalences

where T0 is Feferman's system for explicit mathematics of [5] and [6], (-CA) + (BI) is the usual system of second order arithmetic with the axiom of -comprehension and bar induction and KPi is Kripke-Platek set theory with ∈-induction

for arbitrary formulas and the additional axiom

.

The least standard model of KPi is L(i0) where i0 is the first recursively inaccessible ordinal.

In this paper we are mainly interested in the theory KPi0 which results from KPi by severely restricting the principles of induction. Basically, complete induction on the natural numbers

is allowed only for ∆0-formulas, and (IND∈) is omitted completely.

There has been much work in developing the interconnections between model theory and algebra. Here we look at a particular example, the divisible ordered abelian groups, and show how the indiscernibles are related to the algebraic structure. Now a divisible ordered abelian group is a model of Th (Q, +, 0, <) and so is linearly ordered by <. Thus the theory is unstable and has a large number of models. It is therefore unrealistic to expect that a simple condition will completely determine a model. Instead we would just like to obtain nice algebraic invariants.

Definition. A subset C of a divisible ordered abelian group is a set of (order) indiscernibles iff for every sequence of integers n1,…,nk and for every c1 < … < ck and d1 < … < dk in C

Note that this simplified form of indiscernibility is an immediate consequence of quantifier elimination for the theory. The above definition could be formulated in the language of +, 0, < but we have used subtraction as a matter of convenience. Similarly we may also use rational coefficients. Also note that a set of order indiscernibles is usually defined with respect to some external order. But in this case there are only two possibilities: the ordering inherited from or its reverse. So we will always assume that a set of order indiscernibles has the ordering inherited from . We may sometimes refer to a set of indiscernibles as a sequence of indiscernibles if we want to explicitly mention the ordering associated with the set.

As in Rogers [3], we treat the partial degrees as notational variants of the enumeration degrees (that is, the partial degree of a function is identified with the enumeration degree of its graph). We showed in [1] that there are no minimal partial degrees. The purpose of this paper is to show that the partial degrees below 0′ (that is, the partial degrees of the Σ2 partial functions) are dense. From this we see that the Σ2 sets play an analagous role within the enumeration degrees to that played by the recursively enumerable sets within the Turing degrees. The techniques, of course, are very different to those required to prove the Sacks Density Theorem (see [4, p. 20]) for the recursively enumerable Turing degrees.

Notation and terminology are similar to those of [1]. In particular, We, Dx, 〈m, n〉, ψe are, respectively, notations for the e th r.e. set in a given standard listing of the r.e. sets, the finite set whose canonical index is x, the recursive code for (m, n) and the e th enumeration operator (derived from We). Recursive approximations etc. are also defined as in [1].

Theorem 1. If B and C are Σ2sets of numbers, and B ≰e C, then there is an e-operator Θ with

Proof. We enumerate an e-operator Θ so as to satisfy the list of conditions:

Let {Bs ∣ s ≥ 0}, {Cs ∣ s ≥ 0} be recursive sequences of approximations to B, C respectively, for which, for each х, х ∈ B ⇔ (∃s*)(∀s ≥ s*)(х ∈ Bs) and х ∈ C ⇔ (∃s*)(∀s ≥ s*)(х ∈ Cs).

We prove several independence results relevant to an old question in the folklore of set theory. These results complement those in [Sh, Chapter XIII, §4]. The question is the following. Suppose V ⊨ “ZFC + CH” and r is a real not in V. Must V[r] ⊨ CH? To avoid trivialities assume = .

We answer this question negatively. Specifically we find pairs of models (W, V) such that W ⊨ ZFC + CH, V = W[r], r a real, = and V ⊨ ¬CH. Actually we find a spectrum of such pairs using ZFC up to “ZFC + there exist measurable cardinals”. Basically the nicer the pair is as a solution, the more we need to assume in order to construct it.

The relevant results in [Sh, Chapter XIII] state that if a pair (of inner models) (W, V) satisfies (1) and (2) then there is an inaccessible cardinal in L; if in addition V ⊨ 2ℵ0 > ℵ2 then 0# exists; and finally if (W, V) satisfies (1), (2) and (3) with V ⊨ 2ℵ0 > ℵω, then there is an inner model with a measurable cardinal.

Definition 1. For a pair (W, V) we shall consider the following conditions:

(1) V = W[r], r a real, = , W ⊨ ZFC + CH but CH fails in V.

(2) W ⊨ GCH.

(3) W and V have the same cardinals.

It will be shown that in the lattice of recursively enumerable sets one can define elementarily with parameters a structure isomorphic to (), i.e. isomorphic to the lattice of sets together with a unary predicate selecting out exactly the sets.

We prove the following Main Theorem: ZF + AD + V ⇒ L(R) ⇒ DC. As a corollary we have that Con(ZF + AD) ⇒ Con(ZF + AD + DC). Combined with the result of Woodin that Con(ZF + AD) ⇒ Con(ZF + AD + ¬ ACω) it follows that DC (as well as ACω) is independent relative to ZF + AD. It is finally shown (jointly with H. Woodin) that ZF + AD + ¬DCR, where DCR is DC restricted to reals, implies the consistency of ZF + AD + DC, in fact implies R# (i.e. the sharp of L(R)) exists.

This paper deals with locales and their spaces of points in intuitionistic analysis or, if you like, in (Grothendieck) toposes. One of the important aspects of the problem whether a certain locale has enough points is that it is directly related to the (constructive) completeness of a geometric theory. A useful exposition of this relationship may be found in [1], and we will assume that the reader is familiar with the general framework described in that paper.

We will consider four formal spaces, or locales, namely formal Cantor space C, formal Baire space B, the formal real line R, and the formal function space RR being the exponential in the category of locales (cf. [3]). The corresponding spaces of points will be denoted by pt(C), pt(B), pt(R) and pt(RR). Classically, these locales all have enough points, of course, but constructively or in sheaves this may fail in each case. Let us recall some facts from [1]: the assertion that C has enough points is equivalent to the compactness of the space of points pt(C), and is traditionally known in intuitionistic analysis as the Fan Theorem (FT). Similarly, the assertion that B has enough points is equivalent to the principle of (monotone) Bar Induction (BI). The locale R has enough points iff its space of points pt(R) is locally compact, i.e. the unit interval pt[0, 1] ⊂ pt(R) is compact, which is of course known as the Heine-Borel Theorem (HB). The statement that RR has enough points, i.e. that there are “enough” continuous functions from R to itself, does not have a well-established name. We will refer to it (not very imaginatively, I admit) as the principle (EF) of Enough Functions.

We define a hierarchy on the regular types of an ω-stable nonmultidimensional theory, using generalised notions of algebraic and strongly minimal formulae. As an application we show that any resplendent model of an ω-stable finite-dimensional theory is saturated.

In classical prepositional calculus for each proposition A(p) the following holds: ⊢A(p) ↔ A3(p). In this paper we consider what remains of this in the intuitionistic case. It turns out that for each proposition A(p) the following holds: there is an n ∈ N such that

As a byproduct of the proof we give some theorems which may be useful elsewhere in propositional calculus.

In this paper, we establish an extended joint consistency theorem for an infinite family of free modal logics with equality. The extended joint consistency theorem incorporates the Craig and Lyndon interpolation lemmas and the Robinson joint consistency theorem. In part, the theorem states that two theories which are jointly unsatisfiable are separated by a sentence in the vocabulary common to both theories.

Our family of free modal logics includes the free versions of I, M, and S4 studied by Leblanc [5, Chapters 8 and 9], supplemented with equality as in [3]. In the relational semantics for these logics, there is no restriction on the accessibility relation in I, while in M(S4) the restriction is reflexivity (refiexivity and transitivity). We say that a restriction on the accessibility relation countenances backward-looping if it implies a sentence of the form ∀x1 …xn(x1Rx2 &…&xn ⊃ xkRxj) (1 ≤ j < k ≤ n ≥ 2), where the xi (1 ≤ i ≤ n) are distinct individual variables. Just as reflexivity and transitivity do not countenance backward-looping, neither do any of the restrictions in our family of free modal logics. (The above terminology is derived from the effect of such restrictions on Kripke tableaux constructions.) The Barcan formula, its converse, the Fitch formula, and the formula T ≠ T′ ⊃ □T ≠ T′ do not hold in our logics.

A subset X of a structure S is called free in S if ∀x ∈ Xx ∉ S[X − {x}]; here, S[Y] is the substructure of S generated from Y by the functions of S. For κ, λ, μ cardinals, let Frμ(κ, λ) be the assertion:

We show that Frω(ωω, ω), the free-subset property for ωω, is equiconsistent with the existence of a measurable cardinal (2.2,4.4). This answers a question of Devlin [De].

In the first section of this paper we prove some combinatorial facts about Frμ(κ, λ); in particular the first cardinal κ such that Frω(κ, ω) is weakly inaccessible or of cofinality ω (1.2). The second section shows that, under Frω(ωω, ω), ωω is measurable in an inner model. For the convenience of readers not acquainted with the core model κ, we first deduce the existence of 0# (2.1) using the inner model L. Then we adapt the proof to the core model and obtain that ωω is measurable in an inner model. For the reverse direction, we essentially apply a construction of Shelah [Sh] who forced Frω(ωω, ω) over a ground model which contains an ω-sequence of measurable cardinals. We show in §4 that indeed a coherent sequence of Ramsey cardinals suffices. In §3 we obtain such a sequence as an endsegment of a Prikry sequence.

The most common way of proving decidability in propositional modal logic is to shew that the system in question has the finite model property. This is not however the only way. Gabbay in [4] proves the decidability of many modal systems using Rabin's result in [8] on the decidability of the second-order theory of successor functions. In particular [4, pp. 258-265] he is able to prove the decidability of a system which lacks the finite model property. Gabbay's system is however complete, in the sense of being characterized by a class of frames, and the question arises whether there is a decidable modal logic which is not complete. Since no incomplete modal logic has the finite model property [9, p. 33], any proof of decidability must employ some such method as Gabbay's. In this paper I use the Gabbay/Rabin technique to prove the decidability of a finitely axiomatized normal modal propositional logic which is not characterized by any class of frames. I am grateful to the referee for suggesting improvements in substance and presentation.

The terminology I am using is standard in modal logic. By a frame is understood a pair 〈W, R〉 in which W is a class (of “possible worlds”) and R ⊆ W2. To avoid confusion in what follows, a frame will henceforth be referred to as a Kripke frame. By contrast, a general frame is a pair 〈, Π〉 in which is a Kripke frame and Π is a collection of subsets of W closed under the Boolean operations and satisfying the condition that if A is in Π then so is R−1 “A. A model on a frame (of either kind) is obtained by adding a function V which assigns sets of worlds to propositional variables. In the case of a general frame we require that V(p) ∈ Π.

By the word problem for some class of algebraic structures we mean the problem of determining, given a finite set E of equations between words (i.e. terms) and an additional equation x = y, whether x = y must hold in all structures satisfying each member of E. In 1947 Post [P] showed the word problem for semigroups to be undecidable. This result was strengthened in 1950 by Turing, who showed the word problem to be undecidable for cancellation semigroups,i.e. semigroups satisfying the cancellation property

Novikov [N] eventually showed the word problem for groups to be undecidable.

(Many flaws in Turing's proof were corrected by Boone [B]. Even after his corrections, at least one problem remains; the sentence on line 16 of p. 502 of [T] does not follow if one relation is principal and the other is a commutation relation. A corrected and somewhat simplified version of Turing's proof can be built on the construction given here.)

In 1966 Gurevich [G] showed the word problem to be undecidable for finite semigroups. However, this result on finite structures has not been extended to cancellation semigroups or groups; indeed it is easy to see that a finite cancellation semigroup is a group, so both questions are the same. We do not here settle the word problem for finite groups, but we do show that the word problem is undecidable for finite semigroups with zero (that is, having an element 0 such that x0 = 0x = 0 for all x) satisfying an approximation to the cancellation property (1).

We examine the multiplicity of complementation amongst subspaces of V∞. A subspace V is a complement of a subspace W if V ∩ W = {0} and (V ∪ W)* = V∞. A subspace is called fully co-r.e. if it is generated by a co-r.e. subset of a recursive basis of V∞. We observe that every r.e. subspace has a fully co-r.e. complement.

Theorem. If S is any fully co-r.e. subspace then S has a decidable complement.

We give an analysis of other types of complements S may have. For example, if S is fully co-r.e. and nonrecursive, then S has a (nonrecursive) r.e. nowhere simple complement.

We impose the condition of immunity upon our subspaces.

Theorem. Suppose V is fully co-r.e. Then V is immune iff there exist M1, M2 ∈ L(V∞), with M1supermaximal and M2k-thin, such that M1, ⊕ V = M2 ⊕ V = V∞.

Corollary. Suppose V is any r.e. subspace with a fully co-r.e. immune complement W(e.g., V is maximal or V is h-immune). Then there exist an r.e. supermaximal subspace M and a decidable subspace D such that V ⊕ W = M ⊕ W = D ⊕ W = V∞.

We indicate how one may obtain many further results of this type. Finally we examine a generalization of the concepts of immunity and soundness. A subspace V of V∞ is nowhere sound if (i) for all Q ∈ L(V∞) if Q ⊃ V then Q = V∞, (ii) V is immune and (iii) every complement of V is immune. We analyse the existence (and ramifications of the existence) of nowhere sound spaces.

In this paper we introduce a new hierarchy of sets and operators which we call the REA hierarchy for “recursively enumerable in and above”. The hierarchy is generated by composing (possibly) transfinite sequences of the pseudo-jump operators considered in Jockusch and Shore [1983]. We there studied pseudo-jump operators defined by analogy with the Turing jump as ones taking a set A to A ⊕ for some index e. We would now call these 1-REA operators and will extend them to α-REA operators for recursive ordinals α in analogy with the iterated Turing jump operators (A → A(α) for α < and Kleene's hyperarithmetic hierarchy. The REA sets will then, of course, be the results of applying these operators to the empty set. They will extend and generalize Kleene's H sets but will still be contained in the class of set singletons thus providing us with a new richer subclass of the set singletons which, as we shall see, is related to the work of Harrington [1975] and [1976] on the problems of Friedman [1975] about the arithmetic degrees of such singletons. Their degrees also give a natural class extending the class H of Jockusch and McLaughlin [1969] by closing it off under transfinite iterations as well as the inclusion of [d, d′] for each degree d in the class. The reason for the class being closed under this last operation is that the REA operators include all operators and so give a new hierarchy for them as well as the sets. This hierarchy also turns out to be related to the difference hierarchy of Ershov [1968], [1968a] and [1970]: every α-r.e. set is α-REA but each level of the REA hierarchy after the first extends all the way through the difference hierarchy although never entirely encompassing even the next level of the difference hierarchy.

A countable group is ℵ0-categorical if it is characterized up to isomorphism among all countable groups by its first order theory. We investigate countable ℵ0-categorical groups which are characteristically simple and have proper subgroups of finite index. We determine such groups up to isomorphism (Corollary 4 to Proposition 3), and we show that their theories are finitely axiomatizable (Proposition 2).

Let K be a finite nonabelian simple group. In §3 we construct from K two denumberable groups, K* and K#, and use them to classify countable direct limits of finite Cartesian powers of K (Proposition 1). The results of §3 are used in §4 to show that K* and K# are ℵ0-categorical and to find axioms for their theories. By Corollary 2 to Proposition 2, K* and K# are counterexamples to the conjecture of U. Feigner [3, p. 309] that ℵ0-categorical groups are FC-solvable. Also in §4 the results mentioned in the first paragraph are proved. §2 is devoted to preliminary lemmas about finite Cartesian powers of K.

A characteristic subgroup of a group G is a subgroup which is mapped onto itself by all automorphisms of G. G is characteristically simple if its only characteristic subgroups are itself and the identity subgroup. Each characteristic subgroup of a group is a union of orbits of the automorphism group of the group. If G is ℵ0-categorical, then as the automorphism group of G has only a finite number of orbits, G has only a finite number of characteristic subgroups.

Let λ be an ordinal less than or equal to an infinite cardinal κ. For S ⊂ κ, [S]λ denotes the collection of all order type λ subsets of S. A set X ⊂ [κ]λ will be called Ramsey iff there exists p ∈ [κ]κ such that either [p]λ ⊂ X or [p]λ ∩ X = ∅. The set p is called homogeneous for X.

The infinite Ramsey theorem implies that all subsets of [ω]n are Ramsey for n < ω. Using the axiom of choice, one can define a non-Ramsey subset of [ω]ω. In [GP], Galvin and Prikry showed that all Borel subsets of [ω]ω are Ramsey, where one topologizes [ω]ω as a subspace of Baire space. Silver [S] proved that analytic sets are Ramsey, and observed that this is best possible in ZFC.

When κ > ω, the assertion that all subsets of [κ]n are Ramsey is a large cardinal hypothesis equivalent to κ being weakly compact (and strongly inaccessible). Again, is not possible in ZFC to have all subsets of [κ]ω Ramsey. The analogy to the Galvin-Prikry theorem mentioned above was established by Kleinberg, extending work by Kleinberg and Shore in [KS]. The set [κ]ω is given a topology as a subspace of κω, which has the usual product topology, κ taken as discrete. It was shown that all open subsets of [κ]ω are Ramsey iff κ is a Ramsey cardinal (that is, κ → (κ)<ω).

In this note we examine the spaces [κ]λ for κ ≥ λ ≥ ω. We show that κ Ramsey implies all open subsets of [κ]λ are Ramsey for λ < κ, and that if κ is measurable, then all open subsets of [κ]κ are Ramsey. Let us remark here that we can with the same methods prove these results with “κ-Borel” in the place of “open”, where the κ-Borel sets are the smallest collection containing the opens and closed under complementation and intersections of length less than κ. Also, although here we consider just subsets of [κ]λ, it is no more difficult to show that partitions of [κ]λ into less than κ many κ-Borel sets have, under the appropriate hypothesis, size κ homogeneous sets.

The Gödel class with identity (GCI) is the class of closed, prenex quantificational formulas whose prefixes have the form ∀∀∃ … ∃ and whose matrices contain arbitrary predicate letters and the identity sign “ = ”, but do not contain function signs or individual constants. The ∀∀∃ … ∃ class without identity was shown solvable over fifty years ago ([4], [12], [17]); slightly later, that class was shown to possess the stronger property of finite controllability ([5], [18]). (A class of formulas is solvable iff it is decidable for satisfiability; it is finitely controllable iff every satisfiable formula in it has a finite model.) At the end of [5], Gödel claims that the finite controllability of the GCI can be shown “by the same method” as he employed to show this for the class without identity. This claim has been questioned for nearly twenty years; in §1 below we give a brief history of investigations into it. The major result of this paper shows Gödel to have been mistaken: the GCI is unsolvable. §2 contains the basic construction, which yields a satisfiable formula in the GCI that lacks finite models. This formula may easily be exploited to encode undecidable problems into the GCI.

Questions about the constructive or effective character of particular arguments arise in several areas of classical mathematics, such as in the theory of recursive functions and in numerical analysis. Some philosophers have advocated Lewis's S4 as the proper logic in which to formalize such epistemic notions. (The fundamental work on this is Hintikka [4].) Recently there have been studies of mathematical theories formalized with S4 as the underlying logic so that these epistemic notions can be expressed. (See Shapiro [7], Myhill [5], and Goodman [2]. The motivation for this work is discussed in Goodman [3].) The present paper is a contribution to the study of the simplest of these theories, namely first-order arithmetic as formalized in S4. Following Shapiro, we call this theory epistemic arithmetic (EA). More specifically, we show that EA is a conservative extension of Hey ting's arithmetic HA (ordinary first-order intuitionistic arithmetic). The question of whether EA is conservative over HA was raised but left open in Shapiro [7].

The idea of our proof is as follows. We interpret EA in an infinitary propositional S4, pretty much as Tait, for example, interprets classical arithmetic in his infinitary classical propositional calculus in [8]. We then prove a cut-elimination theorem for this infinitary propositional S4. A suitable version of the cut-elimination theorem can be formalized in HA. For cut-free infinitary proofs, there is a reflection principle provable in HA. That is, we can prove in HA that if there is a cut-free proof of the interpretation of a sentence ϕ then ϕ is true. Combining these results shows that if the interpretation of ϕ is provable in EA, then ϕ is provable in HA.

From the moment choice sequences appear in Brouwer's writings, they do so as elements of a spread. This led Kreisel to take the so-called axiom of spreaddata as the basic axiom in a formal theory of choice sequences (Kreisel [1965, pp. 133–136]). This axiom expresses the idea that to be given a choice sequence means to be given a spread to which the choice sequence belongs. Subsequently, however, it was discovered that there is a formal clash between this axiom and closure of the domain of choice sequences under arbitrary (lawlike) continuous operations (Troelstra [1968]). For this reason, the formal system CS was introduced (Kreisel and Troelstra [1970]), in which spreaddata is replaced by analytic data. In this system CS, the domain of choice sequences is closed under all continuous operations, and therefore it provides a workable basis for intuitionistic analysis. But the problem whether the axiom of spreaddata is compatible with closure of the domain of choice sequences under the continuous operations from a restricted class, which is still rich enough to validate the typical axioms of continuous choice, remained open. It is precisely this problem that we aim to discuss in this paper.

Recall that a spread is a (lawlike, inhabited) decidable subtree S of the tree N< N of all finite sequences, having all branches infinite: