We prove (in ZFC Set Theory) that all infinite games whose winning sets are of the following forms are determined:

(1) (A − S) ∪ B, where A is , , and the game whose winning set is B is “strongly determined” (meaning that all of its subgames are determined).

(2) A Boolean combination of sets and sets smaller than the continuum.

This also enables us to show that strong determinateness is not preserved under complementation, improving a result of Morton Davis which required the continuum hypothesis to prove this fact.

Various open questions related to the above results are discussed. Our main conjecture is that (2) above remains true when is replaced by “Borel”.

The theory of cylindric algebras (CA's) is the algebraic theory of first order logics. Several ideas about logic are easier to formulate in the frame of CA-theory. Such are e.g. some concepts of abstract model theory (cf. [1] and [10]–[12]) as well as ideas about relationships between several axiomatic theories of different similarity types (cf. [4] and [10]). In contrast with the relationship between Boolean algebras and classical propositional logic, CA's correspond not only to classical first order logic but also to several other ones. Hence CA-theoretic results contain more information than their counterparts in first order logic. For more about this see [1], [3], [5], [9], [10] and [12].

Here we shall use the notation and concepts of the monographs Henkin-Monk-Tarski [7] and [8]. ω denotes the set of natural numbers. CAα denotes the class of all cylindric algebras of dimension α; by “a CAα” we shall understand an element of the class CAα. The class Dcα ⊆ CAα was defined in [7]. Note that Dcα = 0 for α ∈ ω. The classes Wsα, and Csα were defined in 1.1.1 of [8], p. 4. They are called the classes of all weak cylindric set algebras, regular cylindric set algebras and cylindric set algebras respectively. It is proved in [8] (I.7.13, I.1.9) that ⊆ CAα. (These inclusions are proper by 7.3.7, 1.4.3 and 1.5.3 of [8].)

It was proved in 2.3.22 and 2.3.23 of [7] that every simple, finitely generated Dcα is generated by a single element. This is the algebraic counterpart of a property of first order logics (cf. 2.3.23 of [7]). The question arose: for which simple CAα's does “finitely generated” imply “generated by a single element” (see p. 291 and Problem 2.3 in [7]). In terms of abstract model theory this amounts to asking the question: For which logics does the property described in 2.3.23 of [7] hold? This property is roughly the following. In any maximal theory any finite set of concepts is definable in terms of a single concept. The connection with CA-theory is that maximal theories correspond to simple CA's (the elements of which are the concepts of the original logic) and definability corresponds to generation.

An infinite-dimensional vector space V∞ over a recursive field F is called fully effective if V∞ is a recursive set identified with ω upon which the operations of vector addition and scalar multiplication are recursive functions, identity is a recursive relation, and V∞ has a dependence algorithm, that is a uniformly effective procedure which when applied to x, a1,…,an, ∈ V∞ determines whether or not x is an element of {a1,…,an}* (the subspace generated by {a1,…,an}). The study of V∞, and of its lattice of r.e. subspaces L(V∞), was introduced in Metakides and Nerode [15]. Since then both V∞ and L(V∞) (and many other effective algebraic systems) have been studied quite intensively. The reader is directed to [5] and [17] for a good bibliography in this area, and to [15] for any unexplained notation and terminology.

In [15] Metakides and Nerode observed that a study of L(V∞) may in some ways be modelled upon a study of L(ω), the lattice of r.e. sets. For example, they showed how an e-state construction could be modified to produce an r.e. maximal subspace, where M ∈ L(V∞) is maximal if dim(V∞/M) = ∞ and, for all W ∈ L(V∞), if W ⊃ M then either dim(W/M) < ∞ or dim(V∞/W) < ∞.

However, some of the most interesting features of L(V∞) are those which do not have analogues in L(ω). Our concern here, which is probably one of the most striking characteristics of L(V∞), falls into this category. We say M ∈ L(V∞) is supermaximal if dim(V∞/M) = ∞ and for all W ∈ L(V∞), if W ⊃ M then dim(W/M) < ∞ or W = V∞. These subspaces were discovered by Kalantari and Retzlaff [13].

My aim here is to investigate the role of global quantifiers—quantifiers ranging over the entire universe of sets—in the formalization of Zermelo-Fraenkel set theory. The use of such quantifiers in the formulas substituted into axiom schemata introduces, at least prima facie, a strong element of impredicativity into the thapry. The axiom schema of replacement provides an example of this. For each instance of that schema enlarges the very domain over which its own global quantifiers vary. The fundamental question at issue is this: How does the employment of these global quantifiers, and the choice of logical principles governing their use, affect the strengths of the axiom schemata in which they occur?

I shall attack this question by comparing three quite different formalizations of the intuitive principles which constitute the Zermelo-Fraenkel system. The first of these, local Zermelo-Fraenkel set theory (LZF), is formalized without using global quantifiers. The second, global Zermelo-Fraenkel set theory (GZF), is the extension of the local theory obtained by introducing global quantifiers subject to intuitionistic logical laws, and taking the axiom schema of strong collection (Schema XII, §2) as an additional assumption of the theory. The third system is the conventional formalization of Zermelo-Fraenkel as a classical, first order theory. The local theory, LZF, is already very strong, indeed strong enough to formalize any naturally occurring mathematical argument. I have argued (in [3]) that it is the natural formalization of naive set theory. My intention, therefore, is to use it as a standard against which to measure the strength of each of the other two systems.

A fundamental problem in the theory of abelian groups is to determine the structure of Ext(A, Z) for arbitrary abelian groups A. This problem was raised by L. Fuchs in 1958, and since then has been the center of considerable activity and progress.

We briefly summarize the present state of this problem. It is a well-known fact that

where tA denotes the torsion subgroup of A. Thus the structure problem for Ext(A, Z) breakdown to the two distinct cases, torsion and torsion free groups. For a torsion group T,

which is compact and reduced, and its structure is known explicitly [12].

For torsion free A, Ext(A, Z) is divisible; hence it has a unique representation

Thus Ext(A, Z) is characterized by countably many cardinal numbers, which we denote as follows: ν0(A) is the rank of the torsion free part of Ext(A, Z), and νp(A) are the ranks of the p-primary parts of Ext(A, Z), Extp(A, Z).

If A is free it is an elementary fact that Ext(A, Z) = 0. The second named author has shown [16] that in the presence of V = L the converse is also true. For countable torsion free, nonfree A, C. Jensen [13] has shown that νp(A) is either finite or and νp(A) ≤ ν0(A). Therefore, the case for uncountable, nonfree, torsion free groups A remains to be studied.

Let 〈, ≤ 〉 be the usual structure of the degrees of unsolvability and 〈, ≤ 〉 the structure of the T-degrees of partial functions defined in [7]. We prove that every countable distributive lattice with a least element can be isomorphically embedded as an initial segment of 〈, ≤ 〉: as a corollary, the first order theory of 〈, ≤ 〉 is recursively isomorphic to that of 〈, ≤ 〉. We also show that 〈, ≤ 〉 and 〈, ≤ 〉 are not elementarily equivalent.

This paper consists of two parts. In §1 we mention the first strongly compact cardinal. Magidor proved in [6] that it can be the first measurable and it can be also the first supercompact. In [2], Apter proved that Con(ZFC + there is a supercompact limit of supercompact cardinals) implies Con(ZFC + the first strongly compact cardinal κ is ϕ(κ)-supercompact + no α < κ is ϕ(α)-supercompact) for a formula ϕ which satisfies certain conditions.

We shall get almost the same conclusion as Apter's theorem assuming only one supercompact cardinal. Our notion of forcing is the same as in [2] and a trick makes it possible.

In §2 we study a kind of fine ultrafilter on Pκλ investigated by Menas in [7], where κ is a measurable limit of strongly compact cardinals. He showed that such an ultrafilter is not normal in some case (Theorems 2.21 and 2.22 in [7]). We shall show that it is not normal in any case (even if κ is supercompact). We also prove that it is weakly normal in some case.

We work in ZFC and much of our notation is standard. But we mention the following: the letters α,β,γ… denote ordinals, whereas κ,λ,μ,… are reserved for cardinals. R(α) is the collection of sets rank <α. φM denotes the realization of a formula φ to a class M. Except when it is necessary, we drop “M”. For example, M ⊩ “κ is φ(κ)-supercompact” means “κ is φM(κ)-supercompact in M”. If x is a set, |x| is its cardinality, Px is its power set, and . If also x ⊆ OR, denotes its order type in the natural ordering. The identity function with the domain appropriate to the context is denoted by id. For the notation concerning ultrapowers and elementary embeddings, see [11]. When we talk about forcing, “⊩” will mean “weakly forces” and “p < q” means “p is stronger than q”.

Any attempt to give “foundations”, for category theory or any domain in mathematics, could have two objectives, of course related.

(0.1) Noncontradiction: Namely, to provide a formal frame rich enough so that all the actual activity in the domain can be carried out within this frame, and consistent, or at least relatively consistent with a well-established and “safe” theory, e.g. Zermelo-Frankel (ZF).

(0.2) Adequacy, in the following, nontechnical sense:

(i) The basic notions must be simple enough to make transparent the syntactic structures involved.

(ii) The translation between the formal language and the usual language must be, or very quickly become, obvious. This implies in particular that the terminology and notations in the formal system should be identical, or very similar, to the current ones. Although this may seem minor, it is in fact very important.

(iii) “Foundations” can only be “foundations of a given domain at a given moment”, therefore the frame should be easily adaptable to extensions or generalizations of the domain, and, even better, in view of (i), it should suggest how to find meaningful generalizations.

(iv) Sometimes (ii) and (iii) can be incompatible because the current notations are not adapted to a more general situation. A compromise is then necessary. Usually when the tradition is very strong (ii) is predominant, but this causes some incoherence for the notations in the more general case (e.g. the notation f(x) for the value of a function f at x obliges one, in category theory, to denote the composition of arrows (f, g) → g∘f, and all attempts to change this notation have, so far, failed).

Some propositional logics (e.g. the classical system) can be characterized by a finite model, while others (e.g. Heyting's) which have the finite model property (FMP) can be characterized by an infinite set of finite models. Still others (e.g. certain extensions of Heyting's logic) which lack the FMP can only be characterized by a set of models, at least one of which is infinite. Yet all these logics admit finite models even though they may not be characterized by them. (For example, they all admit the 2-element Boolean algebra as a model in the sense that all their theorems are valid on that algebra when the propositional connectives are interpreted in the usual manner.) The object of the present paper is to give a (not too artificial) example of a propositional logic which is consistent and which admits only infinite models. It therefore lacks the FMP in a very strong sense. Such a propositional logic, I shall call hyperinfinite. The existence of hyperinfinite logics was already plausible from a result in abstract algebra which says that there are varieties of algebras of which the only finite element is the trivial algebra (see [3]).

I wish to thank Professor A. S. Troelstra, Amsterdam, for comments on an early version of this paper. The constructive criticism of two anonymous referees has also been useful.

The hyperinfinite propositional logic to be described is obtained from Positive Logic—the negative-free part of Heyting's logic—by adjoining certain axioms which govern the use of a unary modal connective.

Assuming the existence of a supercompact cardinal, we construct a model of ZFC + (There exists a nonsplitting stationary subset of ). Answering a question of Uri Abraham [A], [A-S], we prove that adding a real to the world always makes stationary

Contrary to what is stated in Lemma 7.1 of [8], it is shown that some Boolean algebras of finitary logic admit finitely additive probabilities that are not σ-additive. Consequences of Lemma 7.1 are reconsidered. The concept of a C-σ-additive probability ℬ (where ℬ and C are Boolean algebras, and ℬ ⊆ C) is introduced, and a generalization of Hahn's extension theorem is proved. This and other results are employed to show that every S̄(L)-σ-additive probability on s̄(L) can be extended (uniquely, under some conditions) to a σ-additive probability on S̄(L), where L belongs to a quite extensive family of first order languages, and S̄(L) and s̄(L) are, respectively, the Boolean algebras of sentences and quantifier free sentences of L.

We give here alternative definitions for the notions that S. Shelah has introduced in recent papers: the dimensional order property and the depth of a theory. We will also give a proof that the depth of a countable theory, when denned, is an ordinal recursive in T.

We study the effects of Vopěnka's principle on properties of model theoretic logics. We show that Vopěnka's principle is equivalent to the assumption that every finitely generated logic has a compact cardinal. We show also that it is equivalent to the assumption that every such logic has a global Hanf number.

In [6] Gödel observed that intuitionistic propositional logic can be interpreted in Lewis's modal logic (S4). The idea behind this interpretation is to regard the modal operator □ as expressing the epistemic notion of “informal provability”. With the work of Shapiro [12], Myhill [10], Goodman [7], [8], and Ščedrov [11] this simple idea has developed into a successful program of integrating classical and intuitionistic mathematics.

There is one question quite central to the above program that has remained open. Namely:

In the present paper we give an affirmative answer to this question.

The main ingredient in our proof is the construction of an interpretation of epistemic set theory into intuitionistic set theory which is inverse to the Gödel translation. This is accomplished in two steps. First we observe that Funayama's theorem is constructively provable and apply it to the power set of 1. This provides an embedding of the set of propositions into a complete topological Boolean algebra . Second, in a fashion completely analogous to the construction of Boolean-valued models of classical set theory, we define the -valued universe V(). V() gives a model of epistemic set theory and, since we use a constructive metatheory, this provides an interpretation of epistemic set theory into intuitionistic set theory.

It is proven that the following statement:

“there exists a club C ⊆ κ such that every α ∈ C is an inaccessible cardinal in L and, for every δ a limit point of C, C ∩ δ is almost contained in every club of δ of L”

is equiconsistent with a weakly compact cardinal if δ = ℵ1, and with a weakly compact cardinal of order 1 if δ = ℵ2.

In this paper, we prove a result that J. Y. Girard has conjectured for the last few years. That is,

This result is part of a study of ordinal notations. For the proof, we use some concepts introduced in Parts I, II and III of this article (see [GV1], [GV2] and [V]); in particular, in Part III (see [V]) we defined the category GAR of gardens (of type Ω), and we have shown that there is an isomorphism between GAR and the category DIL Ω of dilators which send Ω into Ω. To describe this isomorphism, we defined a functor SYN from GAR to DIL Ω and a functor DEC from DIL Ω to GAR which are inverse to each other. The functor SYN is constructed by induction on the height x of the garden Jx, iterating, at each step of cofinality Ω, the operation UN (unification of variables; see [G1, Chapter 3]). Here, we define the functor S⊿ from GAR to DIL Ω: this construction is very close to that of SYN, but we iterate, at each step of cofinality Ω, the operation ⊿ (identification of variables; see [G, Chapter 3]). Technically, the definition of S⊿ is easier and more natural than that of SYN (as ⊿ is easier than UN) but is not inversible (as ⊿); for example, if we consider the (regular) garden GεΩ+1 introduced in [GV2],S⊿(GεΩ+1) is exactly the dilator N = ⋃0<n<ωNn, with N1 = 2 + Id, and , but we are incapable of expressing SYN(GεΩ+1).

Before stating the results we would like to thank the referee for reorganizing the whole paper and changing its original logical terminology to an algebraical terminology.

Let Z* be a proper elementary extension of the integral domain Z. For any b ∈ Z*∖Z we define Z〈b〉 as the following subring of Z*:

It is easy to prove that the natural divisibility relation on Z〈b〉 is the restriction to Z〈b〉 of the divisibility on Z* (see Lemma 1.1 below). The much stronger statement that Z〈b〉 is algebraically closed in Z* does not hold for all b. Similarly, there are many b's such that Z〈b〉 is not a recursive ring, e.g. if the set of standard primes dividing b is not recursive then Z〈b〉 cannot be a recursive ring under any bijection with N.

Our main result is the following.

Theorem. There exists an element b ∈ Z*∖Z such that

(1) Z〈b〉 is algebraically closed in Z*,

(2) Z〈b〉 is a recursive ring, and

(3) each infinite element of Z〈b〉 is divisible by a standard integer > 1 (so the only “primes” of Z〈b〉 are the standard ones).

An easy consequence of (1) is that certain induction axioms are satisfied by Z〈b〉. This explains the title of the paper. In the last section of the paper, §3, we shall say more about this.

The starting point of this work was Saracino and Wood's description of the finitely generic abelian ordered groups [S-W].

We generalize the result of Saracino and Wood to a class ∑UH of subdirect products of substructures of elements of a class ∑, which has some relationships with the discriminator variety V(∑t) generated by ∑. More precisely, let ∑ be an elementary class of L-algebras with theory T. Burris and Werner have shown that if ∑ has a model companion then the existentially closed models in the discriminator variety V(∑t) form an elementary class which they have axiomatized. In general it is not the case that the existentially closed elements of ∑UH form an elementary class. For instance, take for ∑ the class ∑0 of linearly ordered abelian groups (see [G-P]).

We determine the finitely generic elements of ∑UH via the three following conditions on T:

(1) There is an open L-formula which says in any element of ∑UH that the complement of equalizers do not overlap.

(2) There is an existentially closed element of ∑UH which is an L-reduct of an element of V(∑t) and whose L-extensions respect the relationships between the complements of the equalizers.

(3) For any models A, B of T, there exists a model C of TUH such that A and B embed in C.

(Condition (3) is weaker then “T has the joint embedding property”. It is satisfied for example if every model of T has a one-element substructure. Condition (3) implies that ∑UH has the joint embedding property and therefore that the class of finitely generic elements of ∑UH is complete.)

Let M be a term of the type free λ-calculus and let be a set of occurrences of redexes in M. A reduction sequence from M which first contracts a member of and afterwards only residuals of is called a development (of M with respect to ). The finite developments theorem says that developments are always finite.

There are several proofs of this theorem in the literature. A plausible strategy is to define some kind of measure for pairs (M, ), which—if M′ results from M by contracting a redex occurrence in and ′ is the set of residuals of in M′— decreases in passing from (M, ) to (M′, ′). This procedure is followed as a matter of fact in the proofs in Hyland [4] and in Barendregt [1] (both are covered in Klop [5]). If, as in the latter proof, the natural numbers are used as measures, then the measure of (M, ) will actually denote an upper bound of the number of reduction steps in a development of M with respect to .

In the present proof we straightforwardly define for each pair (M, ) a natural number, which can easily be seen to indicate the exact number of reduction steps in a development of maximal length of M with respect to .

Predicate-functor logic, as founded by W. V. Quine ([1960], [1971], [1976], [1981]), is first-order predicate logic without individual variables. Instead, adverbs or predicate functors make explicit the permutations and replications of argument-places familiarly indicated by shifting variables about. For the history of this approach, see Quine [1971, 309ff.]. With the evaporation of variables, individual constants naturally assimilate to singleton predicates or adverbs, leaving no logical subjects whatever of type 0. The orphaned “predicates” may then be taken simply as terms in the sense of traditional logic: class and relational terms on model-theoretic semantics, schematic terms on Quine's denotational or truth-of semantics. Predicate-functor logic thus stands forth as the pre-eminent first-order term logic, as distinct from propositional-quantificational logic. By the same token, it might with some justification qualify as “first-order combinatory logic”, with allowance for some categorization of the sort eschewed in general combinatory logic, the ultimate term logic.

Over the years, Quine has put forward various choices of primitive predicate functors for first-order logic with or without the full theory of identity. Moreover, he has provided translations of quantificational into predicate-functor notation and vice versa ([1971, 312f.], [1981, 651]). Such a translation does not of itself establish semantic completeness, however, in the absence of a proof that it preserves deducibility.

An axiomatization of predicate-functor logic was first published by Kuhn [1980], using primitives rather like Quine's. As Kuhn noted, “The axioms and rules have been chosen to facilitate the completeness proof” [1980, 153]. While this expedient simplifies the proof, however, it limits the depth of analysis afforded by the axioms and rules. Mindful of this problem, Kuhn ([1981] and [1983]) boils his axiom system down considerably, correcting certain minor slips in the original paper.