Let Pr be Presburger's arithmetic, i.e., the complete theory of the structure (ω, +). Lipshitz and Nadel showed in [4] that a countable model of Pr is recursively saturated iff it can be expanded to a model of Peano arithmetic, PA. As a starting point for an introductory discussion, let us mention one more fact about countable recursively saturated models of Pr. If is such a model then the following is readily seen (as explained also in §1):

Any two countable recursively saturated elementary endextensions of are isomorphic.

If we drop “countable” from the assumption of this statement, we can still say that the two models are ∞ω-equivalent. Must they be isomorphic if both have cardinality ℵ1? Certainly not, since one of the models can be ω 1-like while the other is not. Once we realized this much, let us concentrate on ω 1-like structures. We prove in §3:

Theorem. Any countable recursively saturated model of Pr has isomorphism types of ω 1-like recursively saturated elementary endextensions. Only one of these is the isomorphism type of a structure that can be expanded to a model of PA.

The key technical result is proven in §2. It says that an as above has precisely two countable recursively saturated elementary endextensions which are nonisomorphic over .

One of the typical preservation theorems in a first order classical predicate logic with equality L is the following theorem due to J. Łoś [4] and A. Tarski [9] (also cf. [1, p. 139]).

Theorem A (Łoś-Tarski). For any sentences A and B in L, the following two conditions (i) and (ii) are equivalent.

(i) Every extension of any model of A is a model of B.

(ii) The two sentences A ⊃ C and C ⊃ B are provable in L for some existential sentence C in L.

In [2], S. Feferman obtained a similar preservation theorem for outer extensions. In the following, we assume that L has a fixed binary predicate symbol <. Then Σ-formulas are formulas in L which are constructed from atomic formulas and their negations by applying ∧ (conjunctions), ∨ (disjunctions), ∀x < y (bounded universal quantifications), and ∃ (existential quantifications). An extension of an L-structure is said to be an outer extension of if ⊨ a < b and b ϵ ∣∣ imply a ϵ ∣∣ for any elements a, b in ∣∣.

Theorem B (Feferman). For any sentences A and B in L, the following two conditions (i) and (ii) are equivalent.

(i) Every outer extension of any model of A is a model of B.

(ii) The two sentences A ⊃ C and C ⊃ B are provable in L for some Σ-sentence C in L.

We give an alternative proof of the stability of separably closed fields of fixed Éršov invariant to the one given in [W]. We show that in case the Éršov invariant is finite, the theory is in fact equational. We also characterize the independence relation in those theories.

It was J. E. Baumgartner who in [1] proved that when a weakly compact cardinal is Lévy-collapsed to ω2 the new ω2 inherits some of the large cardinal properties; e.g. if S is a stationary set of ω-limits in ω2 then for some α < ω2, S ∩ α is stationary in α. Later S. Shelah extended this to the following theorem: if a supercompact cardinal κ is Lévy-collapsed to ω2, then in the resulting model the following holds: if S ⊆ λ is a stationary set of ω-limits and cf(λ) ≥ ω2 then there is an α. < λ such that S ∩ α is stationary in α, i.e. stationary reflection holds for countable cofinality (see [1] and [3]). These theorems are important prototypes of small cardinal compactness theorems; many applications and generalizations can be found in the literature. One might think that these results are true for sets with an uncountable cofinality μ as well, i.e. when an appropriate large cardinal is collapsed to μ++. Though this is true for Baumgartner's theorem, there remains a problem with Shelah's result. The point is that the lemma stating that a stationary set of ω-limits remains stationary after forcing with an ω2-closed partial order may be false in the case of μ-limits in a cardinal of the form λ+ with cf(λ) < μ, as was shown in [8] by Shelah. The problem has recently been solved by Baumgartner, who observed that if a universal box-sequence on the class of those ordinals with cofinality ≤ μ exists, the lemma still holds, and a universal box-sequence of the above type can be added without destroying supercompact cardinals beyond μ.

This paper is concerned with recursive structures and the persistance of an effective notion of categoricity. The terminology and notational conventions are standard. We will devote the rest of this paragraph to a cursory review of some of the assumed conventions. If θ is a formula, then θk denotes θ if k is zero, and ¬θ if k is one. If A is a sequence with domain a subset of ω, then A∣ n denotes the sequence obtained by restricting the domain of A to n. For an effective first order language L, let {ci ∣i<ω} be distinct new constants, and let {θ i ∣i<ω} be an effective enumeration of all sentences in the language L ∪ {ci ∣j<ω}. An infinite L-structure is recursive iff it has universe ω and the set is recursive, where cn is interpreted by n. In general we say that a set of formulas is recursive if the set of its indices with respect to an enumeration such as above is recursive. The ∃-diagram of a structure is recursive if the structure is recursive and the set and θ i is an existential sentence} is also recursive. The definition of “the ∀∃-diagram of is recursive” is similar.

We show that relative to the consistency of a supercompact cardinal does not imply . The model-theoretic transfer property ⟨ℵ1, ℵ0⟩ → ⟨ℵω + 1, ℵω⟩ does not imply , and it is consistent to have an ultrafilter on ℵω + 1 which is λ-indecomposible for all ω < λ < ℵω.

Fred Richman conjectured that the following principle is not constructive:

(*) If A is a decidable subset of the set N of natural numbers and if, for every decidable subset B of N, either A ⊆ B or A ⊆ N − B, then, for some n ∈ N, A ⊆ {n}.

Richman's conjecture was motivated by the problem of uniqueness of divisible hulls of abelian groups in constructive algebra. Richman showed that a countable discrete abelian p-group G has a unique (up to isomorphism over G) divisible hull if the subgroup pG is decidable. He also showed that the converse implies.

We confirm the nonconstructive nature of by showing (in §1) that it is not provable in intuitionistic set theory, IZF. Thus, in the models we construct, there are countable discrete abelian p-groups G whose divisible hulls are unique but whose subgroups pG are not decidable.

Our models do not satisfy further conditions imposed by Richman, namely Church's Thesis and Markov's Principle, so the full conjecture remains an open problem. We do, however, show (in §2) how to embellish our first model so that the fan theorem (i.e., compactness of 2N) fails. (Church's Thesis implies the stronger statement that the negation of the fan theorem holds.)

Our models will be constructed by the method of sheaf semantics [1], [3]. That is, we shall construct Grothendieck topoi in whose internal logic fails.

The aim of this paper is to give, using the Kripke semantics for intuitionism, a representation of finitely generated free Heyting algebras. By means of the representation we determine in a constructive way some set of “special elements” of such algebras. Furthermore, we show that many algebraic properties which are satisfied by the free algebra on one generator are not satisfied by free algebras on more than one generator.

By a “partly numerical structure” (p.n.s.) we shall here mean a quadruple , where M is a set, ω = the natural numbers, ω ⊆ M, and are disjoint sets, is a set of relations (of various positive integral arities) on M, and is a set of functions (of various positive integral arities) with arguments and values in M. Thus, in calculated disharmony with common practice, we do not (except as noted below, in connection with naming the elements of ω) fix a similarity type as part of our notion of a “structure”. Suppose a finitary first-order language (with identity) has been specified, with constant symbols , n ∈ ω, and with exactly enough relation and function symbols of each arity to enable us to interpret in . We wish to consider the variation in the degree (relative to a fixed Gödel-numbering of ) of the complete -theory of as we vary the way in which elements of ∪ are assigned as interpretations to the relation and function symbols of . We shall in fact, therefore, be concerned exclusively with p.n.s.'s for which ∪ is countable. More: we assume to be such that we can effectively tell, uniformly in n > 0, exactly how many n-ary relations has and exactly how many n-ary functions has.

A nonnegative integer is called a number, a collection of numbers a set and a collection of sets a class. We write ε for the set of all numbers, o for the empty set, N(α) for the cardinality of α, ⊂ for inclusion and ⊂+ for proper inclusion. Let α, β 1,…, β k be subsets of some set υ. Then α′ stands for υ−α and β 1 … β k for β 1 ∩ … ∩ β k . For subsets α 1, …, α r of υ we write:

Note that α 0 = υ, hence s 0 = N(υ). If the set υ is finite, the classical inclusion-exclusion principle (abbreviated IEP) states

In this paper we generalize (a) and(b) to the case where α 1, …, α r are subsets of some countable but isolated set υ. Then the role of the cardinality N(α) of the set α is played by the RET (recursive equivalence type) Req α of α. These generalizations of (a) and (b) are proved in §3. Since they involve recursive distinctness, this notion is discussed in §2. In §4 we consider a natural extension of “the sum of the elements of a finite set σ” to the case where σ is countable. §5 deals with valuations, i.e., certain mappings μ from classes of isolated sets into the collection Λ of all isols which permit us to further generalize IEP by substituting μ(α) for Req α.

In this article we shall construct intuitionistic analogues to the main systems of classical tense logic. Since each classical modal logic can be gotten from some tense logic by one of the definitions

• (i) □ p ≡ p ∧ Gp ∧ Hp, ◇p ≡ p ∨ Fp ∨ Pp; or,

• (ii) □ p ≡ p ∧ Gp, ◇p = p ∨ Fp

(see [5]), we shall find that our intuitionistic tense logics give us analogues to the classical modal logics as well.

We shall not here discuss the philosophical issues raised by our logics. Readers interested in the intuitionistic view of time and modality should see [2] for a detailed discussion.

In §2 we define the Kripke models for IKt, the intuitionistic analogue to Lemmon's system Kt. We then prove the completeness and decidability of this system (§§3–5). Finally, we extend our results to other sorts of tense logic and to modal logic.

In the language of IKt, we have: sentence-letters p, q, r, etc.; the (intuitionistic) connectives ∧, ∨, →, ¬; and unary operators P (“it was the case”), F (it will be the case”), H (“it has always been the case”) and G (“it will always be the case”). Formulas are defined inductively: all sentence-letters are formulas; if X is a formula, so are ¬X, PX, FX, HX, and GX; if X and Y are formulas, so are X ∧ Y, X ∨ Y, and X → Y. We shall see that, in contrast to classical tense logic, F and P cannot be defined in terms of G and H.

All structures to be considered here have universe ω, and all languages come equipped with Gödel numberings. If is a structure, then D(), the open diagram of , can be thought of as a subset of ω, and it makes sense to talk about the Turing degree deg(D()). This depends on the presentation as well as the isomorphism type of .

For example, consider the ordering = (ω, <). For any B ⊆ ω, it is possible to code B in a copy of as follows: Let π be the permutation of ω such that for each n ∈ ω, π leaves 2n and 2n + 1 fixed if n ∈ B and switches 2n with 2n + 1 if n ∉ B. Let be the copy of such that ≃π. Then n ∈ B iff the sentence 2n < 2n + 1 is in D(). In §4, this idea will be used to show that for any structure that is not completely trivial, {deg(D()): ≃ } is closed upwards.

It would be satisfying to have a way of assigning Turing degrees to structures such that the degree assigned to a given structure measured the recursion-theoretic complexity of the isomorphism type and was independent of the presentation. Jockusch suggested the following.

Forcing extensions yield models of ZFC in which a long sequence of club subsets of ω1, has the following property: every subsequence of size ℵ1, has a finite intersection.

A many-valued logic version of the Craig-Lyndon interpolation theorem has been given by Gill [1] and Miyama [2]. The former dealt with three-valued logic and the latter generalized it to M-valued logic with M ≥ 3. The purpose of this paper is to improve the form of Miyama's version of the interpolation theorem. The system used in this paper is a many-valued analogue (Takahashi [4], Rousseau [3]) of Gentzen's logical calculus LK. Let T = {1,…, M} be the set of truth values. An M-tuple (Γ 1,…, Γ M ) of sets of formulas is called a sequent, which is regarded as valid if for any valuation (in canonical sense) there is a truth value μ Є T such that the set Γμ contains a formula of the value μ with respect to the valuation. (In the next section, and thereafter, we change the format of the sequent for typographical reasons.) Miyama's result is as follows (in representative form):

• (I) If a sequent ({A}, ∅,…, ∅, {B}) is valid, then there is a formula D such that

• (i) every predicate or propositional variable occurring in D occurs in A and B, and

• (ii) the sequents {{A}, ∅,…, ∅, {D}) and (D}, ∅,…, ∅, {B}) are both valid.

• What shall be proved in this paper is the following (in representative form):

• (II) If a sequent ({A 1}, {A 2}, …, {AM }) is valid, then there is a formula D such that

• (i) every predicate or propositional variable occurring in D occurs in at least two of the formulas A 1,…, AM , and

• (ii) the following M sequents are valid:

• ({A 1},{D},…,{D}),({D},{A 2},…,{D}),…,({D},{D},…,{AM }).

• Clearly the former can be obtained as a corollary of the latter.

Myhill [12] extended the ideas of Shapiro [15], and proposed a system of epistemic set theory IST (based on modal S4 logic) in which the meaning of the necessity operator is taken to be the intuitive provability, as formalized in the system itself. In this setting one works in classical logic, and yet it is possible to make distinctions usually associated with intuitionism, e.g. a constructive existential quantifier can be expressed as (∃x) □ …. This was first confirmed when Goodman [7] proved that Shapiro's epistemic first order arithmetic is conservative over intuitionistic first order arithmetic via an extension of Gödel's modal interpretation [6] of intuitionistic logic.

Myhill showed that whenever a sentence □A ∨ □B is provable in IST, then A is provable in IST or B is provable in IST (the disjunction property), and that whenever a sentence ∃x.□A(x) is provable in IST, then so is A(t) for some closed term t (the existence property). He adapted the Friedman slash [4] to epistemic systems.

Goodman [8] used Epistemic Replacement to formulate a ZF-like strengthening of IST, and proved that it was a conservative extension of ZF and that it had the disjunction and existence properties. It was then shown in [13] that a slight extension of Goodman's system with the Epistemic Foundation (ZFER, cf. §1) suffices to interpret intuitionistic ZF set theory with Replacement (ZFIR, [10]). This is obtained by extending Gödel's modal interpretation [6] of intuitionistic logic. ZFER still had the properties of Goodman's system mentioned above.

This paper contains an example of a decidable theory which has

1) only a countable number of countable models (up to isomorphism);

2) a decidable saturated model; and

3) a countable homogeneous model that is not decidable.

By the results in [1] and [2], this can happen if and only if the set of types realized by the homogeneous model (the type spectrum of the model) is not .

If Γ and Σ are types of a theory T, define Γ ◁ Σ to mean that any model of T realizing Γ must realize Σ. In [3] a decidable theory is constructed that has only countably many countable models, only recursive types, but whose countable saturated model is not decidable. This is easy to do if the restriction on the number of countable models is lifted; the difficulty arises because the set of types must be recursively complex, and yet sufficiently related to control the number of countable models. In [3] the desired theory T is such that

is a linear order with order type ω*. Also, the set of complete types of T is not . The last feature ensures that the countable saturated model is not decidable; the first feature allows the number of countable models to be controlled.

In their paper [3], Hajnal and Komjáth define the following combinatorial principle:

Definition 1.1. Suppose κ is an infinite cardinal and n < ω. Then Hn(κ) is the statement: There is a function F: [κ]n → [[κ]ω]≤ω such that

(a) ∀A ∈[κ]n ∀Y ∈ F(A)(Y ⊆ min (A)), and

(b) .

Hn(κ) is related to a more general principle introduced by Hajnal and Nagy in [4]. For applications of these principles to free sets for set mappings and Ramsey games we refer the reader to [3] and [4].

In [3] Hajnal and Komjáth prove the consistency of ZFC + GCH + ∀n ∈ ω(Hn + 1(ωn + 1)), relative to an ω-Mahlo cardinal. They conjecture that L is a model of this theory, and suggest that the proof might require higher gap morasses. The first few cases of this conjecture are known to be true; it is easy to see that if CH holds then H1 (ω1) is true, and Laver proved that V = L implies H2(ω2). In this paper we go one step further and prove V = L → H3(ω3). Unfortunately our methods do not appear to give Hn (ωn) for n ≥ 4.

Most of our notation is standard. If X is any set and κ is a cardinal number then [X]κ is the set of subsets of X with cardinality κ, and [X]≤κ is the set of subsets of X with cardinality ≤ κ. If X is a set of ordinals then tp(X) is the order type of X.

In this paper I show how to develop stability theory within the context of the topological logic first introduced by McKee [Mc 76], Garavaglia [G 78] and Ziegler [Z 76]. I then study some specific applications to topological modules; in particular I prove two quantifier élimination theorems, one a generalization of a result of Garavaglia.

In the first section I present a summary of basic results on topological model theory, mostly taken from the book of Flum and Ziegler [FZ 80]. This is done primarily to fix notation, but I also introduce the notion of an Lt-elementary substructure. The important point with this concept, as with many others, appears to be to allow only individuals to appear as parameters, not open sets.

In the second section I begin the study of stability theory for Lt. I first develop a translation of the topological language Lt into an ordinary first-order language L*. The first main theorem is (2.3), which shows that the translation is faithful to the model-theoretic content of Lt, and provides the necessary tools for studying Lt theories in the context of ordinary first-order logic. The translation allows me to consider individual stability theory for Lt: the stability-theoretic study of those types of Lt in which only individual variables occur freely and in which only individuals occur as parameters. I originally developed this stability theory entirely within Lt; the fact that the theorems and their proofs were virtually identical to those in ordinary first order logic suggested the reduction from Lt to L*.

We will give a simple philosophical “proof” of the negation of Cantor's continuum hypothesis (CH). (A formal proof for or against CH from the axioms of ZFC is impossible; see Cohen [1].) We will assume the axioms of ZFC together with intuitively clear axioms which are based on some intuition of Stuart Davidson and an old theorem of Sierpiński and are justified by the symmetry in a thought experiment throwing darts at the real number line. We will in fact show why there must be an infinity of cardinalities between the integers and the reals. We will also show why Martin's Axiom must be false, and we will prove the extension of Fubini's Theorem for Lebesgue measure where joint measurability is not assumed. Following the philosophy—if you reject CH you are only two steps away from rejecting the axiom of choice (AC)—we will point out along the way some extensions of our intuition which contradict AC.

Let p be a set. A function Φ is uniformly Σ 1(p) in every admissible set if there is a Σ 1 formula ϕ in the parameter p so that ϕ defines Φ in every Σ 1-admissible set which includes p. A theorem of Van de Wiele states that if Φ is a total function from sets to sets then Φ is uniformly Σ 1 in every admissible set if and only if it is E-recursive. A function is ESp -recursive if it can be generated from the schemes for E-recursion together with a selection scheme over the transitive closure of p. The selection scheme is exactly what is needed to insure that the ESP -recursively enumerable predicates are closed under existential quantification over the transitive closure of p. Two theorems are established: a) If the transitive closure of p is countable then a total function on sets is ESp -recursive if and only if it is uniformly Σ 1(p) in every admissible set. b) For any p, if Φ is a function on the ordinal numbers then Φ is ESP -recursive if and only if it is uniformly Σ 1(p) in every admissible set.