A great deal of the interest in the theory of models of sets of sentences in the lower predicate calculus is due to the increasing success with which this theory has been applied to problems from algebra (see, for example [6, particularly pp. 90–103]).

The existence of nonstandard models prevents this language from being altogether appropriate for application to analysis.

In Anderson and Belnap [8] there was developed a semantics for first degree entailments (fde), i.e., entailments A → B between formulas A and B involving only truth-functions (defined in terms of “or” and “not”) and quantifiers. The key ideas were (i) the notion of a frame ⟨P, FIP, I⟩, where P is a set of (intensional) propositions closed under negation and multiple disjunction, I is a domain of individuals, and FIP is the set of functions from I into P; and (ii) the semantic relation of cons (consequence), as obtaining between a set of propositions taken conjunctively, and a set taken disjunctively; and (iii) the notion of an atomic frame, i.e., a frame generated by a set of propositions X closed under negation, such that for any disjoint subclasses Y and Z of X, Y does not bear cons to Z.

In this paper we consider a fonnal system of second-order Peano arithmetic with a rule of inference stronger than the ω-rule [3]. We also consider the relation to a class of models for analysis (i.e. second-order arithmetic) which lies between the class of ω-models and the class of β-models [5].

The notation used is largely that of [3] and [5]. We assume that the reader has some familiarity with at least the ideas of the former. The formal system (A) of Peano arithmetic employed in [3] includes the comprehension axioms and the second-order induction axiom.

In this paper, we shall state some results concerning ℵ0-complete cardinals and the transcendency of cardinals which is proposed by G. Takeuti [6]. The main purpose of this paper is to prove the following theorems.

Theorem 1. Let ℵ1, be the first ℵ0-complete cardinal number. Then the transcendency of cardinals holds for Δ11-functions of ℵ1-language.

In [3], Takeuti developed the theory of ordinal numbers (ON) and constructed a model of Zermelo-Fraenkel set theory (ZF), using the primitive recursive relation ∈ of ordinal numbers. He proved:

(1) If A is a ZF-provable formula, then its interpretation A0 in ON is ON-provable;

(2) Let B be a sentence of ordinal number theory. Then B is a theorem of ON if and only if the natural translation B* of B in set theory is a theorem of ZF;

(3) (V = L)° holds in ON.

It will be shown later in this paper that the class of all Hilbert spaces is not an elementary class (in the wider sense) in the lower predicate calculus. It is not difficult to find a type and a sentence in Lω1ω1 (of that type) whose models are precisely the Hilbert spaces (slightly altered to include in their domains the real or complex numbers).

In this paper we continue our study, begun in [5], of the connection between ultraproducts and saturated structures. If D is an ultrafilter over a set I, and is a structure (i.e., a model for a first order predicate logic ℒ), the ultrapower of modulo D is denoted by D-prod. The ultrapower is important because it is a method of constructing structures which are elementarily equivalent to a given structure (see Frayne-Morel-Scott [3]). Our ultimate aim is to find out what kinds of structure are ultrapowers of . We made a beginning in [5] by proving that, assuming the generalized continuum hypothesis (GCH), for each cardinal α there is an ultrafilter D over a set of power α such that for all structures , D-prod is α+-saturated.

It is shown in [1] that an ultraproduct of finite sets can be of arbitrarily large cardinality, but if it is infinite then it must have at least the power of the continuum. In this paper we shall take a closer look at the cardinality of ultraproducts of finite sets. Our results were announced without proof in [5]. A discussion of the cardinality of ultraproducts of infinite sets, and another theorem about ultraproducts of finite sets, can be found in [3].

It is well known that L, the first-order predicate calculus, is unsuitable for characterizing certain mathematical structures. Since some of those structures are quite often considered by mathematicians, extensions of L have been considered which (partially) solve the above problem. Roughly speaking, the extensions of L fall into two main categories:

(I) extensions obtained by allowing higher order variables, and

(II) extensions obtained by allowing infinitely long formulas.

A theory T is axiomatizable by a schema if there is a formula Γ, involving symbols of T plus a new relation symbol R, such that the set of all (universal closures of) instances of Γ in T is a set of axioms for T. (It is understood that, if R has n places, an instance of Γ in T is obtained by properly substituting for R in Γ a formula of T which has n selected free variables and is allowed to have any number of other free variables as parameters.) Obviously, the notion is unchanged if finitely many Γ's, each involving several new R's, are allowed instead. All theories we consider are assumed to be theories in the first-order logic with equality (as in [8]), to have finitely many nonlogical symbols, and to be recursively axiomatizable.

In [1, pp. 82–84] L. Åqvist considers a modal system which he calls S3.5 obtained by adding to S3 the axiom ∼□p⊃□∼□p. This system becomes S5 when the rule ├A→├□A is added to it. S3.5 is put forward to stand to S5 as S3 stands to S4 and S2 to T. In this note we show how a natural extension of the modelling for S3 in [2] can give a suitable semantics for S3.5.1

An S3 model2 is an ordered quadruple (GKRφ) where Κ is a set, G ε K and R is transitive and quasi-reflexive over K.

We are concerned here with the set theory given in [1], which we call BL (Bernays-Levy). This theory can be given an elegant syntactical presentation which allows most of the usual axioms to be deduced from the reflection principle. However, it is more convenient here to take the usual Von Neumann-Bernays set theory [3] as a starting point, and to regard BL as arising from the addition of the schema

where S is the formal definition of satisfaction (with respect to models which are sets) and ┌φ┐ is the Gödel number of φ which has a single free variable X.

The “traditional” method of presenting the subject-matter of symbolic logic involves setting down, first of all, a basis for a propositional calculus—which basis might be a system of natural deduction, an axiom system, or a rule concerning tautologous formulas. The next step, ordinarily, consists of the introduction of quantifiers into the symbol-set of the system, and the stating of axioms or rules for quantification. In this paper I shall propose a system somewhat different from the ordinary; this system has rules for quantification and is, indeed, equivalent to classical quantification theory. It departs from the usual, however, in that it has no primitive quantifiers.

In §1 is given a characterization of strongly hypersimple sets in terms of weak arrays which is in appearance more restrictive than the original definition. §1 also includes a new characterization of hyperhypersimple sets. This one is interesting because in §2 a characterization of dense simple sets is shown which is identical in all but the use of strong arrays instead of weak arrays. Another characterization of hyperhypersimple sets, in terms of descending sequences of sets, is given in §3. Also a theorem showing strongly contrasting behavior for simple sets is presented. In §4 a r-maximal set which is not contained in any maximal set is constructed.

§1. In [2], as part of an analogy between the concepts of recursive emimerability and regressiveness, Dekker showed that the intersection of any two regressive sets which are recursively equivalent is a regressive set. In [1], McLaughlin and the author showed that the intersection of two regressive sets, if infinite, has an infinite regressive subset. However, the following theorem shows that the analogy is not complete in this instance.

Theorem. There exist two regressive sets whose intersection is not regressive.

In this paper it will be shown that the ordering of the recursively enumerable (r.e.) sets under inclusion modulo finite differences (m.f.d.), the ordering of the II111 sets under inclusion m.f.d., and the ordering of the metarecursively enumerable (meta-r.e.) sets under inclusion m.f.d. are all distinct. In fact, it will be shown that the three orderings are pairwise elementarily inequivalent when interpreted in the obvious way in a first order language with one binary relation “≥.” Our result answers a question of Hartley Rogers, Jr. [3, p. 203]. All necessary background material may be found in [2] and [4].

In the literature, a type Γ is a set of formulas ϕ(ν0) in the one free variable ν0. A model M realizes Γ if there is an element a of M such that

M ╞ ϕ [a] for all ϕ ∈ Γ.

A model M omits Γ if no element of M realizes Γ. Evidently, the two statements:

(*) M realizes Γ

and

(**) M omits Γ

The class of primitive recursive functions may be defined as the closure of certain initial functions, namely the zero, successor and identity functions, under two schemes, namely composition (sometimes called “substitution”) and recursion. For a detailed definition the reader is referred to any standard work, for instance p. 219 of [2], by Kleene.