Definition. For ordinals α ≤ κ, = [X]α = {p ⊆ X ∣ ot (p) = α}. For α ≤, κ, κ a cardinal, holds iff for every partition F: [Κ]κ → A, there is an X ∈ [Κ]κ with F constant on [X]α. X is called homogeneous for F. When A = 2 the subscript is omitted.

It has been known since the sixties that for finite exponents all such properties are equivalent, i.e., κ→ (κ)2 iff for all n < ω, β < κ. The infinite case is much more difficult, though it is not actually known to be different. The usual methods for dealing with partitions fail, and as the weakest of these properties violates the Axiom of Choice, many techniques are unavailable.

It is extremely unlikely that an infinite exponent can be increased, but infinite subscripts are generally possible.

Theorem. If ∥ α ∥ · 2 ≤ α, then κ → (κ)α implies for all λ < κ.

In §1, we show the subscript can be any λ < κ. In §2 we extend this to 2λ. In §3 we discuss the possibilities with limited amounts of well-ordered choice, and apply the theorem to the problem of obliging ordinals. Except in §3, we assume no choice. For the story of finite exponents, see [1]. For the relationship between infinite-exponent partition relations and choice, see [4].

One of the main results of Gödel [4] and [5] is that, if M is a transitive set such that 〈M, Є 〉 is a model of ZF (Zermelo-Fraenkel set theory) and α is the least ordinal not in M, then 〈Lα, Є 〉 is also a model of ZF.

In this note we shall use the Jensen uniformisation theorem to show that results analogous to the above hold for certain subsystems of ZF. The subsystems we have in mind are those that are formed by restricting the formulas in the separation and replacement axioms to various levels of the Levy hierarchy.

This is all done in §1. In §2 we proceed to establish the exact order relationships which hold among the ordinals of the minimal models of some of the systems discussed in §1. Although the proofs of these latter results will not require any use of the uniformisation theorem, we will find it convenient to use some of the more elementary results and techniques from Jensen's fine-structural theory of L. We thus provide a brief review of the pertinent parts of Jensen's works in §0, where a list of general preliminaries is also furnished.

We remark that some of the techniques which we use in the present paper have been used by us previously in [6] to prove various results about β-models of analysis. Since β-models for analysis are analogous to transitive models for set theory, this is not surprising.

The work [11] deals with questions of first-order definability in algebraic function fields. In particular, it exhibits new cases in which the field of constant functions is definable, and it investigates the phenomenon of definable transcendental elements. We fix some of its proofs and make additional observations concerning definable closure in these fields.

Carnap (in [1], p. 566) has remarked that his measure function m* is fitting for finite languages using a fixed number of one-place predicates, i.e., for any sentence i, m*(i) is the same in all such restricted finite languages in which i occurs. The main purpose of this brief note is to show by means of a counter-example that m* does not have the intuitively desirable property of fittingness when we consider languages using two-place predicates.

We first state for finite languages a general theorem (related to the results in [2]) which guides the construction of large numbers of counter-examples.

Theorem. Let be the symmetric group of n letters. In a language which consists of a finite number of predicates and n individual names, the number of state descriptions in a structure description is equal to the index of some subgroup G in .

Proof. Let Σ be a structure description (of a language satisfying our hypothesis) and let S be an arbitrary state description in Σ. Let π be a permutation in , and let π*S be the state description which results from S by applying the permutation π to the individual names of . It is easily verified that for every π in π*S ∊ Σ. Also, for π, ψ in ,

We define the set G of permutations as follows :

.

The system K′ of “extended basic logic” lacks a principle of extensionality. In this paper a system KE′ will be presented which is like K′ in many respects but which does possess a fairly strong principle of extensionality by way of rule 6.37 below. It will be shown that KE′ is free from contradiction. KE′ is especially well suited for formalizing the theory of numbers presented in my paper, On natural numbers, integers, and rationals. The methods used there can be applied even more directly here because of the freedom of KE′ from type restrictions, but the details of such a derivation of a portion of mathematics will not be presented in this paper. It is evident, moreover, that KE′ contains at least as much of mathematical analysis as does K′, and perhaps considerably more. The method of carrying out proofs in KE′ is closely similar to that used in my book Symbolic logic, and could be expressed in similar notation.

It is extremely difficult to make general statements about functional completeness. (For the main reference on the subject see Post [2].) In this paper we restrict ourselves to the case of unary functions in a finite valued logic, and prove a result concerning minimal functionally complete sets, along with a necessary and sufficient condition for completeness. A basic familiarity with group theory would be helpful.

Assuming AD, we show that all of the ordinals below $\delta _5^1$ represented by descriptions (c.f. [2], but also defined below) are cardinals. Using this analysis we also get a simple representation for the cardinal structure below $\delta _5^1$. As an application, we compute the cofinalitites of all cardinals below $\delta _5^1$.

In this paper, we consider Foreman’s maximality principle, which says that any nontrivial forcing notion either adds a new real or collapses some cardinals. We prove the consistency of some of its consequences. We observe that it is consistent that every c.c.c. forcing adds a real and that for every uncountable regular cardinal κ, every κ-closed forcing of size 2<κ collapses some cardinal.

The fundamental role of the restricted calculus of predicates in applications of symbolic logic, and particularly in Hubert's Beweistheorie as summed up by Hilbert and Bernays, makes it important that this logical calculus should be accurately defined. The first standard formulation of the calculus was that of Hilbert and Ackermann's Grundzüge der theoretischen Logik. This employed (in the first three editions) a finite set of axioms and rules of derivation, with rules of substitution included. A reaction by Hilbert and Ackermann's successors to persistent difficulty encountered with the rules of substitution has been to omit these rules, and instead enlarge the set of axioms and the other rules of derivation so as to encompass all possible substitutions. Such an enlargement seems to me to be undesirable. As an alternative, this note is designed to put the original approach of Hilbert and Ackermann for once and for all on a sound basis.

The lambda calculus was conceived by Church, and to some extent by Curry (see [1] and [2]), as a general theory of functions, more precisely as a formalization of two basic functional concepts: application and abstraction. This claim to universality was impossible to maintain, for it soon became apparent that functions in the calculus are required to have properties which are by no means universal. Self-application is certainly a very special feature, but more relevant from our point of view is the existence of fixed points, which is definitely a nonuniversal property of functions. This situation was not ignored, but was considered rather as a form of inconsistency very much related to the paradoxes of set theory (for example, see the discussion of the paradoxical combinator in [2]).

More recently, the lambda calculus has been studied by people in computer science, and as a result the algorithmic nature of the system has been more explicitly recognized. Eventually, the right approach was taken by D. Scott (see [4] and [5]) who restricted the objects of the calculus to continuous functionals over complete lattices, which can be combined in structures where application and abstraction are properly interpreted. It soon became clear that complete lattices were not necessary, and more general structures were introduced. On the other hand the requirement of continuity was essential for the construction, and has remained a basic feature. It is one of the purposes of this paper to show that in the general situation continuity can be replaced by monotonicity.

The work of Scott was extended by Wadsworth [6], who gave a precise formulation of the operational semantics, and proved the equivalence to Scott's interpretation.