In this article an application of extended fields of real numbers to the proof of theorems in the theory of cooperative games will be presented. The proofs set forth below involve the use of A. Robinson's theory of nonstandard analysis and are metamathematical in character. Alternative proofs utilizing ordinary topological methods can in fact be carried out quite briefly. However, attempts to apply nonstandard analysis to game theory are relatively novel. For this reason these results may be of interest not only insofar as they present new information on the theory of the kernel of a cooperative game, but also in that they serve to demonstrate the possibility of effectively exploiting nonstandard analysis as a tool for future investigation in this area. It may well turn out that nonstandard analysis could serve as the most natural vehicle through which concepts defined for games with a finite number of players might be extended to games with a continuum of players.

In this paper we prove some theorems concerning measures on complete Boolean algebras. Among other things, in §I of this paper, we construct a counterexample to the following conjecture of W. Luxemburg: Every measure on a nonatomic hyperstonian Boolean algebra is normal. (See [3, p. 57].) This result is expressed by Theorem 1, §I. In order to construct this example we have to suppose that a real-valued measurable cardinal exists. This hypothesis is independent of the usual axioms of set theory. Luxemburg proved that our assumption is necessary. Our second result is stated in Theorem 2 near the end of the paper.

One of the standard ways of postulating large cardinal axioms is to consider elementary embeddings, j, from the universe, V, into some transitive submodel, M. See Reinhardt–Solovay [7] for more details. If j is not the identity, and κ is the first ordinal moved by j, then κ is a measurable cardinal. Conversely, Scott [8] showed that whenever κ is measurable, there is such j and M. If we had assumed, in addition, that , then κ would be the κth measurable cardinal; in general, the wider we assume M to be, the larger κ must be.

In [8] J. A. Robinson introduced a complete refutation procedure called resolution for first order predicate calculus. Resolution is based on ideas in Herbrand's Theorem, and provides a very convenient framework in which to search for a proof of a wff believed to be a theorem. Moreover, it has proved possible to formulate many refinements of resolution which are still complete but are more efficient, at least in many contexts. However, when efficiency is a prime consideration, the restriction to first order logic is unfortunate, since many statements of mathematics (and other disciplines) can be expressed more simply and naturally in higher order logic than in first order logic. Also, the fact that in higher order logic (as in many-sorted first order logic) there is an explicit syntactic distinction between expressions which denote different types of intuitive objects is of great value where matching is involved, since one is automatically prevented from trying to make certain inappropriate matches. (One may contrast this with the situation in which mathematical statements are expressed in the symbolism of axiomatic set theory.).

As was first mentioned in [3, §5], if A is any – set, A is the union of two disjoint – sets B(0), B(1). In metarecursion theory this is proven as follows. Let ƒ be a one-to-one metarecursive function whose range is A, let R be an unbounded metarecursive set whose complement is also unbounded, and set B(0) = f(R), B(1) = f(). The corresponding fact of ordinary recursion theory, namely that any r.e. but not recursive set can be split into two other such sets, was proved by Friedberg [2, Theorem 1], using a clever priority argument. Sacks [7, Corollary 2] then showed that any r.e. but not recursive set is the union of two disjoint r.e. sets neither of which was recursive in the other, a much stronger result. In this paper we attempt to prove the analogous result for – sets A, but succeed only in the case A is simple; i.e., the complement of A contains no infinite subset. As a corollary we show the metadegrees are dense, a fact already announced by Sacks [8, Corollary 1], but only proven by him for nonzero metadegrees.

In [2] Vaught showed that if T is a complete theory formalized in the first-order predicate calculus, then it is not possible for T to have exactly (up to isomorphism) two countable models. In this note we extend his methods to obtain a theorem which implies the above.

First some definitions. We define Fn(T) to be the set of well-formed formulas (wffs) in the language of T whose free variables are among x1 x2, …, xn. An n-type of T is a maximal consistent set of wffs of Fn(T); equivalently, a subset P of Fn(T) is an n-type of T if there is a model M of T and elements a1, a2, …, an of M such that M ⊧ ϕ(a1, a2, …, an) for every ϕ ∈ P. In the latter case we say that 〈a1, a2, …, an〉 ony realizes P in M. Every set of wffs of Fn(T) which is consistent with T can be extended to an n-type of T.

The notion of algebraic closedness plays an important part in the theory of commutative fields. The corresponding notion in the theory of ordered fields is (not only intuitively but in a sense which can be made precise in a metamathematical framework, compare [4]) that of a real closed ordered field. Several suggestions have been made (see [2] and [8]) for the formulation of corresponding concepts in the theory of groups and in the theory of skew fields (division rings, noncommutative fields). Here we present a concept of this kind, which preserves the principal metamathematical properties of algebraically closed commutative fields and which applies to a wide class of first order theories K, including the theories of commutative and of skew fields and the theories of commutative and of general groups.

In [1], various formal proof systems for infinitary formulas were defined. The formal proof system is the result of extending the basic predicate calculus by adding a collection Σ of axiom schemes and a collection Ω of rules of inference. Let Taut be the collection of all infinitary prepositional tautologies, considered as axiom schemes. Let ΩI consist of all the quantificational rules of independent choices. We will show, in §2 (see Theorem 2.1), that (Taut; 0) is not complete for L∞ω (i.e., infinitary finite-quantifier) sentences; that is, we will exhibit an L∞ω sentence ϕ such that ¬ϕ is true in all models, but ¬ϕ is not provable in (Taut; 0). (The unprovability is shown by a weak forcing version of Boolean general models.) This answers a question of Karp in [1,12.1(i)]. In §4, we will show that our ϕ is “ complete for L∞ω ) sentences.”

It is natural, given the usual iterative description of the universe of sets, to investigate set theories which in some way take account of the unfinished character of the universe. We do not here consider any arguments aimed at justifying one system over another, or at clarifying the basic philosophy. Rather, we look at an obvious candidate which is similar to a system discussed by L. Pozsgay in [1]. Pozsgay sketched the development of the ordinary theorems in such a system and attempted to show it equiconsistent with ZF. In this paper we show that the consistency of the system we call IZF can be proved in the usual ZF set theory.

In his paper [1] Chang provides among other things answers to questions of the following type: Given two models and of powers α and β, respectively, what is the least λ such that implies His proofs are by induction on the quantifier rank of formulas and they use an idea which in the case of ordinary first-order language goes back to Ehrenfeucht and Fraïssé. But, as we show, one can easily prove that if λ is big compared with κ and with the cardinality of the universe of the structure , then every L∞κ-formula is equivalent modulo the set of all Lλκ-sentences which hold in to a Lλκ-formula. From this, Chang's results follow immediately. The same method can be applied to similar problems concerning generalized languages.

Intuitively, a topological space is a set together with some notion of “nearness”. Classically, this notion of nearness is usually described by a collection of open sets. With the development of Non-standard analysis by Abraham Robinson [11], we have an alternative way to describe this notion. If X is any set and *X is a nonstandard extension of X then we can describe a topology on X by means of a relationship of “infinitely close” on some points of X. In many ways this latter approach is more intuitive and leads to more straightforward proofs [11] than the classical approach. The first section of this paper explores the connections between these two approaches. In the main section of the paper we extend the notion of “infinitely close” to infinite points of *X. In the final section we give a very intuitive characterization of the compact-open topology (when the domain is locally compact) and use it to give quick, straight-forward proofs of the usual facts about this topology [1], [2]. This last section can be read independently of the second section.

We use the same terminology and notations as in Part I of this paper [3], but numerals in square brackets refer to the references at the end of the present part. The word “space” will be used in the sense of “subspace of ŪF”. A. G. Hamilton [6] proved that any two α-bases of any α-space are recursively equivalent (see also [5]). This means that dimαV, introduced in [3] only for an isolic α-space V, can be defined for anyα-space V. Several results of [3] can therefore be strengthened. These and some other improvements are listed in §8. It is proved in §9 that every α-subspace of an isolic α-space is again an isolic α-space.

“From two integers k, l one passes immediately to kl; this process leads in a few steps to numbers which are far larger than any occurring in experience, e.g., 67(257729).

Intuitionism, like ordinary mathematics, claims that this number can be represented by an arabic numeral. Could not one press further the criticism which intuitionism makes of existential assertions and raise the question: What does it mean to claim the existence of an arabic numeral for the foregoing number, since in practice we are not in a position to obtain it?