On June 4, 1925, Hilbert delivered an address to the Westphalian Mathematical Society in Miinster; that was, as a quick calculation will convince you, almost exactly sixty years ago. The address was published in 1926 under the title Über das Unendliche and is perhaps Hilbert's most comprehensive presentation of his ideas concerning the finitist justification of classical mathematics and the role his proof theory was to play in it. But what has become of the ambitious program for securing all of mathematics, once and for all? What of proof theory, the very subject Hilbert invented to carry out his program? The Hilbertian ambition reached out too far: in its original form, the program was refuted by Gödel's Incompleteness Theorems. And even allowing more than finitist means in metamathematics, the Hilbertian expectations for proof theory have not been realized: a constructive consistency proof for second-order arithmetic is still out of reach. (And since that theory provides a formal framework for analysis, it was considered by Hilbert and Bernays as decisive for proof theory.) Nevertheless, remarkable progress has been made. Two separate, but complementary directions of research have led to surprising insights: classical analysis can be formally developed in conservative extensions of elementary number theory; relative consistency proofs can be given by constructive means for impredicative parts of second order arithmetic. The mathematical and metamathematical developments have been accompanied by sustained philosophical reflections on the foundations of mathematics. This indicates briefly the main themes of the contributions to the symposium; in my introductory remarks I want to give a very schematic perspective, that is partly historical and partly systematic.

§0. Introduction. What follows is a write-up of my contribution to the symposium “Hilbert's Program Sixty Years Later” which was sponsored jointly by the American Philosophical Association and the Association for Symbolic Logic. The symposium was held on December 29,1985 in Washington, D. C. The panelists were Solomon Feferman, Dag Prawitz and myself. The moderator was Wilfried Sieg. The research which I discuss here was partially supported by NSF Grant DMS-8317874.

I am grateful to the organizers of this timely symposium on an important topic. As a mathematician I particularly value the opportunity to address an audience consisting largely of philosophers. It is true that I was asked to concentrate on the mathematical aspects of Hilbert's program. But since Hilbert's program is concerned solely with the foundations of mathematics, the restriction to mathematical aspects is really no restriction at all.

Hilbert assigned a special role to a certain restricted kind of mathematical reasoning known as finitistic. The essence of Hilbert's program was to justify all of set-theoretical mathematics by means of a reduction to finitism. It is now well known that this task cannot be carried out. Any such possibility is refuted by Gödel's theorem. Nevertheless, recent research has revealed the feasibility of a significant partial realization of Hilbert's program. Despite Gödel's theorem, one can give a finitistic reduction for a substantial portion of infinitistic mathematics including many of the best-known nonconstructive theorems. My purpose here is to call attention to these modern developments.

In the Symposium on Hilbert's Program to which the following was contributed, I was asked to talk about the metamathematical aspects of Hilbert's Program (H.P.), while the other two speakers (Simpson and Prawitz) were to deal with the mathematical and philosophical aspects respectively. However, more so than for other foundational schemes, these three aspects of H.P., both as originally conceived and in its subsequent developments, are intimately linked.

Here I shall survey a body of proof-theoretical results stemming from H.P., but organized in a way that is closely tied to various reductive foundational aims, albeit going beyond those advanced by Hilbert. I believe this view of reductive proof-theory (not original with me) helps one to better understand what has been achieved thereby than other, more familiar accounts.

We show that if κ is an inaccessible cardinal then P κλ splits into λ<κ many disjoint stationary subsets. We also show that if P κλ carries a strongly saturated ideal then the nonstationary ideal cannot be λ+-saturated.

The present note outlines an answer to a question listed as open in our recent survey article [1], familiarity with which is assumed.

Let Γ be the class of all X ⊆ ω such that X is reducible to for some some arithmetical and some positive with Y ⊆ p(Y) for all Y.

We prove that cons(ZF) implies cons(ZF + Borel conjecture + there exists a Ramsey ultrafilter). We also prove some results on strong measure zero sets from the existence of generalized Luzin sets. We study the relationships between strong measure zero sets and rapid filters on ω.

§0. Introduction and material background. The present paper is devoted to the study of intermediate propositional logics, and it is based on [Be, §§1 and 2].

§2 (§§0 and 1 are introductory) concerns the axiomatization of finite logics. In the literature several effective procedures to axiomatize finite logics are present (cf., for instance, [MK] and [Wr]), but, in each case, the number of propositional variables which are used is redundant. In this direction, Theorem 2.2 provides (a) a criterion to determine, given a finite logic L, the least n such that L is axiomatizable by formulas in n variables, and (b) an effective axiomatization by an n-formula. As a corollary we obtain a negative answer to Problem 7.10 of [Ho/On], showing that there is no connection between the slice to which L belongs and the number of propositional variables necessary to axiomatize L.

The principal results of the paper are in §3. In fact, a great deal of research has been done on the correspondence between conditions on the relation of Kripke-structures from one side, and axioms added to Int from the other. In this section we (a) introduce the concept of finitely separable class of Kripke-frames, and show, by means of several examples, that this concept is “wide”, in the sense that all the most studied classes of frames determined by semantical conditions are finitely separable; (b) show that each finitely separable class is axiomatizable, and that the axioms can be found by means of semantical considerations only; and (c) establish the finite model property for all the finitely separable logics.

Baur a défini la notion d'extension séparée de corps valués et montré que toute extension d'un corps maximal est séparée. Nous prouvons que, si (K, υ) est henselien et de caractéristique résiduelle nulle, alors (K, υ) ⊂ (L, w) est séparée ssi L est linéairement disjoint sur K de toute extension immédiate de K.

Separated and immediate extensions of valued fields. The notion of separated extension of valued fields was introduced by Baur. He showed that extensions of maximal fields are separated. We prove that, when (K, υ) is Henselian with residual characteristic 0, then (K, υ) ⊂ (L, w) is separated iff L is linearly disjoint over K from each immediate extension of K.

If θ is any singular cardinal of cofinality ω 1, we produce a forcing extension in which MA holds below θ but fails at θ. The failure is due to a partial order which splits a gap of size θ in .

Une bijection p de M 2 dans M est dite sans cycles, ou encore localement libre, si pour tout terme t(x 1,…,xn ) formé à partir de p, et qui n'est pas une simple variable, on a, pour tous a 1,…,an de M, t(a 1,… an ) ≠ a 1; toutes ces “fonctions-paires” sont élémentairement équivalentes, à condition, bien sûr, que l'ensemble M soit non vide, et leur théorie (parfois traduite dans un autre langage) a été considérée par divers auteurs.

Ainsi, A. Mal'cev [Mal'cev 1962] a décrit toutes les théories d'algèbre localement libres, associées à un langage fonctionnel arbitraire; O. Belegradek [Belegradek] a étudié dans le détail leurs propriétés de stabilité; F. Lucas [Lucas 1978] avait également considéré des fonctions-paires localement libres; c'est encore une fonction-paire qui a été utilisée par J. F. Pabion [Pabion 1982], pour ω1-saturer, mais pas davantage, les modèles de l'Arithmétique; il a été remarqué, dans [Poizat 1984], combien il était utile pour cela de disposer d'une théorie non superstable sans propriété de recouvrement fini; dans un tout autre contexte, et pour de tout autres raisons, cette théorie de fonctions-paires a également été utilisée dans [Poizat 1986]; tous ces emplois supposent qu'on y maîtrise bien l'élimination des quanteurs.

We are interested in regressive isols, recursive functions, and the extensions of recursive functions to the isols. One of the nicest concepts that has been applied to the study of these notions is of an infinite series of isols. J. C. E. Dekker introduced infinite series of isols in [3]. With this concept one may associate with each number theoretic function u and regressive isol B a value in the isols to correspond to the series

When u is chosen as a recursive function, or as a recursive combinatorial function, many of the sums that one associates with familiar finite series may be generalized to infinite series. For example, if B is any regressive isol, then

The results presented in our paper were motivated by an interest in extending infinite series to a setting where the terms being summed may be infinite isols. In our paper, we do this in a special way, as will be described below. We would first like to briefly comment on some facts about the concept of defining an infinite series in the isols.

The theory of separably closed fields of a fixed characteristic and a fixed imperfectness degree is clearly recursively axiomatizable. Ershov [1] showed that it is complete, and therefore decidable. Later it became clear that this theory also has the prime extension property in a suitable language (cf. [4, Proposition 1]); hence it admits quantifier elimination. The purpose of this work is to give an explicit, primitive recursive procedure for such quantifier elimination in the case of a finite imperfectness degree.

Let K be an algebraic number field and IK the ring of algebraic integers in K. *K and *IK denote enlargements of K and IK respectively. Let x Є *K – K. In this paper, we are concerned with algebraic extensions of K(x) within *K. For each x Є *K – K and each natural number d, YK(x,d) is defined to be the number of algebraic extensions of K(x) of degree d within *K. x Є *K – K is called a Hilbertian element if YK(x,d) = 0 for all d Є N, d > 1; in other words, K(x) has no algebraic extension within *K. In their paper [2], P. C. Gilmore and A. Robinson proved that the existence of a Hilbertian element is equivalent to Hilbert's irreducibility theorem. In a previous paper [9], we gave many Hilbertian elements of nonstandard integers explicitly, for example, for any nonstandard natural number ω, 2 ω P ω and 2 ω (ω 3 + 1) are Hilbertian elements in *Q, where p ω is the ωth prime number.

§0. Introduction. Ask a beginning philosophy of mathematics student why we believe the theorems of mathematics and you are likely to hear, “because we have proofs!” The more sophisticated might add that those proofs are based on true axioms, and that our rules of inference preserve truth. The next question, naturally, is why we believe the axioms, and here the response will usually be that they are “obvious”, or “self-evident”, that to deny them is “to contradict oneself” or “to commit a crime against the intellect”. Again, the more sophisticated might prefer to say that the axioms are “laws of logic” or “implicit definitions” or “conceptual truths” or some such thing.

Unfortunately, heartwarming answers along these lines are no longer tenable (if they ever were). On the one hand, assumptions once thought to be self-evident have turned out to be debatable, like the law of the excluded middle, or outright false, like the idea that every property determines a set. Conversely, the axiomatization of set theory has led to the consideration of axiom candidates that no one finds obvious, not even their staunchest supporters. In such cases, we find the methodology has more in common with the natural scientist's hypotheses formation and testing than the caricature of the mathematician writing down a few obvious truths and preceeding to draw logical consequences.

The central problem in the philosophy of natural science is when and why the sorts of facts scientists cite as evidence really are evidence. The same is true in the case of mathematics. Historically, philosophers have given considerable attention to the question of when and why various forms of logical inference are truth-preserving. The companion question of when and why the assumption of various axioms is justified has received less attention, perhaps because versions of the “self-evidence” view live on, and perhaps because of a complacent if-thenism.

In this paper we prove three theorems about first-order theories that are categorical in a higher power. The first theorem asserts that such a theory either is totally categorical or there exist prime and minimal models over arbitrary base sets. The second theorem shows that such theories have a natural notion of dimension that determines the models of the theory up to isomorphism. From this we conclude that I(T,ℵα,) = ℵ0 + ∣α∣ where ℵα = the number of formulas modulo T-equivalence provided that T is not totally categorical. The third theorem gives a new characterization of these theories.

Three classes of decidable discrete linear orders with varying degrees of effectiveness are investigated. We consider how a classical order type may lie in relation to these three classes, and we characterize by their order types elements of these classes that have effective nontrivial self-embeddings.

§0. Introduction. We started our study of filters of partitions in [15]. We shall here restrict ourselves to the consideration of filters on (ω)ω, the set of all infinite partitions of ω. §1 is an attempt to elucidate the connection between filters on (ω)ω and filters over ω. Given a filter H over ω, we define two filters FH and GH on (ω)ω, and we characterize p-points, rare ultrafilters and Ramsey ultrafilters in terms of properties of the associated filters of partitions.

The remainder of the paper is devoted to the study of those filters that can be associated with Hindman's theorem and its extensions. Let us introduce some notation. Suppose * is an associative operation on ω, and let a subset A of ω and an ordinal α with 0 < α ≤ ω be given.

This paper is meant to be a comment on Beth's definability theorem. In it we shall make the following points.

Implicit definability as mentioned in Beth's theorem for first-order logic is a special case of a more general notion of uniqueness. If α is a nonlogical constant, T α a set of sentences, α* an additional constant of the same syntactical category as α and T α, a copy of T α with α* instead of α, then for implicit definability of α in T α one has, in the case of predicate constants, to derive α(x 1,…,xn ) ↔ α*(x 1,…,xn ) from T α ∪ T α*, and similarly for constants of other syntactical categories. For uniqueness one considers sets of schemata S α and derivability from instances of S α ∪ S α* in the language with both α and α*, thus allowing mixing of α and α* not only in logical axioms and rules, but also in nonlogical assumptions. In the first case, but not necessarily in the second one, explicit definability follows. It is crucial for Beth's theorem that mixing of α and α* is allowed only inside logic, not outside. This topic will be treated in §1.

Let the structural part of logic be understood roughly in the sense of Gentzen-style proof theory, i.e. as comprising only those rules which do not specifically involve logical constants. If we restrict mixing of α and α* to the structural part of logic which we shall specify precisely, we obtain a different notion of implicit definability for which we can demonstrate a general definability theorem, where a is not confined to the syntactical categories of nonlogical expressions of first-order logic. This definability theorem is a consequence of an equally general interpolation theorem. This topic will be treated in §§2, 3, and 4.

A Boolean product construction is used to give examples of existentially closed algebras in the universal Horn class ISP(K) generated by a universal class K of finitely subdirectly irreducible algebras such that Γa(K) has the Fraser-Horn property. If ⟦a ≠ b⟧ ∩ ⟦c ≠ d ⟧ = ∅ is definable in K and K has a model companion of K-simple algebras, then it is shown that ISP(K) has a model companion. Conversely, a sufficient condition is given for ISP(K) to have no model companion.