We show that one can solve Post's Problem by constructing generic sets in the usual set theoretic framework applied to tiny universes. This method leads to a new class of recursively enumerable sets: r.e. generic sets. All r.e. generic sets are low and simple and therefore of Turing degree strictly between 0 and 0′. Further they supply the first example of a class of low recursively enumerable sets which are automorphic in the lattice ℰ of recursively enumerable sets with inclusion. We introduce the notion of a promptly simple set. This describes the essential feature of r.e. generic sets with respect to automorphism constructions.

We show that Shelah’s Eventual Categoricity Conjecture for successors follows from the existence of class many strongly compact cardinals. This is the first time the consistency of this conjecture has been proven. We do so by showing that every AEC with LS(K) below a strongly compact cardinal κ is < κ-tame and applying the categoricity transfer of Grossberg and VanDieren [11]. These techniques also apply to measurable and weakly compact cardinals and we prove similar tameness results under those hypotheses. We isolate a dual property to tameness, called type shortness, and show that it follows similarly from large cardinals.

Lindenbaum's construction of a matrix for a propositional calculus, in which the wffs themselves are taken as elements and the theorems as the designated elements, immediately establishes two general results: that every prepositional calculus is many-valued, and that every many-valued propositional calculus is also ℵ0-valued. These results are however concerned exclusively with theoremhood, the inferential structure of the calculus being relevant only incidentally, in that it may serve to determine the set of theorems. We therefore ask what happens when deducibility is taken into consideration on a par with theoremhood. The answer is that in general the Lindenbaum construction is no longer adequate and both results fail.

The bounded proper forcing axiom BPFA is the statement that for any family of ℵ1 many maximal antichains of a proper forcing notion, each of size ℵ1, there is a directed set meeting all these antichains.

A regular cardinal κ is called ∑1-reflecting, if for any regular cardinal χ, for all formulas φ, “H(χ) ⊨ ‘φ’” implies “∃δ < κ, H(δ) ⊨ ‘φ’”.

We investigate several algebraic consequences of BPFA, and we show that the consistency strength of the bounded proper forcing axiom is exactly the existence of a ∑1-reflecting cardinal (which is less than the existence of a Mahlo cardinal).

We also show that the question of the existence of isomorphisms between two structures can be reduced to the question of rigidity of a structure.

We deal in the following with certain theories S, by which we mean sets of sentences closed under logical deduction. The basic logic is understood to be the classical one, but we place no restriction on the orders of the variables to be used. However, we do assume that we can at least express certain notions from classical first-order number theory within these theories. In particular, there should correspond to each primitive recursive function ξ a formula φ(χ), where ‘x’ is a variable ranging over natural numbers, such that for each numeral ñ, φ(ñ) expresses in the language of S that ξ(η) = 0. Such formulas, when obtained say by the Gödel method of eliminating primitive recursive definitions in favor of arithmetical definitions in +. ·. are called PR-formulas (cf. [1] §2 (C)).

This paper presents several results whose common element is their connection with Hubert's tenth problem.1 The one that seems most striking (perhaps because it can be stated with a minimum of technical apparatus) is this: there does not exist an algorithm for determining whether or not a polynomial (in n variables) represents every integer.2

In addition, the present paper (i) gives a simple characterization of the diophantine sets (of positive, non-negative, and arbitrary integers), and (ii) gives a rigorous proof of Myhill's theorem [6] that there is no decision method for statements of the form

It is shown that axiomatic extensions of intuitionistic propositional calculus defining univocally new connectives, including those proposed by Gabbay, are strongly complete with respect to valuations in Heyting algebras with additional operations. In all cases, the double negation of such a connective is equivalent to a formula of intuitionistic calculus. Thus, under the excluded third law it collapses to a classical formula, showing that this condition in Gabbay's definition is redundant. Moreover, such connectives can not be interpreted in all Heyting algebras, unless they are already equivalent to a formula of intuitionistic calculus. These facts relativize to connectives over intermediate logics. In particular, the intermediate logic with values in the chain of length n may be “completed” conservatively by adding a single unary connective, so that the expanded system does not allow further axiomatic extensions by new connectives.

The set of all words in the alphabet {l, r} forms the full binary tree T. If x ∈ T then xl and xr are the left and the right successors of x respectively. We consider the monadic second-order language of the full binary tree with the two successor relations. This language allows quantification over elements of rand over arbitrary subsets of T. We prove that there is no monadic second-order formula ϕ*(X, y) such that for every nonempty subset X of T there is a unique y ∈ X that satisfies ϕ*(X, y) in T.

A positive answer to a question of M. van Lambalgen and D. Zambella whether there exist nonrecursive sets that are low for the class of random sets is obtained. Here a set A is low for the class RAND of random sets if RAND = RANDA.

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.

Let ℜ be an expansion of a dense linear order (R, <) without endpoints having the intermediate value property, that is, for all a, b ∈ R, every continuous (parametrically) definable function f: [a, b] → R takes on all values in R between f(a) and f(b). Every expansion of the real line (ℝ, <), as well as every o-minimal expansion of (R, <), has the intermediate value property. Conversely, some nice properties, often associated with expansions of (ℝ, <) or with o-minimal structures, hold for sets and functions definable in ℜ. For example, images of closed bounded definable sets under continuous definable maps are closed and bounded (Proposition 1.10).

Of particular interest is the case that ℜ expands an ordered group, that is, ℜ defines a binary operation * such that (R, <, *) is an ordered group. Then (R, *) is abelian and divisible (Proposition 2.2). Continuous nontrivial definable endo-morphisms of (R, *) are surjective and strictly monotone, and monotone nontrivial definable endomorphisms of (R, *) are strictly monotone, continuous and surjective (Proposition 2.4). There is a generalization of the familiar result that every proper noncyclic subgroup of (ℝ, +) is dense and codense in ℝ: If G is a proper nontrivial subgroup of (R, *) definable in ℜ, then either G is dense and codense in R, or G contains an element u such that (R, <, *, e, u, G) is elementarily equivalent to (ℚ, <, +, 0, 1, ℤ), where e denotes the identity element of (R, *) (Theorem 2.3).

Here is an outline of this paper. First, we deal with some basic topological results. We then assume that ℜ expands an ordered group and establish the results mentioned in the preceding paragraph. Some examples are then given, followed by a brief discussion of analytic results and possible limitations. In an appendix, an explicit axiomatization (used in the proof of Theorem 2.3) is given for the complete theory of the structure (ℚ, <, +, 0, 1, ℤ).

This paper contains some results concerning the completeness of a first-order system of infinite valued logic

There are under consideration two distinct notions of completeness corresponding to the two notions of validity (see Definition 3) and strong validity (see Definition 4). Both notions of validity, whether based on the unit interval [0, 1] or based on linearly ordered MV-algebras, use the element 1 as the designated truth value. Originally, it was thought by many investigators in the field that one should be able to prove that the set of valid sentences is recursively enumerable. It was first proved by Rutledge in [9] that the set of valid sentences in the monadic first-order infinite valued logic is recursively enumerable.

In [4] Alonzo Church introduced an elegant and expressive formulation of type theory with λ-conversion. In [8] Henkin introduced the concept of a general model for this system, such that a sentence A is a theorem if and only if it is true in all general models. The crucial clause in Henkin's definition of a general model ℳ is that for each assignment φ of values in ℳ to variables and for each wff A, there must be an appropriate value of A in ℳ. Hintikka points out in [10, p. 3] that this constitutes a rather strong requirement concerning the structure of a general model. Henkin draws attention to the problem of constructing nonstandard models for the theory of types in [9, p. 324].

We shall use a simple idea of combinatory logic to find a characterization of general models which does not directly refer to wffs, and which is easier to work with in certain contexts. This characterization can be applied, with appropriate minor and obvious modifications, to a variety of formulations of type theory with λ-conversion. We shall be concerned with a language ℒ with extensionality in which there is no description or selection operator, and in which (for convenience) the sole primitive logical constants are the equality symbols Qoαα for each type α.

Most Gentzen systems arising in logic contain few axiom schemata and many rule schemata. Hilbert systems, on the other hand, usually contain few proper inference rules and possibly many axioms. Because of this, the two notions tend to serve different purposes. It is common for a logic to be specified in the first instance by means of a Gentzen calculus, whereupon a Hilbert-style presentation ‘for’ the logic may be sought—or vice versa. Where this has occurred, the word ‘for’ has taken on several different meanings, partly because the Gentzen separator ⇒ can be interpreted intuitively in a number of ways. Here ⇒ will be denoted less evocatively by ⊲.

In this paper we aim to discuss some of the useful ways in which Gentzen and Hilbert systems may correspond to each other. Actually, we shall be concerned with the deducibility relations of the formal systems, as it is these that are susceptible to transformation in useful ways. To avoid potential confusion, we shall speak of Hilbert and Gentzen relations. By a Hilbert relation we mean any substitution-invariant consequence relation on formulas—this comes to the same thing as the deducibility relation of a set of Hilbert-style axioms and rules. By a Gentzen relation we mean the fully fledged generalization of this notion in which sequents take the place of single formulas. In the literature, Hilbert relations are often referred to as sentential logics. Gentzen relations as defined here are their exact sequential counterparts.

In this paper we prove the strong normalization theorem for full first order classical N.D. (natural deduction)—full in the sense that all logical constants are taken as primitive. We also give a syntactic proof of the normal form theorem and (weak) normalization for the same system.

The theorem has been stated several times, and some proofs appear in the literature. The first proof occurs in Statman [1], where full first order classical N.D. (with the elimination rules for ∨ and ∃ restricted to atomic conclusions) is embedded in a system for second order (propositional) intuitionistic N.D., for which a strong normalization theorem is proved using strongly impredicative methods.

A proof of the normal form theorem and (weak) normalization theorem occurs in Seldin [1] as an extension of a proof of the same theorem for an N.D.-system for the intermediate logic called MH.

The proof of the strong normalization theorem presented in this paper is obtained by proving that a certain kind of validity applies to all derivations in the system considered.

The notion “validity” is adopted from Prawitz [2], where it is used to prove the strong normalization theorem for a restricted version of first order classical N.D., and is extended to cover the full system. Notions similar to “validity” have been used earlier by Tait (convertability), Girard (réducibilité) and Martin-Löf (computability).

In Prawitz [2] the N.D. system is restricted in the sense that ∨ and ∃ are not treated as primitive logical constants, and hence the deductions can only be seen to be “natural” with respect to the other logical constants. To spell it out, the strong normalization theorem for the restricted version of first order classical N.D. together with the well-known results on the definability of the rules for ∨ and ∃ in the restricted system does not imply the normalization theorem for the full system.

This paper develops in certain directions the work of Meyer in [3], [4], [5] and [6] (see also Routley [10] and Asenjo [11]). In those works, Peano's axioms for arithmetic were formulated with a logical base of the relevant logic R, and it was proved finitistically that the resulting arithmetic, called R#, was absolutely consistent. It was pointed out that such a result escapes incautious formulations of Gödel's second incompleteness theorem, and provides a basis for a revived Hilbert programme. The absolute consistency result used as a model arithmetic modulo two. Modulo arithmetics are not ordinarily thought of as an extension of Peano arithmetic, since some of the propositions of the latter, such as that zero is the successor of no number, fail in the former. Consequently a logical base which, unlike classical logic, tolerates contradictory theories was used for the model. The logical base for the model was the three-valued logic RM3 (see e.g. [1] or [8]), which has the advantage that while it is an extension of R, it is finite valued and so easier to handle.

The resulting model-theoretic structure (called in this paper RM32) is interesting in its own right in that the set of sentences true therein constitutes a negation inconsistent but absolutely consistent arithmetic which is an extension of R#. In fact, in the light of the result of [6], it is an extension of Peano arithmetic with a base of a classical logic, P#. A generalisation of the structure is to modulo arithmetics with the same logical base RM3, but with varying moduli (called RM3i here).

This paper treats of semantical systems S of sufficient strength so that for any set W definable in S (in a sense which will be made precise), there must exist a sentence X which is true in S if and only if it is an element of W. We call such an X a Tarski sentence for W. It is the sentence which (in a purely extensional sense) says of itself that it is in W. If W is the set of all expressions not provable in some syntactical system C, then X is the Gödel sentence which is true (in S) if and only if it is not provable (in C). We provide a novel method for the construction of these sentences, which yields sentences particularly simple in structure. The method is applicable to a variety of systems, including a form of elementary arithmetic, and some systems of protosyntax self applied. In application to the former, we obtain an extremely simple and direct proof of a theorem, which is essentially Tarski's theorem that the truth set of elementary arithmetic is not arithmetically definable.

The crux of our method is in the use of a certain function, the ‘norm’ function, which replaces the classical use of the diagonal function. To give a heuristic idea of the norm function, let us define the norm of an expression E (of informal English) as E followed by its own quotation.

In this paper we shall prove

Theorem 0.1. Suppose there is a singular strong limit cardinal κ such that □κ fails; then AD holds in L(R).

See [10] for a discussion of the background to this problem. We suspect that more work will produce a proof of the theorem with its hypothesis that κ is a strong limit weakened to ∀α < κ (αω < κ), and significantly more work will enable one to drop the hypothesis that K is a strong limit entirely. At present, we do not see how to carry out even the less ambitious project.

Todorcevic [23] has shown that if the Proper Forcing Axiom (PFA) holds, then □κ fails for all uncountable cardinals κ. Thus we get immediately:

It has been known since the early 90's that PFA implies PD, that PFA plus the existence of a strongly inaccessible cardinal implies ADL(ℝ) and that PFA plus a measurable yields an inner model of ADℝ containing all reals and ordinals. As we do here, these arguments made use of Tororcevic's work, so that logical strength is ultimately coming from a failure of covering for some appropriate core models.

In late 2000, A. S. Zoble and the author showed that (certain consequences of) Todorcevic's Strong Reflection Principle (SRP) imply ADL(ℝ). (See [22].) Since Martin's Maximum implies SRP, this gave the first derivation of ADL(ℝ) from an “unaugmented” forcing axiom.

Tarski's writings on the concepts of truth and logical consequence rank among the most influential works in both logic and philosophy of the twentieth century. Because of this, it would be impossible to give a careful and accurate account of how far that influence reaches and of the complex route by which it spread. In logic, Tarski's methods of defining satisfaction and truth, as well as his work pioneering general model-theoretic techniques, have been entirely absorbed into the way the subject is presently done; they have become part of the fabric of contemporary logic, material presented in the initial pages of every modern textbook on the subject. In philosophy, the influence has been equally pervasive, extending not only to work in semantics and the philosophies of logic and language, but to less obviously allied areas such as epistemology and the philosophy of science as well.

Rather than try to chart the wide-ranging influence of these writings or catalog the important research they have inspired, I will concentrate on various confusions and misunderstandings that continue to surround this work. For in spite of the extensive attention the work has received in the past fifty years, especially in the philosophical literature, misunderstandings of both conceptual and historical sorts are still remarkably widespread. Indeed in the philosophical community, recent reactions to Tarski's work on truth range from Karl Popper's “intense joy and relief” at Tarski's “legitimation” of the notion [1974, p. 399], to Hilary Putnam's assessment that “as a philosophical account of truth, Tarski's theory fails as badly as it is possible for an account to fail” [1985, p. 64]. Opinions have not exactly converged.

Several extensions of the basic modal language are characterized in terms of interpolation. Our main results are of the following form: Language L′ is the least expressive extension of L with interpolation. For instance, let ℳ(D) be the extension of the basic modal language with a difference operator [7], First-order logic is the least expressive extension of ℳ(D) with interpolation. These characterizations are subsequently used to derive new results about hybrid logic, relation algebra and the guarded fragment.