We introduce stable canonical rules and prove that each normal modal multi-conclusion consequence relation is axiomatizable by stable canonical rules. We apply these results to construct finite refutation patterns for modal formulas, and prove that each normal modal logic is axiomatizable by stable canonical rules. We also define stable multi-conclusion consequence relations and stable logics and prove that these systems have the finite model property. We conclude the paper with a number of examples of stable and nonstable systems, and show how to axiomatize them.

As Gödel [6] has pointed out, there is a certain interchangeability between the intuitionistic notion of proof and the notion of constructive functional of finite type. He achieves this interchange in the direction from logic to functionals by his functional interpretation of Heyting arithmetic H in a free variable theory T of primitive recursive functionals of finite type. In the present paper we shall extend Gödel's functional interpretation to the case in which H and T are extended by adding an abstract notion of constructive ordinal. In other words, we obtain the Gödel functional interpretation of an intuitionistic theory U of numbers (i.e., nonnegative integers) and constructive ordinals in a free variable theory V of finite type over both numbers and constructive ordinals. This allows us to obtain an analysis of noniterated positive inductive definitions [8].

The notion of constructive ordinal to be treated is as follows. There is given a function J which embeds the nonnegative integers in the constructive ordinals. A constructive ordinal of the form Jn is said to be minimal. There is also given a function δ which associates to each constructive ordinal Z and number n a constructive ordinal δZn which we denote by Zn. When Z is nonminimal, each Zn is called an immediate predecessor of Z. The basic principle for forming constructive ordinals says: for every function f from numbers n to constructive ordinals, there exists a constructive ordinal Z such that Zn = fn for all n. The principle of transfinite induction with respect to constructive ordinals says: if a property Q(Z) of constructive ordinals Z holds for minimal Z, and if ∀nQ(Zn) → Q(Z) holds for all Z, then Q(Z) holds for all Z.

We give a solution stated in the title to problem 3 of part 1 of the problems listed in the book of Eklof and Mekler [2], p. 453. There, in pp. 241-242, this is discussed and proved in some cases. The existence of strongly λ-free ones was proved earlier by the criteria in [5] and [3]. We can apply a similar proof to a large class of other varieties in particular to the variety of (non-commutative) groups.

Let X be a compact metric space. A closed set K ⊆ X is located if the distance function d(x, K) exists as a continuous real-valued function on X; weakly located if the predicate d(x, K) > r is allowing parameters. The purpose of this paper is to explore the concepts of located and weakly located subsets of a compact separable metric space in the context of subsystems of second order arithmetic such as RCA0, WKL0 and ACA0. We also give some applications of these concepts by discussing some versions of the Tietze extension theorem. In particular we prove an RCA0 version of this result for weakly located closed sets.

It is well known that classical propositional logic can be interpreted in intuitionistic prepositional logic. In particular Glivenko's theorem states that a formula is provable in the former iff its double negation is provable in the latter. We extend Glivenko's theorem and show that for every involutive substructural logic there exists a minimum substructural logic that contains the first via a double negation interpretation. Our presentation is algebraic and is formulated in the context of residuated lattices. In the last part of the paper, we also discuss some extended forms of the Koltnogorov translation and we compare it to the Glivenko translation.

The logic KM is the smallest normal modal logic that includes the McKinsey axiom

It is shown here that this axiom is not valid in the canonical frame for KM, answering a question first posed in the Lemmon-Scott manuscript [Lemmon, 1966].

The result is not just an esoteric counterexample: apart from interest generated by the long delay in a solution being found, the problem has been of historical importance in the development of our understanding of intensional model theory, and is of some conceptual significance, as will now be explained.

The relational semantics for normal modal logics first appeared in [Kripke, 1963], where a number of well-known systems were shown to be characterised by simple first-order conditions on binary relations (frames). This phenomenon was systematically investigated in [Lemmon, 1966], which introduced the technique of associating with each logic L a canonical frame which invalidates every nontheorem of L. If, in addition, each L-theorem is valid in , then L is said to be canonical. The problem of showing that L is determined by some validating condition C, meaning that the L-theorems are precisely those formulae valid in all frames satisfying C, can be solved by showing that satisfies C—in which case canonicity is also established. Numerous cases were studied, leading to the definition of a first-order condition Cφ associated with each formula φ of the form

where Ψ is a positive modal formula.

The stability of each of the theories of separably closed fields is proved, in the manner of Shelah's proof of the corresponding result for differentially closed fields. These are at present the only known stable but not superstable theories of fields. We indicate in §3 how each of the theories of separably closed fields can be associated with a model complete theory in the language of differential algebra. We assume familiarity with some basic facts about model completeness [4], stability [7], separably closed fields [2] or [3], and (for §3 only) differential fields [8].

Combining techniques of the first author and Shelah with ideas of Magidor, we show how to get a model in which, for fixed but arbitrary finite n, the first n strongly compact cardinals k1..…kn are so that ki; for i = 1..…n is both the ith measurable cardinal and supercompact. This generalizes an unpublished theorem of Magidor and answers a question of Apter and Shelah.

In 1960 G. F. Rose [R] made the following definition: A function f: ω → ω is (m, n)-computable, where 1 ≤ m ≤ n, iff there exists a recursive function R: ωn → ωn such that, for all n-tuples (x1,…, xn) of distinct natural numbers,

J. Myhill (see [McN, p. 393]) asked if f had to be recursive if m was close to n; B. A. Trakhtenbrot [T] responded by showing in 1963 that f is recursive whenever 2m > n. This result is optimal, because, for example, the characteristic function of any semirecursive set is (1,2)-computable. Trakhtenbrot's work was extended by E. B. Kinber [Ki1], using similar techniques. In 1986 R. Beigel [B] made a powerful conjecture, much more general than the above results. Partial verification, falling short of a full proof, appeared in [O]. Using new techniques, M. Kummer has recently established the conjecture, which will henceforth be referred to as the cardinality theorem (CT). It is the goal of this paper to show the connections between these various theorems, to review the methods used by Trakhtenbrot, and to use them to prove a special case of CT strong enough to imply Kinber's theorem (see §3). We thus have a hierarchy of results, with CT at the top. We will also include a discussion of Kummer's methods, but not a proof of CT.

A number of classical theories are interpreted in analogous theories that are based on intuitionistic logic. The classical theories considered include subsystems of first- and second-order arithmetic, bounded arithmetic, and admissible set theory.

Indescribability is closely related to the reflection principles of Zermelo-Fränkel set theory. In this axiomatic setting the universe of all sets stratifies into a natural cumulative hierarchy (Vα: α ϵ On) such that any formula of the language for set theory that holds in the universe already holds in the restricted universe of all sets obtained by some stage.

The axioms of ZF prove the existence of many ordinals α such that this reflection scheme holds in the world Vα. Hanf and Scott noticed that one arrives at a large cardinal notion if the reflecting formulas are allowed to contain second order free variables to which one assigns subsets of Vα. For a given collection Ω of formulas in the ϵ language of set theory with higher type variables and a unary predicate symbol they define an ordinal α to be Ω indescribable if for all sentences Φ in Ω and A ⊆ Vα

Since a sufficient coding apparatus is available, this definition is (for the classes of formulas that we are going to consider) equivalent to the one that one obtains by allowing finite sequences of relations over Vα, some of which are possibly k-ary. We will be interested mainly in certain standardized classes of formulas: Let (, respectively) denote the class of all formulas in the language introduced above whose prenex normal form has n alternating blocks of quantifiers of type m (i.e. (m + 1)th order) starting with ∃ (∀, respectively) and no quantifiers of type greater than m. In Hanf and Scott [1961] it is shown that in ZFC, indescribability is equivalent to inaccessibility and indescribability coincides with weak compactness.

We develop approximate counting of sets definable by Boolean circuits in bounded arithmetic using the dual weak pigeonhole principle (dWPHP(PV)), as a generalization of results from [15]. We discuss applications to formalization of randomized complexity classes (such as BPP, APP, MA, AM) in PV1 + dWPHP(PV).

Of the several methods for proving the completeness of sets of axioms for the prepositional calculus perhaps the simplest is due to Kalmár, although it does not appear to be widely known. In this paper we generalize Kalmár's method to indicate how to obtain a complete axiomatization of any fragment of the propositional calculus which includes material implication. We shall carry through the description and proofs for the case where, in addition to a symbol for implication, there is just one other primitive truth-function symbol. For systems in which there are no other function symbols, or more than one other such symbol, notational changes but not conceptual changes will be required.

In reverse mathematics, one formalizes theorems of ordinary mathematics in second order arithmetic and attempts to discover which set theoretic axioms are required to prove these theorems. Often, this project involves making choices between classically equivalent definitions for the relevant mathematical concepts. In this paper, we consider a number of equivalent definitions for the notions of well quasi-order and better quasi-order and examine how difficult it is to prove the equivalences of these definitions.

As usual in reverse mathematics, we work in the context of subsystems of second order arithmetic and take RCA0 as our base system. RCA0 is the subsystem formed by restricting the comprehension scheme in second order arithmetic to formulas and adding a formula induction scheme for formulas. For the purposes of this paper, we will be concerned with fairly weak extensions of RCA0 (indeed strictly weaker than the subsystem ACA0 which is formed by extending the comprehension scheme in RCA0 to cover all arithmetic formulas) obtained by adjoining certain combinatorial principles to RCA0. Among these, the most widely used in reverse mathematics is Weak König's Lemma; the resulting theory WKL0 is extensively documented in [11] and elsewhere.

We give three other combinatorial principles which we use in this paper. In these principles, we use k to denote not only a natural number but also the finite set {0, …, k − 1}.

Zest for both system and objectivity is the formal logician's original sin. He pays for it by constant frustrations and by living ofttimes the life of an intellectual outcaste. The task of squeezing a large body of stubborn facts into a more or less rigid system can be a painful one, especially since the facts of mathematics are among the most stubborn of all facts. Moreover, the more general and abstract we get, the farther removed we are from the raw mathematical experience. As intuition ceases to operate effectively, we fall into many unexpected traps. The formal logician gets little sympathy for his frustrations. He is regarded as too rigid by his philosophical colleagues and too speculative by his mathematical friends. The life of an intellectual outcaste may be a result partly of temperament and partly of the youthfulness of the logic profession. The unfortunate lack of wide appeal of logic may, however, be prolonged partly on account of the fact that very little of the well-established techniques of mathematics seems applicable to the treatment of serious problems of logic.

The axiomatic method is well suited to provide results which are both exact and systematic. How attractive would it be if we could get an axiom system in which all the axioms and deductions were intuitively clear and all theorems of mathematics were provable? Such a system would undoubtedly satisfy Descartes who admits solely intuition and deduction, which are, for him, the only “mental operations by which we are able, wholly without fear of illusion, to arrive at the knowledge of things.” Indeed, according to Descartes, intuition and deduction “are the most certain routes to knowledge, and the mind should admit no others. All the rest should be rejected as suspect of error and dangerous.”

One of the basic results in the theory of λ-conversion is the Church-Rosser Theorem, which says that, using certain rules for conversion and reduction of λ-formulae, any two interconvertible formulae can both be reduced to one formula. (I will not explain this in detail, as λ-conversion is described fully in Church's [2], where the Church-Rosser Theorem is Theorem 7 XXVII; see also Chapter 4 of Curry and Feys' [3].) The first part of the present paper contains an abstract form of this theorem.

We give a new, simplified and detailed account of the correspondence between levels of the Sherali–Adams relaxation of graph isomorphism and levels of pebble-game equivalence with counting (higher-dimensional Weisfeiler–Lehman colour refinement). The correspondence between basic colour refinement and fractional isomorphism, due to Tinhofer [22; 23] and Ramana, Scheinerman and Ullman [17], is re-interpreted as the base level of Sherali–Adams and generalised to higher levels in this sense by Atserias and Maneva [1] and Malkin [14], who prove that the two resulting hierarchies interleave. In carrying this analysis further, we here give (a) a precise characterisation of the level k Sherali–Adams relaxation in terms of a modified counting pebble game; (b) a variant of the Sherali–Adams levels that precisely match the k-pebble counting game; (c) a proof that the interleaving between these two hierarchies is strict. We also investigate the variation based on boolean arithmetic instead of real/rational arithmetic and obtain analogous correspondences and separations for plain k-pebble equivalence (without counting). Our results are driven by considerably simplified accounts of the underlying combinatorics and linear algebra.

Consider the predicate of natural numbers defined by:

where R is recursive. If, as usual, the variable ƒ ranges over ωω (the set of functions from natural numbers to natural numbers) then this is just the usual normal form for Π11 sets. If, however, ƒ ranges over 2ω (the set of functions from ω into {0, 1}) then every such predicate is recursively enumerable.3 Thus the second type of formula is generally ignored. However, the reduction just mentioned requires proof, and the proof uses some form of the Brower-König Infinity Lemma.

Let G be a separated (Hausdorff) topological group and let *G be an enlargement of G (see [8]). Thus, *G (i) possesses the same formal properties as G in the sense explained in [8], and (ii) every set of subsets {Aν } of G with the finite intersection property—i.e. such that every nonempty finite subset of {Aν } has a nonempty intersection—satisfies ∩*Aν ≠ ø, where the *Aν are the extensions of the Aν in *G, respectively.

We present a cut elimination argument that witnesses the conservativity of the compositional axioms for truth (without the extended induction axiom) over any theory interpreting a weak subsystem of arithmetic. In doing so we also fix a critical error in Halbach’s original presentation. Our methods show that the admission of these axioms determines a hyper-exponential reduction in the size of derivations of truth-free statements.