Modal reduction principles (MRPs) are modal formulas of the following form: Mp → Np, where M, N are (possibly empty) sequences of modal operators (i.e. □ or ◊). The notation M, N will be used to abbreviate such an MRP. We study a certain semantic correspondence between modal formulas and relational properties and obtain two main results. (1) On transitive semantic structures every MRP corresponds to a first-order relational property. (2) For the general case a syntactic criterion exists for distinguishing modal formulas with corresponding first-order properties from the others.

In this paper we define the notions of invariant implicit definability (i.i.d.) and semi-invariant implicit definability (s.i.i.d.) on ε-models, A, of a certain weak set theory. These notions are intended to be the analogs of recursiveness and recursive enumerability, respectively. Following Barwise, with each A is associated an infinitary language whose formulas are elements of A.

The extent to which we can grasp the content of a (non-logical) theory, expressing it economically by means of an axiom system or basis, varies greatly. In this paper we shall investigate what degree of economy, or at least regularity, can be achieved for all recursively axiomatizable first-order theories. A useful approach, also of interest in its own right, turns out to be the study of bases for subtheories, where a subtheory of a given theory consists of those theorems from which certain predicate symbols are absent. These predicate symbols might be thought of as the formal counterparts of the “purely theoretical” terms employed by a science, the theory corresponding to the science itself and the subtheory to its “observational consequences”. Roughly speaking, the types of operations involving such predicate symbols will be reduced to a minimum, so that their syntactical role in deductions will emerge more clearly.

A subspace V of an infinite dimensional fully effective vector space V∞ is called decidable if V is r.e. and there exists an r.e. W such that V ⊕ W = V∞. These subspaces of V∞ are natural analogues of recursive subsets of ω. The set of r.e. subspaces forms a lattice L(V∞) and the set of decidable subspaces forms a lower semilattice S(V∞). We analyse S(V∞) and its relationship with L(V∞). We show:

Proposition. Let U, V, W ∈ L(V∞) where U is infinite dimensional andU ⊕ V = W. Then there exists a decidable subspace D such that U ⊕ D = W.

Corollary. Any r.e. subspace can be expressed as the direct sum of two decidable subspaces.

These results allow us to show:

Proposition. The first order theory of the lower semilattice of decidable subspaces, Th(S(V∞), is undecidable.

This contrasts sharply with the result for recursive sets.

Finally we examine various generalizations of our results. In particular we analyse S*(V∞), that is, S(V∞) modulo finite dimensional subspaces. We show S*(V∞) is not a lattice.

Consider the following propositions:

(A) Every uncountable subset of contains an uncountable chain or antichain (with respect to ⊆).

(B) Every uncountable Boolean algebra contains an uncountable antichain (i.e., an uncountable set of pairwise incomparable elements).

Until quite recently, relatively little was known about these propositions. The oldest result, due to Kunen [4] and the author independently, asserts that if the Continuum Hypothesis (CH) holds, then (A) is false. In fact there is a counter-example 〈Aα: α < ω1〉 such that α < β implies Aβ −Aα is finite. Kunen also observed that Martin's Axiom (MA) + ¬CH implies that no such counterexample 〈Aα: α < ω1〉 exists.

Much later, Komjáth and the author [2] showed that ◊ implies the existence of several kinds of uncountable Boolean algebras with no uncountable chains or antichains. Similar results (but motivated quite differently) were obtained independently by Rubin [5]. Berney [3] showed that CH implies that (B) is false, but his algebra has uncountable chains. Finally, Shelah showed very recently that CH implies the existence of an uncountable Boolean algebra with no uncountable chains or antichains.

Except for Kunen's result cited above, the only result in the other direction was the theorem, due also to Kunen, that MA + ¬CH implies that any uncountable subset of with no uncountable antichains must have both ascending and decending infinite sequences under ⊆.

In 1936 Alonzo Church proposed the following thesis: Every effectively computable number-theoretic function is general recursive. The classical mathematician can easily give examples of nonrecursive functions, e.g. by diagonalizing a list of all general recursive functions. But since no such function has been found which is effectively computable, there is as yet no classical evidence against Church's Thesis.

The intuitionistic mathematician, following Brouwer, recognizes at least two notions of function: the free-choice sequence (or ordinary number-theoretic function, thought of as the ever-finite but ever-extendable sequence of its values) and the sharp arrow (or effectively definable function, all of whose values can be specified in advance).

We introduce new proof systems for propositional logic, simple deduction Frege systems, general deduction Frege systems, and nested deduction Frege systems, which augment Frege systems with variants of the deduction rule. We give upper bounds on the lengths of proofs in Frege proof systems compared to lengths in these new systems. As applications we give near-linear simulations of the propositional Gentzen sequent calculus and the natural deduction calculus by Frege proofs. The length of a proof is the number of lines (or formulas) in the proof.

A general deduction Frege proof system provides at most quadratic speedup over Frege proof systems. A nested deduction Frege proof system provides at most a nearly linear speedup over Frege system where by “nearly linear” is meant the ratio of proof lengths is O(α(n)) where α is the inverse Ackermann function. A nested deduction Frege system can linearly simulate the propositional sequent calculus, the tree-like general deduction Frege calculus, and the natural deduction calculus. Hence a Frege proof system can simulate all those proof systems with proof lengths bounded by O(n . α(n)). Also we show that a Frege proof of n lines can be transformed into a tree-like Frege proof of O(n log n) lines and of height O(log n). As a corollary of this fact we can prove that natural deduction and sequent calculus tree-like systems simulate Frege systems with proof lengths bounded by O(n log n).

In this paper the following theorem is proved regarding groups of finite Morley rank which are perfect central extensions of quasisimple algebraic groups.

Theorem 1. Let G be a perfect group of finite Morley rank and let C0be a definable central subgroup of G such that G/C0 is a universal linear algebraic group over an algebraically closed field; that is G is a perfect central extension of finite Morley rank of a universal linear algebraic group. Then C0 = 1.

Contrary to an impression which exists in some circles, the center of the universal extension of a simple algebraic group, as an abstract group, is not finite in general. Thus the finite Morley rank assumption cannot be omitted.

Corollary 1. Let G be a perfect group of finite Morley rank such that G/Z(G) is a quasisimple algebraic group. Then G is an algebraic group. In particular, Z(G) is finite([4], Section 27.5).

An understanding of central extensions of quasisimple linear algebraic groups which are groups of finite Morley rank is necessary for the classification of tame simple K*-groups of finite Morley rank, which constitutes an approach to the Cherlin-Zil’ber conjecture. For this reason the theorem above and its corollary were proven in [1] (Theorems 4.1 and 4.2) under the assumption of tameness, which simplifies the argument considerably. The result of the present paper shows that this assumption can be dropped. The main line of argument is parallel to that in [1]; the absence of the tameness assumption will be countered by a model-theoretic result and results from K-theory. The model-theoretic result places limitations on definability in stable fields, and may possibly be relevant to eliminating certain other uses of tameness.

In this paper, we give a constructive nonstandard model of intuitionistic arithmetic (Heyting arithmetic). We present two axiomatisations of the model: one finitary and one infinitary variant. Using the model these axiomatisations are proven to be conservative over ordinary intuitionistic arithmetic. The definition of the model along with the proofs of its properties may be carried out within a constructive and predicative metatheory (such as Martin-Löf's type theory). This paper gives an illustration of the use of sheaf semantics to obtain effective proof-theoretic results.

The axiomatisations of nonstandard intuitionistic arithmetic (to be called HAI and HAIω respectively) as well as their model are based on the construction in [5] of a sheaf model for arithmetic using a site of filters. In this paper we present a “minimal” version of this model, built instead on a suitable site of provable filter bases. The construction of this site can be viewed as an extension of the well-known construction of the classifying topos for a geometric theory which uses “syntactic sites”. (Such sites can in fact be used to prove semantical completeness of first order logic in a strictly constructive framework, see [6].)

We should mention that for classical nonstandard arithmetics there are several nonconstructive methods of proving conservativity over arithmetic, e.g. the compactness theorem, Mac Dowell–Specker's theorem [3].

Tarski published his first geometry paper, [24b], in 1924. As is well known, the area of the union of two disjoint figures is the sum of the areas of these two figures. This observation is the basis of a method for proving that two figures, say A and B, have the same area: if we can divide each of the two figures A and B into a finite number of pairwise disjoint subfigures A1,…,An and B1,…,Bn such that for every i, figures Ai and Bi are congruent (we say that two such figures are equivalent by finite decomposition), then figures A and B have the same area. The method is by no means universal. For example a disc and a rectangle can never be equivalent by finite decomposition, even if they have the same area. Hilbert [1922, Kapitel IV] proved from his axiom system the so-called De Zolt axiom:

If a polygon V is a proper subset of a polygon W then they are not equivalent by a finite decomposition.

Hilbert's proof is elementary but difficult. In [24b] Tarski gave an easy but nonelementary proof of a stronger version of the De Zolt axiom:

If a polygon V is a proper subset of a polygon W then they are not equivalent by finite decomposition into any figures.

It is shown that the first-order theory ThR(A) of a countable module over an arbitrary countable ring R is ℵ0-categorical if and only if Ai finite, n ∈ ω, κi ≤ ω. Furthermore, ThR(A) is ℵ0-categorical for all R-modules A if and only if R is finite and there exist only finitely many isomorphism classes of indecomposable R-modules.

We investigate various ways of introducing axioms for randomness in set theory. The results show that these axioms, when added to ZF, imply the failure of AC. But the axiom of extensionality plays an essential role in the derivation, and a deeper analysis may ultimately show that randomness is incompatible with extensionality.

We prove that for every family I of closed subsets of a Polish space each set can be covered by countably many members of I or else contains a nonempty set which cannot be covered by countably many members of I. We prove an analogous result for κ-Souslin sets and show that if A# exists for any A ⊂ ωω, then the above result is true for sets. A theorem of Martin is included stating that this result is also true for weakly homogeneously Souslin sets. As an application of our results we derive from them a general form of Hurewicz's theorem due to Kechris. Louveau, and Woodin and a theorem of Feng on the open covering axiom. Also some well-known theorems on finding “big” closed sets inside of “big” and are consequences of our results.

A recursive enumerator for a function h is an algorithm f which enumerates for an input x finitely many elements including h(x). f is a k(n)-enumerator if for every input x of length n. h(x) is among the first k(n) elements enumerated by f. If there is a k(n)-enumerator for h then h is called k(n)-enumerable. We also consider enumerators which are only A-recursive for some oracle A.

This paper is concerned with finding a fairly simple system of logic which is “basic” in the sense that every system of logic is definable in it. If a “system”is regarded as being a class of propositions, rather than a class of sentences, then every class of propositions which is a system should be definable in the basic logic. The system of logic proposed in this paper will not be proved to be a basic logic, but strong evidence that it is basic will be given at the end of the paper. Evidence will also be given that the class of propositions which are not provable in the system is not definable in any system of logic. It will be established that the decision problem is unsolvable for the system. Notable characteristics of the system are its lack of negation and universal quantification, and its similarity to systems proposed by Church, Curry, and Rosser.

By a “system” will be meant a class of propositions the membership of which can be specified by ordinary recursive methods, so that if the membership of a given proposition in the class can be established at all, such membership can be established in a finite number of steps. (This is not the same as demanding that a criterion must exist for determining, in a finite number of steps, whether or not some given proposition is a member of the class. Such a demand would require a solution of the decision problem for every system.)

Our task in the treatment of general set theory will be to give a survey for the purpose of characterizing the different stages and the principal theorems with respect to their axiomatic requirements from the point of view of our system of axioms. The delimitation of “general set theory” which we have in view differs from that of Fraenkel's general set theory, and also from that of “standard logic” as understood by most logicians. It is adapted rather to the tendency of von Neumann's system of set theory—the von Neumann system having been the first in which the possibility appeared of separating the assumptions which are required for the conceptual formations from those which lead to the Cantor hierarchy of powers. Thus our intention is to obtain general set theory without use of the axioms V d, V c, VI.

It will also be desirable to separate those proofs which can be made without the axiom of choice, and in doing this we shall have to use the axiom V*—i.e., the theorem of replacement taken as an axiom. From V*, as we saw in §4, we can immediately derive V a and V b as theorems, and also the theorem that a function whose domain is represented by a set is itself represented by a functional set; and on the other hand V* was found to be derivable from V a and V b in combination with the axiom of choice. (These statements on deducibility are of course all on the basis of the axioms I–III.)

We study the long-standing open problem of giving $\forall {\rm{\Sigma }}_1^b$ separations for fragments of bounded arithmetic in the relativized setting. Rather than considering the usual fragments defined by the amount of induction they allow, we study Jeřábek’s theories for approximate counting and their subtheories. We show that the $\forall {\rm{\Sigma }}_1^b$ Herbrandized ordering principle is unprovable in a fragment of bounded arithmetic that includes the injective weak pigeonhole principle for polynomial time functions, and also in a fragment that includes the surjective weak pigeonhole principle for FPNP functions. We further give new propositional translations, in terms of random resolution refutations, for the consequences of $T_2^1$ augmented with the surjective weak pigeonhole principle for polynomial time functions.

Let k and l be two multiplicatively independent integers, and let L ⊆ ℕn be a l-recognizable set which is not definable in 〈ℕ; +〉. We prove that the elementary theory of 〈ℕ; +, Vk, L〉, where Vk(x) denotes the greatest power of k dividing x, is undecidable. This result leads to a new proof of the Cobham-Semënov theorem.