The Fraenkel-Mostowski method has been widely used to prove independence results among weak versions of the axiom of choice. In this paper it is shown that certain statements cannot be proved by this method. More specifically it is shown that in all Fraenkel-Mostowski models the following hold:

1. The axiom of choice for sets of finite sets implies the axiom of choice for sets of well-orderable sets.

2. The Boolean prime ideal theorem implies a weakened form of Sikorski's theorem.

Freivalds defined an acceptable programming system independent criterion for learning programs for functions in which the final programs were required to be both correct and “nearly” minimal size. i.e.. within a computable function of being purely minimal size. Kinber showed that this parsimony requirement on final programs limits learning power. However, in scientific inference, parsimony is considered highly desirable. A lim-computable function is (by definition) one calculable by a total procedure allowed to change its mind finitely many times about its output. Investigated is the possibility of assuaging somewhat the limitation on learning power resulting from requiring parsimonious final programs by use of criteria which require the final, correct programs to be “not-so-nearly” minimal size, e.g., to be within a lim-computable function of actual minimal size. It is shown that some parsimony in the final program is thereby retained, yet learning power strictly increases. Considered, then, are lim-computable functions as above but for which notations for constructive ordinals are used to bound the number of mind changes allowed regarding the output. This is a variant of an idea introduced by Freivalds and Smith. For this ordinal notation complexity bounded version of lim-computability, the power of the resultant learning criteria form finely graded, infinitely ramifying, infinite hierarchies intermediate between the computable and the lim-computable cases. Some of these hierarchies, for the natural notations determining them, are shown to be optimally tight.

In [2] John Burgess describes predicative versions of Frege's logic and poses the problem of finding their exact arithmetical strength. I prove here that PV, the simplest such theory, is equivalent to Robinson's arithmetical theory Q.

Let L be any finitary language. By restricting our attention to the universal Horn sentences of L and appealing to a semantical notion of logical consequence, we can formulate the universal Horn logic of L. The present paper provides some theorems about universal Horn logic that serve to distinguish it from the full first order predicate logic. Universal Horn equivalence between structures is characterized in two ways, one resembling Kochen's ultralimit theorem. A sharp version of Beth's definability theorem is established for universal Horn logic by means of a reduced product construction. The notion of a consistency property is relativized to universal Horn logic and the corresponding model existence theorem is proven. Using the model existence theorem another proof of the definability result is presented. The relativized consistency properties also suggest a syntactical notion of proof that lies entirely within the universal Horn logic. Finally, a decision problem in universal Horn logic is discussed. It is shown that the set of universal Horn sentences preserved under the formation of homomorphic images (or direct factors) is not recursive, provided the language has at least two unary function symbols or at least one function symbol of rank more than one.

This paper begins with a discussion of how algebraic relations between structures can be used to obtain fragments of a given logic. Only two such fragments seem to be under current investigation: equational logic and universal Horn logic. Other fragments which seem interesting are pointed out.

Let (K, v) be a henselian valued field of characteristic 0. Then K admits a definable partition on each piece of which the leading term of a polynomial in one variable can be computed as a definable function of the leading term of a linear map. The main step in obtaining this partition is an answer to the question, given a polynomial f(x) ∈ K[x], what is v(f(x))?

Two applications are given: first, a constructive quantifier elimination relative to the leading terms, suggesting a relative decision procedure; second, a presentation of every definable subset of K as the pullback of a definable set in the leading terms subjected to a linear translation.

This paper is a sequel to [1]. Our purpose is to exhibit certain basic relationships between (ordered dyadic) trees and regular nest structures. We introduce the notion of a tree being isomorphic to a nest structure, and we study some necessary and sufficient conditions for the existence of such isomorphisms. By virtue of some of our results, Kónig's lemma on infinite trees, and our fundamental lemma of [1] concerning infinite regular nest structures become intimately related — either yields an alternative proof of the other.

In this paper we give an introduction to categorical proof theory, and reinterpret, with improvements, Läuchli's work on abstract realizability restricted to propositional logic (but see [M1] for predicate logic). Partly to make some points of a foundational nature, we have included a substantial amount of background material. As a result, the paper is (we hope) readable with a knowledge of just the rudiments of category theory, the notions of category, functor, natural transformation, and the like. We start with an extended introduction giving the background, and stating what we do with a minimum of technicalities.

In three publications [L1, 2, 3] published in the years 1968, 1969 and 1972, J. Lambek gave a categorical formulation of the notion of formal proof in deductive systems in certain propositional calculi. The theory is also described in the recent book [LS]. See also [Sz].

The basic motivation behind Lambek's theory was to place proof theory in the framework of modern abstract mathematics. The spirit of the latter, at least for the purposes of the present discussion, is to organize mathematical objects into mathematical structures. The specific kind of structure we will be concerned with is category.

In Lambek's theory, one starts with an arbitrary theory in any one of several propositional calculi. One has the (formal) proofs (deductions) in the given theory of entailments A ⇒ B, with A and B arbitrary formulas. One introduces an equivalence relation on proofs under which, in particular, equivalent proofs are proofs of the same entailment; equivalence of proofs is intended to capture the idea of the proofs being only inessentially different. One forms a category whose objects are the formulas of the underlying language of the theory, and whose arrows from A to B, with the latter arbitrary formulas, are the equivalence classes of formal proofs of A ⇒ B.

In [2] T. J. Carlson introduces an approach to ordinal notation systems which is based on the notion of Σ1-elementary substructure. We gave a detailed ordinal arithmetical analysis (see [7]) of the ordinal structure based on Σ1-elementarily as defined in [2]. This involved the development of an appropriate ordinal arithmetic that is based on a system of classical ordinal notations derived from Skolem hull operators, see [6]. In the present paper we establish an effective order isomorphism between the classical and the new system of ordinal notations using the results from [6] and [7]. Moreover, on the basis of a concept of relativization we develop mutual (relatively) elementary recursive assignments which are uniform with respect to the underlying relativization.

In [1] Gandy established the following selection theorem for recursion in type-2 objects.

Theorem. Let F be a normal type-2 object. Then it is possible to select (uniformly and effectively in F) an integer from each nonempty set of integers semirecursive in F .

Notice that this really asserts that the predicates semirecursive in F are closed under existential quantification over type-0. Moschovakis [6] has essentially proven this theorem for F of arbitrary type.

In [2] Grilliot stated a powerful generalization of Gandy's result, namely:

Grilliot's Selection Theorem. Let F be a normal type-(n + 2) object (n an arbitrary integer). Then it is possible to select (uniformly and effectively in F) a nonempty recursive in F subset of each nonempty semirecursive in F set oftype-(n − 1) objects.

Notice again that this actually says that predicates semirecursive in F are closed under quantification over type-(n − 1) objects.

Despite the similarity of these two results, Gandy and Grilliot proposed rather different methods of proof. Furthermore, the proof that Grilliot presented in [2] contains an error which cannot easily be corrected. (We will comment on the nature of this error at the end of §1.) Fortunately, however, Grilliot's theorem is valid. We will present a proof of Grilliot's selection theorem which is a direct generalization of the proof of Gandy's theorem given in [6]. In fact, we will prove a general result (the theorem stated in §2) which subsumes both Gandy's and Grilliot's results.

This paper deals with some structural properties of the sequent calculus and describes strong symmetries between cut-free derivations and derivations, which do not make use of identity axioms. Both of them are discussed from a semantic and syntactic point of view.

Identity axioms and cuts are closely related to the treatment of negation in the sequent calculus, so the results of this article explain some nice symmetries of negation.

We show that the existence of a recursively enumerable set whose Turing degree is neither low nor complete cannot be proven from the basic axioms of first order arithmetic (P −) together with Σ2 -collection (BΣ2). In contrast, a high (hence, not low) incomplete recursively enumerable set can be assembled by a standard application of the infinite injury priority method. Similarly, for each n, the existence of an incomplete recursively enumerable set that is neither lown nor high n-1, while true, cannot be established in P− + BΣn+1 . Consequently, no bounded fragment of first order arithmetic establishes the facts that the highn and lown jump hierarchies are proper on the recursively enumerable degrees.

The literature on modal logic includes a number of general completeness and decidability results. The work of Schiller Joe Scroggs [5], R.A. Bull [1], Kit Fine [2], and Krister Segerberg [6] provide examples.

Scroggs showed that the proper extensions of S5 have the finite model property and are axiomatizable. (Harrop [3] then argued that logics having these properties are decidable.) Bull extended Scroggs' result by showing that the normal extensions of S4.3 have the finite model property. Fine subsequently provided a model-theoretic proof of Bull's result and also proved the axiomatizability of these logics. In a different direction Segerberg proved that every normal logic containing the characteristic axioms of Lewis' systems S4 and S5 is decidable.

The present paper is in this tradition. We extend the results of Scroggs and Segerberg by showing that every normal modal logic containing the S5 axiom has the finite model property, is axiomatizable, and thus is decidable.

Forcing extensions yield models of ZFC in which a long sequence of club subsets of ω1, has the following property: every subsequence of size ℵ1, has a finite intersection.

NFP is the predicative fragment of NF. In this system we do not allow a set to exist if it cannot be defined without using quantifiers ranging over its type or parameters of a higher type. NFI is a less restrictive fragment located between NFP and NF.

We show that NFP is really weaker than NFI; similarly, NFI is weaker than NF. This result will be obtained in the following manner: on the one hand, we will show that NFP can be proved consistent in elementary arithmetic and that second order arithmetic is interpretable in NFI; on the other hand, we will prove the consistency of NFI in third order arithmetic, which is contained in NF.

The paper is divided in four sections. In §1, we define the concepts needed and collect a few results together in such a way that they will be ready for later use. In §2, we will present a model-theoretic (quick) proof of the consistency of NFI (and thus of NFP). The proof will be chosen (it is not the quickest!) so as to motivate in a natural manner the details of the proof-theoretical version of it that will be presented in §3. §4 will be devoted to the axiom of infinity in NFP and NFI.

Motivated by problems that arise in computing degrees of belief, we consider the problem of computing asymptotic conditional probabilities for first-order sentences. Given first-order sentences φ and θ, we consider the structures with domain {1, …, N} that satisfy θ, and compute the fraction of them in which φ is true. We then consider what happens to this fraction as N gets large. This extends the work on 0-1 laws that considers the limiting probability of first-order sentences, by considering asymptotic conditional probabilities. As shown by Liogon'kiĭ [24], if there is a non-unary predicate symbol in the vocabulary, asymptotic conditional probabilities do not always exist. We extend this result to show that asymptotic conditional probabilities do not always exist for any reasonable notion of limit. Liogon'kiĭ also showed that the problem of deciding whether the limit exists is undecidable. We analyze the complexity of three problems with respect to this limit: deciding whether it is well-defined, whether it exists, and whether it lies in some nontrivial interval. Matching upper and lower bounds are given for all three problems, showing them to be highly undecidable.

In this paper we introduce a new natural deduction system for the logic of lattices, and a number of extensions of lattice logic with different negation connectives. We provide the class of natural deduction proofs with both a standard inductive definition and a global graph-theoretical criterion for correctness, and we show how normalisation in this system corresponds to cut elimination in the sequent calculus for lattice logic. This natural deduction system is inspired both by Shoesmith and Smiley's multiple conclusion systems for classical logic and Girard's proofnets for linear logic.

It is shown that if ZF is consistent, then so is ZFC + GCH + “There is a graph with cardinality ℵ2 and chromatic number ℵ2 such that every subgraph of cardinality ≤ ℵ1, has chromatic number ≤ ℵ0”. This partially answers a question of Erdös and Hajnal.

We give here a direct purely combinatorial proof that weak compactness is equivalent to a combinatorial property (2). This property (2) is apparently stronger, and from it, all other usual equivalent definitions and usual properties of weakly compact cardinals can be deduced. So this proof may be useful for books which want to present weakly compact cardinals, but not logic.

A direct simple proof of a weaker implication (e.g., weakly compact μ is not the first inaccessible, and every stationary set has an initial segment which is stationary) was given by Kunen [K] and independently by the author [Sh]. Baumgartner [B] had another proof.

We were motivated by the manuscript of Erdös, Hajnal, Mate and Rado's book on partition calculus, and by conversations with A. Levi who was writing a book on naive set theory.

Notation. Let i, j, α, β, γ, η, σ be ordinals, μ be a cardinal, f, g be functions.

Let cf α be the cofinality of α.

A partially ordered set T is a tree if for any a ∈ T, {b: b < a, b ∈ T} is well ordered; its order type (an ordinal) is called the level of a, and Tα is the set of a ∈ T of level a.