The question of continuity of functions defined on Baire space NN or on the reals has been of great interest to constructivists. We have little hope of answering the question without a fundamental philosophical analysis of the notion of constructive function. However, one can ask whether the continuity of functions can be derived in known intuitionistic formal systems, and answer these questions by mathematical means.

The constructivist does not generally wish to restrict himself to reals (or members of NN ) given by a recursive law; therefore the most natural formal systems have variables for reals, sequences and functions. The axioms of these systems turn out to be compatible with the assumption that every real or sequence is given by a recursive law, and every function by an effective operation. An underivability result, therefore, is only strengthened if this assumption is taken as an axiom; and once this is done, we may as well work with (indices of) effective operations, within arithmetic.

By KLS is meant the assertion that each effective operation on Baire space NN is continuous. In [B1] it is shown that this statement cannot be proved in various intuitionistic formal systems. The continuity of functions on the reals has been of even greater interest to constructivists than the continuity of functions on NN ; by KLS(R) is meant the assertion that each effective operation from the real numbers R to R is continuous. It is the purpose of the present note to show that KLS(R) is also underivable in intuitionistic formal systems.

In the original example of a transfinite hierarchy of degrees of unsolvability, a predicate Ha is associated with each a in the set 0 of ordinal notations. (See Kleene [K1], Spector [S1], and the references there to Davis and Mostowski.) The predicates are defined by means of induction over the partial well-ordering relation ≤0 on the set 0 of notations for the recursive ordinals. The usefulness of this hierarchy of predicates is enhanced by Spector's proof [S1] of the “uniqueness” property: viz. that the degree of Ha depends only on the ordinal |a| for which a is a notation.

We give here alternative definitions for the notions that S. Shelah has introduced in recent papers: the dimensional order property and the depth of a theory. We will also give a proof that the depth of a countable theory, when denned, is an ordinal recursive in T.

Let ω be the set of natural numbers, i.e. {0, 1, 2, 3, …}. A string is a mapping from an initial segment of ω into {0, 1}. We identify a set A ≤ ω with its characteristic function. A set A ≤ ω is called n-generic if it is Cohen-generic for n-quantifier arithmetic. This is equivalent to saying that for every set of strings S, there is a σ < A such that σ ∈ S or (∀ν ≥ σ)(ν ∉ S). By degree we mean Turing degree (of unsolvability). We call a degree n-generic if it has an n-generic representative. For a degree a, D(≤ a) denotes the set of degrees recursive in a.

The relation between generic degrees and minimal degrees has been widely studied. Spector [9] proved the existence of minimal degrees. Shoenfield [8] simplified the proof by using trees. In the construction of a minimal degree, given σ we extend σ to ν so that ν is in the (splitting or nonsplitting) subtree of a given tree. But in the construction of a generic set, given σ we extend σ to ν to meet the given dense set. So these two constructions are quite different. Jockusch [5] showed that any 2-generic degree bounds no minimal degree. Chong and Jockusch [3] showed that any 1-generic degree below 0′ bounds no minimal degree.

We use NF to designate the system known as Quine's New Foundations (see [1] and [2]), and NF + AF to designate the same system with a suitable axiom of infinity adjoined. We use ML to designate the revised system appearing in the third printing of Quine's “Mathematical Logic” (see [3]). This system ML is just the system P proposed by Wang in [4], and essentially includes NF as a part.

The pripcipal results of the present paper are:

A. In NF the axiom of infinity is equivalent to the definability of an ordered pair having the same type as its constituents.

B. Considering the formulas of NF as having translations in the formalism of ML in the sense defined in [4], we find that a formula of NF is provable in NF if and only if its translation is provable in ML. From the fact that the axiom of infinity is provable in ML, one might be tempted to conclude that the axiom of infinity is provable in NF. It is explained why this is not a valid inference. Consequently, it would appear that there is a sense in which NF + AF is stronger than ML.

The following is a contribution to the abstract study of the model theory of modal logics. Historically, individual modal logics have been specified deductively; and, as a result, it has seemed natural to view modal logics as sets of sentences provable in some deductive system. This proof theoretic view has influenced the abstract study of modal logics. For example, Fine [1975] defines a modal logic to be any set of sentences in the modal language L□ which contains all tautologies, all instances of the schema (□(ϕ ⊃ Ψ) ⊃ (□ϕ ⊃ □Ψ)), and which is closed under modus ponens, necessitation and substitution.

Here, however, modal logics are equated with classes of “possible world” interpretations. “Worlds” are taken to be ordered pairs (a, λ), where a is a sentential interpretation and λ is an ordinal. Properties of the accessibility relation are identified with those classes of binary relational systems closed under isomorphisms. The origin of this approach is the study of alternative Kripke semantics for the normal modal logics (cf. Weaver [1973]). It is motivated by the desire that modal logics provide accounts of both logical truth and logical consequence (cf. Corcoran and Weaver [1969]) and the realization that there are alternative Kripke semantics for S4, B and M which give identical accounts of logical truth, but different accounts of logical consequence (cf. Weaver [1973]). It is shown that the Craig interpolation theorem holds for any modal logic which has characteristic modal axioms and whose associated class of binary relational systems is closed under subsystems and finite direct products. The argument uses a back and forth construction to establish a modal analogue of Robinson's theorem. The argument for the interpolation theorem from Robinson's theorem employs modal analogues of the Ehrenfeucht-Fraïssé characterization of elementary equivalence and Hintikka's distributive normal form, and is itself a modal analogue of a first order argument (cf. Weaver [1982]).

Since its introduction in [K1-Po], the upper semilattice of Turing degrees has been an object of fascination to practitioners of the recursion-theoretic art. Starting from relatively simple concepts and definitions, it has turned out to be a structure of enormous complexity and richness. This paper is a contribution to the ongoing study of this structure.

Much of the work on Turing degrees may be formulated in terms of the embeddability of certain first-order structures in a structure whose universe is some set of degrees and whose relations, functions, and constants are natural degree-theoretic ones. Thus, for example, we know that if {P, ≤P) is a partial ordering of cardinality at most ℵ1 which is locally countable—each point has at most countably many predecessors—then there is an embedding

where D is the set of all Turing degrees and <T is Turing reducibility. If (P, ≤P) is a countable partial ordering, then the image of the embedding may be taken to be a subset of R, the set of recursively enumerable degrees. Without attempting to make the notion completely precise, we shall call embeddings of the first sort global, in contrast to local embeddings which impose some restrictions on the image set.

The study of partition relations for cardinal numbers introduced by Erdös and his school in the 1950's has, for the past several years, had a profound impact in logic. Unfortunately, quite early in their development, it was noticed by Rado [1] that the potentially most fruitful class of such relations, infinite exponent partition relations, were always in contradiction with the axiom of choice (AC). As a result, such relations were overlooked. This turned out to be a mistake; for, as has been noticed recently, a close study of infinite exponent partition relations is both interesting and rewarding. For example, there are weakened versions of such relations which are provable in ZF and which have valuable applications in recursion theory and set theory. In addition, the pure theory of these relations, like that of the axiom of determinateness, is fruitful as well as elegant. For more background here one should refer to [3].

At any rate, with a more detailed look at infinite exponent partition relations came a more refined version of Rado's original theorem. Specifically, Rado used the full axiom of choice to carefully construct partitions to violate any desired relation—a more sophisticated look at the actual theory of such relations indicated how one could put together some desired partitions using only well-ordered choice [3].

The distinction between well-ordered choice and full choice is by no means vacuous in this context. For Mathias has shown [4] that the simplest infinite exponent partition relation, ω → (ω)ω, is consistent with countable choice (well-ordered choice of length ℵ0) and, in fact, is consistent with dependent choice.

There seem to be at least two approaches to the problem of generating notations for ordinals: the papers of Bachmann [2], Gerber [5], Feferman [3], and Isles [6] provide examples of one and those of Ackermann [1], Takeuti [10], Kino [7], Schütte [9], and Pfeiffer [8] the other. In both of these the normal forms produced can be regarded as being finite trees “based on” some ordinal π.

Let be the set of recursive functions computable by machines using at most t(x) computation steps on argument x, for all but finitely many inputs x. We call t a name for the complexity class . Suppose we allow our machines to run longer, say h(x, t(x)) steps on argument x, where h is some fixed recursive function. One might expect that, for large enough h, permitting our machines to run longer by an amount h will always allow us to compute new functions, i.e., is a proper subset of . This turns out not to be the case: The “gap theorem” ([2], [3]) implies that for every recursive h there exists a recursive t such that . However, if we restrict our attention to names from a certain subclass of the recursive functions, then we can indeed uniformly increase the size of our -classes. Informally, we call a recursive function t “honest” if some machine computes t(x) in roughly t(x) steps for each argument x. (A precise definition is given in Definition 1 below.) Then according to the “compression theorem” [1], there exists a single recursive function h such that, for every honest t, is a proper subset of . Thus the phenomenon of the gap theorem is avoided by restricting attention to honest functions. It is a surprising consequence of the “honesty theorem” of McCreight and Meyer ([4], [5]) that there is no loss of generality in this restriction. Namely, for any recursive function t there is an honest recursive function t′ such that .

The usefulness of measurable cardinals in set theory arises in good part from the fact that an ultraproduct of wellfounded structures by a countably complete ultrafilter is wellfounded. In the standard proof of the wellfoundedness of such an ultraproduct, one first shows, without any use of the axiom of choice, that the ultraproduct contains no infinite descending chains. One then completes the proof by noting that, assuming the axiom of choice, any partial ordering with no infinite descending chain is wellfounded. In fact, the axiom of dependent choices (a weakened form of the axiom of choice) suffices. It is therefore of interest to ask whether some use of the axiom of choice is needed in order to prove the wellfoundedness of such ultraproducts or whether, on the other hand, their wellfoundedness can be proved in ZF alone. In Theorem 1, we show that the axiom of choice is needed for the proof (assuming the consistency of a strong partition relation). Theorem 1 also contains some related consistency results concerning infinite exponent partition relations. We then use Theorem 1 to show how to change the cofinality of a cardinal κ satisfying certain partition relations to any regular cardinal less than κ, while introducing no new bounded subsets of κ. This generalizes a theorem of Prikry [5].

Two questions on rings of strictly upper triangular matrices arising from B. Rose's work [5] are answered in this paper. An n × n matrix (αi, j) is strictly upper triangular if αi, j = 0 whenever i ≥ j. The ring of strictly upper triangular n × n matrices with entries from a field F will be denoted by Sn(F). Throughout this paper n will be an integer greater than 2. B. Rose [5] has shown that the complete theory of Sn(F) for an algebraically closed field F is ℵ1categorical. The first main result of this paper is that the rings Sn(F) and Sn(K) for fields F and K are isomorphic or elementarily equivalent if and only if F and K are isomorphic or elementarily equivalent, respectively (Corollary 1.6 and Theorem 2.2). This result shortens the proof of B. Rose's categoricity theorem [5, Theorem 7] by avoiding the co-stability considerations; furthermore, this result yields a proof of the converse of this categoricity theorem. The second main result is that the theory of rings of strictly upper triangular n × n matrices over algebraically closed fields is the model-completion of the theory of rings of strictly upper triangular n × n matrices over arbitrary fields (Theorem 2.5). This answers affirmatively the two conjectures at the end of [5].

We prove a number of results about countable Borel equivalence relations with forcing constructions and arguments. These results reveal hidden regularity properties of Borel complete sections on certain orbits. As consequences they imply the nonexistence of Borel complete sections with certain features.

Let Dtt denote the set of truth-table degrees. A bijection π: Dtt → Dtt is an automorphism if for all truth-table degrees x and y we have x ≤tty ⇔ π(x) ≤ttπ(y). We say an automorphism π is fixed on a cone if there is a degree b such that for all x ≥ttb we have π(x) = x. We first prove that for every 2-generic real X we have X′ ≰ttX ⊕ 0′. We next prove that for every real X ≥tt 0′ there is a real Y such that Y ⊕ 0′ ≡ttY′ ≡ttX. Finally, we use this to demonstrate that every automorphism of the truth-table degrees is fixed on a cone.

Let TP be the theory obtained by adding a generic predicate to an o-minimal theory T. We prove that if T admits elimination of imaginaries, then TP also admits elimination of imaginaries.

In this paper we continue our investigation of the Dedekind cardinals which was initiated in [2]. Those results are summarized below. Let ω be the finite cardinals and Δ the Dedekind cardinals. In [7] Myhill defined a class of functions ƒ: Χκω→ω, which he called the combinatorial functions, and which he applied to the study of recursive equivalence types.

In this paper we consider some axiomatic systems of set theory related to the system NF (New Foundations) of Quine. In particular we discuss the possible relations of cardinality between a finite set x and its subset class SC(x) = {y | y ∩ x} and also between x and its unit set class USC(x) = {{y} | y ε x}. Specker [5] has shown that in NF the cardinal of a finite set x can never be the same as the cardinal of SC(x).

Let φ be a monadic second order sentence about a finite structure from a class which is closed under disjoint unions and has components. Compton has conjectured that if the number of n element structures has appropriate asymptotics, then unlabelled (labelled) asymptotic probabilities ν(φ) (μ(φ) respectively) for φ always exist. By applying generating series methods to count finite models, and a tailor made Tauberian lemma, this conjecture is proved under a mild additional condition on the asymptotics of the number of single component -structures. Prominent among examples covered, are structures consisting of a single unary function (or partial function) and a fixed number of unary predicates.