We show that the following results in the classical theory of unbounded linear operators on Hilbert spaces can be proved within the framework of Bishop's constructive mathematics: the Kato-Rellich theorem, the spectral theorem. Stone's theorem, and the self-adjointness of the most common quantum mechanical operators, including the Hamiltonians of electro-magnetic fields with some general forms of potentials.

The purpose of this paper is not to produce a survey of systems with a universal set: we do not yet understand them well enough. Rather it will concentrate on one particular aspect of them: the curious circumstance that there are half-a-dozen or so distinct proofs of ~ AC available in set theories with a universal set. This began to emerge in 1953 when Specker published in [2] a proof of ~ AC in Quine's system NF. Until recently this was an isolated phenomenon and poorly understood. The proof answered few questions and seemed rather ad hoc, thus inviting an investigation to determine whether this was an artefact caused by the particular axioms for NF, or part of a general conflict between the demands of big sets and AC.

We will start with an informal discussion of the genesis of set theories with a universal set, collecting, en route, a number of desiderata for such theories. A number of new refutations of AC in systems meeting some or all of these conditions will then be presented. The conclusion the reader is invited to draw is that any sensible set theory with a universal set will probably have trouble with AC.

Why do set theory with V e V at all? It contradicts conventional wisdom which teaches us that e is wellfounded. It must be said for this doctrine (let us abbreviate it to WOOF) that it has enabled the rapid execution of the logicist programme, so if that were the sole purpose of set theory we could consider ourselves well served. However most pure mathematicians are platonists and study things not because they are useful but simply because they are there. And the belief that V is not there can, after all, only be the result of early conditioning.

In this paper we present a new proof of a decidability result for the firstorder theories of certain subvarieties of Heyting algebras. By a famous result of Grzegorczyk, the full first-order theory of Heyting algebras is undecidable. In contrast, the first-order theory of Boolean algebras and of many interesting subvarieties of Boolean algebras is decidable by a result of Tarski [8]. In fact, Kozen [6] gives a comprehensive quantitative classification of the complexities of the first-order theories of various subclasses of Boolean algebras (including the full variety).

This stark contrast may be reconciled from the standpoint of universal algebra as arising out of the byplay between structure and decidability: A good structure theory entails positive decidability results. Boolean algebras have a well-developed structure theory [5], while the corresponding theory for Heyting algebras is quite meagre. Viewed in this way, we may hope to obtain decidability results if we focus attention on subclasses of Heyting algebras with good structural properties.

K. Idziak and P. M. Idziak [4] have considered an interesting subvariety of Heyting algebras, , which is the variety generated by all linearly-ordered Heyting algebras. This variety is shown to be the largest subvariety of Heyting algebras with a decidable theory of its finite members. However their proof is rather indirect, proceeding via semantic interpretation into the monadic second order theory of trees. The latter is a powerful theory—it interprets many other theories—but is computationally highly infeasible. In fact, by a celebrated theorem of Rabin, its complexity is not bounded by any elementary recursive function. Consequently, the proof of [4], besides being indirect, also gives no information on the quantitative computational complexity of the theory of .

Here we pursue the theme of structure and decidability. We isolate the indecomposable algebras in and use this to prove a theorem on the structure of if -algebras. This theorem relates the -algebras structurally to Boolean algebras. This enables us to bootstrap the known decidability results for Boolean algebras to the variety if .

We study amalgamation properties in a family of abstract elementary classes that we call coloring classes. The family includes the examples previously studied in [3]. We establish that the amalgamation property is equivalent to the disjoint amalgamation property in all coloring classes; find the Hanf number for the amalgamation property for coloring classes; and improve the results of [3] by showing, in ZFC, that the (disjoint) amalgamation property for classes Kα studied in that paper must hold up to ℶα (only a consistency result was previously known).

We say that a first order formula Φ distinguishes a structure M over a vocabulary L from another structure M′ over the same vocabulary if Φ is true on M but false on M′. A formula Φ defines an L-structure M if Φ distinguishes M from any other non-isomorphic L-structure M′. A formula Φ identifies an n-element L-structure M if Φ distinguishes M from any other non-isomorphic n-element L-structure M′.

We prove that every n-element structure M is identifiable by a formula with quantifier rank less than and at most one quantifier alternation, where k is the maximum relation arity of M. Moreover, if the automorphism group of M contains no transposition of two elements, the same result holds for definability rather than identification.

The Bernays-Schönfinkel class consists of prenex formulas in which the existential quantifiers all precede the universal quantifiers. We prove that every n-element structure M is identifiable by a formula in the Bernays-Schönfinkel class with less than quantifiers. If in this class of identifying formulas we restrict the number of universal quantifiers to k, then less than quantifiers suffice to identify M and. as long as we keep the number of universal quantifiers bounded by a constant, at total quantifiers are necessary.

The present note is conceived as a sequel to Mostowki's paper, Über die Unabhängigkeit des Wohlordnungssatzes vom Ordnungsprinzip (which will be quoted as “M.”).

In that paper Mostowski considers a system of axioms , very close to the system of axiomatic set theory of Bernays, but where the axiom of choice is not supposed to hold. Mostowski proves that the well-ordering theorem (Wohlordnungssatz) cannot be derived from the system plus the principle of simple ordering (Ordnungsprinzip).

Recall that a group G has finitely satisfiable generics (fsg) or definable f-generics (dfg) if there is a global type p on G and a small model $M_0$ such that every left translate of p is finitely satisfiable in $M_0$ or definable over $M_0$, respectively. We show that any abelian group definable in a p-adically closed field is an extension of a definably compact fsg definable group by a dfg definable group. We discuss an approach which might prove a similar statement for interpretable abelian groups. In the case where G is an abelian group definable in the standard model $\mathbb {Q}_p$, we show that $G^0 = G^{00}$, and that G is an open subgroup of an algebraic group, up to finite factors. This latter result can be seen as a rough classification of abelian definable groups in $\mathbb {Q}_p$.

In 2007, Terence Tao wrote on his blog an essay about soft analysis, hard analysis and the finitization of soft analysis statements into hard analysis statements. One of his main examples was a quasi-finitization of the infinite pigeonhole principle IPP, arriving at the “finitary” infinite pigeonhole principle FIPP1. That turned out to not be the proper formulation and so we proposed an alternative version FIPP2. Tao himself formulated yet another version FIPP3 in a revised version of his essay.

We give a counterexample to FIPP1 and discuss for both of the versions FIPP2 and FIPP3 the faithfulness of their respective finitization of IPP by studying the equivalences IPP ↔ FIPP2 and IPP ↔ FIPP3 in the context of reverse mathematics ([9]). In the process of doing this we also introduce a continuous uniform boundedness principle CUB as a formalization of Tao's notion of a correspondence principle and study the strength of this principle and various restrictions thereof in terms of reverse mathematics, i.e., in terms of the “big five” subsystems of second order arithmetic.

This paper proposes a generalization of several reducibilities in the sense of recursive function theory. For this purpose two structures are introduced as models of combinatory logic and reducibilities are defined in a rather natural way by means of the application operation in each model. The first model we consider is called the graph model and was introduced by Dana Scott in [11]. Reducibilities in this model are generalizations of enumeration and Turing reducibilities. The second model is called the hypergraph model and induces reducibilities which are generalizations of hyperenumeration and hyperarithmetical reducibilities.

The possibility of formulating reducibilities in this way is not new. For the graph model it is implicit in [7] and for the hypergraph model in [9]. On the other hand it is possible to look at the present method as a variant of the well-known technique of relativization in recursive function theory. We think that this does not exhaust the power of the method, which is conceptually elegant and provides a natural frame for the results of this paper.

In the first part of the paper we discuss the definition and general properties of the models. Then we introduce the reducibilities in the graph model and prove several theorems which are generalizations of properties already Known for enmeration and Turing reducibilities. Next we define reducibilities in the hypergraph model and try to extend the preceding results. For this purpose we prove two theorems showing significant relations between the operators in both models. In fact we prove that each operator in the hypergraph model can be simulated on a comeager set by an operator of the graph model.

Given a finite lexicon L of relational symbols and equality, one may view the collection of all L-structures on the set of natural numbers ω as a space in several different ways. We consider it as: (i) the space of outcomes of certain infinite two-person games; (ii) a compact metric space; and (iii) a probability measure space. For each of these viewpoints, we can give a notion of relative ubiquity, or largeness, for invariant sets of structures on ω. For example, in every sense of relative ubiquity considered here, the set of dense linear orderings on ω is ubiquitous in the set of linear orderings on ω.

In the fourth edition of Hilbert-Ackermann's Grundzüge der Theoretischen Logik (Springer-Verlag: Berlin-Göttingen-Heidelberg 1959) the following assertion is “proved” (p. 131):

Ein Ausdruck, der keine freien Individuenvariablen enthält, ist dann und nur dann allgemeingültig, wenn allgemeingültig ist. Die Ausdrücke, die die Identität enthalten, sind dabei eingeschlossen.

As a counter example one can cite: ∃x(¬∃xFx ∨ Fx), which is universally valid yet the generalized form: ∀x(¬∃xFx ∨ Fx) is not (e.g. let the domain of individuals be the natural numbers and let ‘Fx’ be ‘x is factorable’).

It may be of interest to see where the alleged proof went astray. It is argued (correctly) that any expression can only be derived by the given rules from a formula , or from this formula with any other variable in place of y. Moreover (still correctly) any upper formula of such a formula is of the form (“Upper formula” in the transitive sense of an upper formula of an upper formula being an upper formula). But then it is stated (incorrectly) that if is an upper formula of , then is an upper formula of . To show that this is incorrect we here append a proof of the universally valid formula mentioned above:

We give the natural topological model for ¬BD-ℕ, and use it to show that the closure of spaces with the anti-Specker property under product does not imply BD-ℕ. Also, the natural topological model for ¬BD is presented. Finally, for some of the realizability models known indirectly to falsify BD-ℕ, it is brought out in detail how BD-ℕ fails.

We study automorphism groups of randomizations of separable structures, with focus on the ℵ0-categorical case. We give a description of the automorphism group of the Borel randomization in terms of the group of the original structure. In the ℵ0-categorical context, this provides a new source of Roelcke precompact Polish groups, and we describe the associated Roelcke compactifications. This allows us also to recover and generalize preservation results of stable and NIP formulas previously established in the literature, via a Banach-theoretic translation. Finally, we study and classify the separable models of the theory of beautiful pairs of randomizations, showing in particular that this theory is never ℵ0-categorical (except in basic cases).

We study the effects of Vopěnka's principle on properties of model theoretic logics. We show that Vopěnka's principle is equivalent to the assumption that every finitely generated logic has a compact cardinal. We show also that it is equivalent to the assumption that every such logic has a global Hanf number.

Introduction. In [Sh, Chapter IX], Shelah constructs a model of set theory in which Suslin's hypothesis is true, yet there is an Aronszajn tree which is not special. In his model, we have . He asks whether the same result could be obtained consistently with CH. In this paper, we answer his question in the affirmative.

Let us say that a tree T is S-st-special iff there is a function ƒ with dom(f) = {t ∈ T: rank(t) ∈ S} and for every t1 < t2 both in dom(t) we have f(t2) ≠ f(t1) < rank(t1). In Shelah's model, every tree is S-st-special for some fixed stationary costationary set S. Also, there is some tree T such that T is not S′-st-special whenever S′ – S is stationary. These properties, which are sufficient to ensure that Suslin's hypothesis holds and that T is not special, also hold in the model constructed in this paper. These properties also ensure that every Aronszajn tree has a stationary antichain (i.e., an antichain A such that {rank(t): t ∈ A} is stationary). Hence, it is natural to ask whether there is a model of Suslin's hypothesis in which some Aronszajn tree has no stationary antichain. We answer in the affirmative in [S].

The construction we use owes much to Shelah's approach to the theorem, due to Jensen (see [DJ]), that CH is consistent with Suslin's hypothesis. This is given in [Sh, Chapter V].

A formalized theory is called complete if for each sentence expressible in this theory either the sentence itself or its negation is provable.

A theory is called deciddble if there exists an effective procedure (called decision-procedure) which enables one to decide of each sentence, in a finite number of steps, whether or not it is provable in the theory.

It is known that there exist complete but undecidable theories. There exist, namely, the so called essentially undecidable theories, i.e. theories which are undecidable and remain so after an arbitrary consistent extension of the set of axioms. Using the well-known method of Lindenbaum we can therefore obtain from each such theory a complete and undecidable theory.

The aim of this paper is to prove a theorem which shows that complete theories satisfying certain very general conditions are always decidable. In somewhat loose formulation these conditions are: There exist four effective methods M1, M2, M3, M4, such that

(a) M1 enables us to decide in each case whether or not any given formula is a sentence of the theory;

(b) M2 gives an enumeration of all axioms of the theory;

(c) the rules of inference can be arranged in a sequence R1, R2, … such that if p1, … pk, r are arbitrary sentences of the theory, we can decide by M3 whether or not r results from p1, … pk, by the n-th rule;

(d) M4 enables us to construct effectively the negation of each effectively given sentence.

In order to express these conditions more precisely we shall make use of an arithmetization of the considered theory .

A distinctive feature of modern mathematics is the interaction between its various branches and the blurring of the boundaries between different areas. This is strikingly illustrated in the work of Alfred Tarski. He was a logician first and an algebraist second. His contributions to algebra can be divided into three (ill-defined and overlapping) categories, general algebra, the study of various algebraic structures arising from problems outside algebra, mostly in logic and set theory, and the use of concepts and techniques from logic in the study of algebraic structures. Even more roughly, these three categories could be labeled as pure algebra, applications of algebra to logic, and applications of logic to algebra.

Before Tarski came to the United States in 1939, he had written a series of papers on both the axiomatic and the structural aspects of Boolean algebras, and his inclination to algebraize mathematical problems is well illustrated by his paper [38g], Algebraische Fassung des Massproblems. Many of his later investigations of various types of algebraic structures are inspired by work done in this earlier period. However, beginning around 1940 there is a much greater emphasis on the study of algebra in its various aspects.

The paper [41], On the calculus of relations, is a landmark event in this respect. The object here was to find an axiomatic basis for the arithmetic of binary relations. The axioms that he chose are simple and natural (see Monk [1986]).

We extend the characterization of extremal valued fields given in [2] to the missing case of valued fields of mixed characteristic with perfect residue field. This leads to a complete characterization of the tame valued fields that are extremal. The key to the proof is a model theoretic result about tame valued fields in mixed characteristic. Further, we prove that in an extremal valued field of finite p-degree, the images of all additive polynomials have the optimal approximation property. This fact can be used to improve the axiom system that is suggested in [8] for the elementary theory of Laurent series fields over finite fields. Finally we give examples that demonstrate the problems we are facing when we try to characterize the extremal valued fields with imperfect residue fields. To this end, we describe several ways of constructing extremal valued fields; in particular, we show that in every ℵ1 saturated valued field the valuation is a composition of extremal valuations of rank 1.