We prove that in the Solovay model, every OD equivalence relation, Ε, over the reals, either admits an OD reduction to the equality relation on the set of all countable (of length < ω1) binary sequences, or continuously embeds Ε0, the Vitali equivalence.

If Ε is a (resp. ) relation then the reduction above can be chosen in the class of all Δ1 (resp. Δ2) functions.

The proofs are based on a topology generated by OD sets.

Scientific inquiry is represented as a process of rational hypothesis revision in the face of data. For the concept of rationality, we rely on the theory of belief dynamics as developed in [5, 9]. Among other things, it is shown that if belief states are left unclosed under deductive logic then scientific theories can be expanded in a uniform, consistent fashion that allows inquiry to proceed by any method of hypothesis revision based on “kernel” contraction. In contrast, if belief states are closed under logic, then no such expansion is possible.

Robert Solovay [8] investigated the version of the modal sentential calculus one gets by taking “□ϕ” to mean “ϕ is true in every transitive model of Zermelo-Fraenkel set theory (ZF).” Defining an interpretation to be a function * taking formulas of the modal sentential calculus to sentences of the language of set theory that commutes with the Boolean connectives and sets (□ϕ)* equal to the statement that ϕ* is true in every transitive model of ZF, and stipulating that a modal formula ϕ is valid if and only if, for every interpretation *, ϕ* is true in every transitive model of ZF, Solovay obtained a complete and decidable set of axioms.

In this paper, we stifle the hope that we might continue Solovay's program by getting an analogous set of axioms for the modal predicate calculus. The set of valid formulas of the modal predicate calculus is not axiomatizable; indeed, it is complete .

We also look at a variant notion of validity according to which a formula ϕ counts as valid if and only if, for every interpretation *, ϕ* is true. For this alternative conception of validity, we shall obtain a lower bound of complexity: every set which is in the set of sentences of the language of set theory true in the constructible universe will be 1-reducible to the set of valid modal formulas.

Some of the most striking results in modern set theory have emerged from the study of simply-definable sets of real numbers. Indeed, simple questions like: what are the posible cardinalities?, are they measurable?, do they have the property of Baire?, etc., cannot be answered in ZFC.

When one restricts the attention to the analytic sets, i.e., the continuous images of Borel sets, then ZFC does provide an answer to these questions. But this is no longer true for the projective sets, i.e., all the sets of reals that can be obtained from the Borel sets by taking continuous images and complements. In this paper we shall concentrate on particular projective classes, the , and using forcing constructions we will produce models of ZFC where, for some n, all , sets have some specified property. For the definition and basic facts about the projective classes , and , as well as the Kleene (or lightface) classes , and , we refer the reader to Moschovakis [19].

The first part of the paper is about measure and category. Early in this century, Luzin [16] and Luzin-Sierpiński [17] showed that all analytic (i.e., ) sets of reals are Lebesgue measurable and have the property of Baire.

Let G be a finite group. A Z [G]-lattice is a finitely generated Z-torsionfree module over the group ring Z [G]. There is a general conjecture concerning classes of modules over sufficiently recursive rings, and linking wildness and undecidability. Given a finite group G, Z [G] is sufficiently recursive, and our aim here is just to investigate this conjecture for Z [G]-lattices. In this setting, the conjecture says that

if and only if

In particular, we wish to deal here with the direction from the left to the right, so the one assuring that wildness implies undecidability. Of course, before beginning the analysis of this problem, one should agree upon a sharp definition of wildness for lattices. But, for our purposes, one might alternatively accept as a starting point a general classification of wild Z [G]-lattices when G is a finite p-group for some prime p, based on the isomorphism type of G. This is due to several authors and can be found, for instance, in [3]. It says that, when p is a prime and G is a finite p-group, then

if and only if

.

More precisely, the representation type of Z [G]-lattices is finite when G is cyclic of order ≤ p2, tame domestic when G is the Klein group [1], tame non-domestic when G is cyclic of order 8 [11].

So our claim might be stated as follows.

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].

We give two proofs of strong normalisation for second order classical natural deduction. The first one is an adaptation of the method of reducibility candidates introduced in [9] for second order intuitionistic natural deduction; the extension to the classical case requires in particular a simplification of the notion of reducibility candidate. The second one is a reduction to the intuitionistic case, using a Kolmogorov translation.