Cofinal Indiscernibles and some Applications to New Foundations

We prove a theorem about models with indiscernibles that are cofinal in a given linear order. We apply this theorem to obtain new independence results for Quine's set theory New Foundations, thus solving two open problems in this field. Mathematics Subject Classification : 03C55, 03B15, 03B60, 03E70.     

A theory of sets with the negation of the axiom of infinity

In this paper we introduce a theory of finite sets FST with a strong negation of the axiom of infinity asserting that every set is provably bijective with a natural number. We study in detail the role of the axioms of Power Set, Choice, Regularity in FST, pointing out the relative dependences or independences among them. FST is shown to be provably equivalent to a fragment of Alternative Set Theory. Furthermore, the introduction of FST is motivated in view of a non-standard development. MSC: 03E30, 03E35.     

Ultrapowers as sheaves on a category of ultrafilters

In the paper we investigate the topos of sheaves on a category of ultrafilters. The category is described with the help of the Rudin-Keisler ordering of ultrafilters. It is shown that the topos is Boolean and two-valued and that the axiom of choice does not hold in it. We prove that the internal logic in the topos does not coincide with that in any of the ultrapowers. We also show that internal set theory, an axiomatic nonstandard set theory, can be modeled in the topos.Mathematics Subject Classification (2000): Primary 03G30, 03C20, Secondary 03E05, 03E70, 03H05The author would like to thank the Mittag-Leffler Institute for partial suport.     

Imaginaries in Hilbert spaces

We characterise imaginaries (up to interdefinability) in Hilbert spaces using a Galois theory for compact unitary groups.The authors would like to thank Frank Wagner and the CIRM for their hospitality during the Simpleton 2002 meeting during which the discussions that led to this paper took place.The first author would like to thank Ilan Hirshberg for a few important hintsAt the time of the writing of this paper, the first author was a graduate student with the Équipe de logique mathématique of Université Paris VIIMathematics Subject Classification (2000): 03C45, 03C95     

Computable limits and colimits in categories of partial enumerated sets

Computable limits and colimits are “recursive counterparts” of the suitable classical concepts from category theory. We present mainly some interesting problems related to computable products. Moreover, some “computable counterparts” of well-known classical facts from category theory are given. MSC: 03D45, 18A30.     

The Fields of Real and Complex Numbers with a Small Multiplicative Group

We consider the model theory of the real and complex fieldswith a multiplicative group having the Mann property. Amongthese groups are the finitely generated multiplicative groupsin these fields. As a by-product we obtain some results on groupswith the Mann property in rings of Witt vectors and in fieldsof positive characteristic.k 2000 Mathematics Subject Classification03C10, 03C35, 03C60, 03C64, 03C98, 13K05.     

Hilbert's ϵ-operator in intuitionistic type theories

We investigate Hilbert's ?-calculus in the context of intuitionistic type theories, that is, within certain systems of intuitionistic higher-order logic. We determine the additional deductive strength conferred on an intuitionistic type theory by the adjunction of closed ?-terms. We extend the usual topos semantics for type theories to the ?-operator and prove a completeness theorem. The paper also contains a discussion of the concept of “partially defined” ?-term. MSC: 03B15, 03B20, 03G30.     

Adding Skolem functions to simple theories

We examine the conditions under which we can keep simplicity or categoricity after adding a Skolem function to the theory.Mathematics Subject Classification (2000): 03C45, 03C50     

Precovers,Modalities and Universal Closure Operators in a Topos

In this paper we develop the notion of formal precover in a topos by defining a relation between elements and sets in a local set theory. We show that such relations are equivalent to modalities and to universal closure operators. Finally we prove that these relations are well characterized by a convenient restriction to a particular set. Mathematics Subject Classification: 03G30, 03B20, 03F55.     

A Note on the Theorems of Church-Turing and Trachtenbrot

We sketch proofs of the theorems of Church-Turing and Trachtenbrot using a semi-monomorphic axiomatization. Mathematics Subject Classification : 03B10, 03B25.     

Recursion in Partial Type-1 Objects With Well-Behaved Oracles

We refine the definition of II-computability of [12] so that oracles have a “consistent”, but natural, behaviour. We prove a Kleene Normal Form Theorem and closure of semi-recursive relations under ?¹. We also show that in this more inclusive computation theory Post's theorem in the arithmetical hierarchy still holds. Mathematics Subject Classification: 03D65, 03D75.     

Translations of Logical Formulas and the Equiconsistency Problem

A translation of formulas in a language L₁ to formulas in a language L₂ is a mapping which preserves the parameters and commutes with the substitution prefix, the propositional connectives and the quantifiers. Every translation generates a corresponding transformation of theories in L₁ to theories in L₂. We formulate the equiconsistency problem for such transformations and propose a variant of its solution. First, for a transformation F we find the least theory A(F) in L₁ such that its inclusion in a theory T (in L₁) guarantees equiconsistency of F and F(T), then we propose axiomatizations of A(F) for some F's. Mathematics Subject Classification: 03B10, 03B15, 03F25.     

ITERATIONS OF SATISFACTION CLASSES AND MODELS OF PEANO ARITHMETIC

We consider iterations of satisfaction classes and apply them to construct expansions of models of Peano arithmetic to models of A|Δ+∑-AC. 1991 MSC: 03F35, 03C62.     

A New Modal Lindström Theorem

We prove new Lindstr?m theorems for the basic modal propositional language, and for some related fragments of first-order logic. We find difficulties with such results for modal languages without a finite-depth property, high-lighting the difference between abstract model theory for fragments and for extensions of first-order logic. In addition we discuss new connections with interpolation properties, and the modal invariance theorem. Mathematics Subject Classification (2000): Primary 03B45; Secondary 03C95     

Consequences of Schanuel's condition for zeros of exponential terms

Assuming “Schanuel's Condition” for a certain class of exponential fields, Sturm's technique for polynomials in real closed fields can be extended to more complicated exponential terms in the corresponding exponential field. Hence for this class of terms the exact number of zeros can be calculated. These results give deeper insights into the model theory of exponential fields. MSC: 03C65, 03C60, 12L12.     

Abelian-by-G Groups,for G Finite,from the Model Theoretic Point of View

Let G be a finite group. We prove that the theory af abelian-by-G groups is decidable if and only if the theory of modules over the group ring ?[G] is decidable. Then we study some model theoretic questions about abelian-by-G groups, in particular we show that their class is elementary when the order of G is squarefree. Mathematics Subject Classification: 03C60, 03B25.     

Dissipative Transport: Thermal Contacts and Tunnelling Junctions

The general theory of simple transport processes between quantum mechanical reservoirs is reviewed and extended. We focus on thermoelectric phenomena, involving exchange of energy and particles. The theory is illustrated on the example of two reservoirs of free fermions coupled through a local interaction. We construct a stationary state and determine energy and particle currents with the help of a convergent perturbation series. We explicitly calculate several interesting quantities to lowest order, such as the entropy production rate, the resistance, and the heat conductivity. Convergence of the perturbation series allows us to prove that they are strictly positive under suitable smallness and regularity assumptions on the interaction between the reservoirs. Communicated by Gian Michele Graf submitted 15/01/03, accepted: 25/02/03     

Trivial, strongly minimal theories are model complete after naming constants

We prove that if is any model of a trivial, strongly minimal theory, then the elementary diagram is a model complete -theory. We conclude that all countable models of a trivial, strongly minimal theory with at least one computable model are -decidable, and that the spectrum of computable models of any trivial, strongly minimal theory is .

     

Sweet &; sour and other flavours of ccc forcing notions

We continue developing the general theory of forcing notions built with the use of norms on possibilities, this time concentrating on ccc forcing notions and classifying them.The first author thanks the Hebrew University of Jerusalem for its hospitality during his visits to Jerusalem. His research was also partially supported by a grant from the University Committee on Research of UNOThe research of the second author was partially supported by the Israel Science Foundation. Publication 672Mathematics Subject Classification (2000): Primary 03E35 Secondary; 03E40, 03E05Revised version: 30 September 2003     

Weak Forms of the Axiom of Choice and the Generalized Continuum Hypothesis

In this paper we study some statements similar to the Partition Principle and the Trichotomy. We prove some relationships between these statements, the Axiom of Choice, and the Generalized Continuum Hypothesis. We also prove some independence results. MSC: 03E25, 03E50, 04A25, 04A50.