Mathematics

The {\em tetravalent modal logic} ( TML ) is one of the two logics defined by Font and Rius (\cite{FR2}) (the other is the {\em normal tetravalent modal logic} TML N ) in connection with Monteiro's tetravalent modal algebras. These logics are expansions of the well--known {\em Belnap--Dunn's four--valued logic} that combine a many-valued character (tetravalence) with a modal character. In fact, TML is the logic that preserve degrees of truth with respect to tetravalent modal algebras. As Font and Rius observed, the connection between the logic TML and the algebras is not so good as in TML N , but, as a compensation, it has a better proof-theoretic behavior, since it has a strongly adequate Gentzen calculus (see \cite{FR2}). In this work, we prove that the sequent calculus given by Font and Rius does not enjoy the cut--elimination property. Then, using a general method proposed by Avron, Ben-Naim and Konikowska (\cite{Avron02}), we provide a sequent calculus for TML with the cut--elimination property. Finally, inspired by the latter, we present a {\em natural deduction} system, sound and complete with respect to the tetravalent modal logic.

In this paper, we study Cyclic Henkin Logic CHL, a logic that can be described as provability logic without the third Löb condition, to wit, that provable implies provably provable (aka principle 4). The logic CHL does have full modalised fixed points. We implement these fixed points using cyclic syntax, so that we can work just with the usual repertoire of connectives. The main part of the paper is devoted to developing the logic on cyclic syntax. Many theorems, like the multiple fixed point theorem, become matter of course in this context. We submit that the use of cyclic syntax is of interest even for the study of classical Löb's Logic. We show that a version of the de Jongh-Sambin algorithm can be seen as one half of a synonymy between the theory GL^\circ, i.e.\ CHL plus the third Löb Condition, and ordinary Löb's Logic GL. Our development illustrates that an appropriate computation scheme for the algorithm is guard recursion. We show how arithmetical interpretations work for the cyclic syntax. In an appendix, we give some further information about the arithmetical side of the equation.

We give a new proof of the fact that finite bipartite graphs cannot be axiomatized by finitely many first-order sentences among FINITE graphs. (This fact is a consequence of a general theorem proved by L. Ham and M. Jackson, and the counterpart of this fact for all bipartite graphs in the class of ALL graphs is a well-known consequence of the compactness theorem.) Also, to exemplify that our method is applicable in various fields of mathematics, we prove that neither finite simple groups, nor the ordered sets of join-irreducible congruences of slim semimodular lattices can be described by finitely many axioms in the class of FINITE structures. Since a 2007 result of G. Grätzer and E. Knapp, slim semimodular lattices have constituted the most intensively studied part of lattice theory and they have already led to results even in group theory and geometry. In addition to the non-axiomatizability results mentioned above, we present a new property, called Decomposable Cyclic Elements Property, of the congruence lattices of slim semimodular lattices.

Given a subset of X??R n we can associate with every point x??R n a vector space V of maximal dimension with the property that for some ball centered at x , the subset X coincides inside the ball with a union of lines parallel with V . A point is singular if V has dimension 0 . In an earlier paper we proved that a (R,+,<,Z) -definable relation X is actually definable in (R,+,<,1) if and only if the number of singular points is finite and every rational section of X is (R,+,<,1) -definable, where a rational section is a set obtained from X by fixing some component to a rational value. Here we show that we can dispense with the hypothesis of X being (R,+,<,Z) -definable by assuming that the components of the singular points are rational numbers. This provides a topological characterization of first-order definability in the structure (R,+,<,1) . It also allows us to deliver a self-definable criterion (in Muchnik's terminology) of (R,+,<,1) - and (R,+,<,Z) -definability for a wide class of relations, which turns into an effective criterion provided that the corresponding theory is decidable. In particular these results apply to the class of k??recognizable relations on reals, and allow us to prove that it is decidable whether a k??recognizable relation (of any arity) is l??recognizable for every base l?? .

The ordered structures of natural, integer, rational and real numbers are studied in this thesis. The theories of these numbers in the language of order are decidable and finitely axiomatizable. Also, their theories in the language of order and addition are decidable and infinitely axiomatizable. For the language of order and multiplication, it is known that the theories of N and Z are not decidable (and so not axiomatizable by any computably enumerable set of sentences). By Tarski's theorem, the multiplicative ordered structure of R is decidable also. In this thesis we prove this result directly by quantifier elimination and present an explicit infinite axiomatization. The structure of Q in the language of order and multiplication seems to be missing in the literature. We show the decidability of its theory by the technique of quantifier elimination and after presenting an infinite axiomatization for this structure, we prove that it is not finitely axiomatizable.

In our previous work, we proposed the logic obtained from full non-associative Lambek calculus by adding a sort of linear-logical modality. We call this logic non-associative non-commutative intuitionistic linear logic ( NACILL , for short). In this paper, we establish the decidability and undecidability results for various extensions of NACILL . Regarding the decidability results, we show that the deducibility problems for several extensions of NACILL with the rule of left-weakening are decidable. Regarding the undecidability results, we show that the provability problems for all the extensions of non-associative non-commutative classical linear logic by the rules of contraction and exchange are undecidable.

Beklemishev introduced an ordinal notation system for the Feferman-Schütte ordinal Γ 0 based on the autonomous expansion of provability algebras. In this paper we present the logic BC (for Bracket Calculus). The language of BC extends said ordinal notation system to a strictly positive modal language. Thus, unlike other provability logics, BC is based on a self-contained signature that gives rise to an ordinal notation system instead of modalities indexed by some ordinal given a priori. The presented logic is proven to be equivalent to RC Γ 0 , that is, to the strictly positive fragment of GLP Γ 0 . We then define a combinatorial statement based on BC and show it to be independent of the theory ATR 0 of Arithmetical Transfinite Recursion, a theory of second order arithmetic far more powerful than Peano Arithmetic.

Consider an o-minimal structure on the real field. Let M be a definable C r manifold, where r is a nonnegative integer. We first demonstrate an equivalence of the category of definable C r vector bundles over M with the category of finitely generated projective modules over the ring C r df (M) . Here, the notation C r df (M) denotes the ring of definable C r functions on M . We also show an equivalence of the category of definable C r bilinear spaces over M with the category of bilinear spaces over the ring C r df (M) . The main theorems of this paper are homotopy theorems for definable C r vector bundles and definable C r bilinear spaces over M . As an application, we show that the Grothendieck rings K 0 ( C r df (M)) , K 0 ( C 0 df (M)) and the Witt ring W( C r df (M)) are all isomorphic.

For G an algebraic group definable over a model of ACVF , or more generally a definable subgroup of an algebraic group, we study the stable completion G ? of G , as introduced by Loeser and the second author. For G connected and stably dominated, assuming G commutative or that the valued field is of equicharacteristic 0, we construct a pro-definable G -equivariant strong deformation retraction of G ? onto the generic type of G . For G=S a semiabelian variety, we construct a pro-definable S -equivariant strong deformation retraction of S ? onto a definable group which is internal to the value group. We show that, in case S is defined over a complete valued field K with value group a subgroup of R , this map descends to an S(K) -equivariant strong deformation retraction of the Berkovich analytification S an of S onto a piecewise linear group, namely onto the skeleton of S an . This yields a construction of such a retraction without resorting to an analytic (non-algebraic) uniformization of S . Furthermore, we prove a general result on abelian groups definable in an NIP theory: any such group G is a directed union of ??-definable subgroups which all stabilize a generically stable Keisler measure on G .

We show that every definable nested family of closed and bounded subsets of a P -minimal field K has non-empty intersection. As an application we answer a question of Darnière and Halupczok showing that P -minimal fields satisfy the "extreme value property": for every closed and bounded subset U⊆K and every interpretable continuous function f:U→ Γ K (where Γ K denotes the value group), f(U) admits a maximal value. Two further corollaries are obtained as a consequence of their work. The first one shows that every interpretable subset of K× Γ n K is already interpretable in the language of rings, answering a question of Cluckers and Halupczok. This implies in particular that every P -minimal field is polynomially bounded. The second one characterizes those P -minimal fields satisfying a classical cell preparation theorem as those having definable Skolem functions, generalizing a result of Mourgues.