Costas Dimitracopoulos

Logic Colloquium 2005 (Lecture Notes in Logic)

The Annual European Meeting of the Association for Symbolic Logic, generally known as the Logic Colloquium, is the most prestigious annual meeting in the field. Many of the papers presented there are invited surveys of recent developments. Highlights of this volume from the 2005 meeting include three papers on different aspects of connections between model theory and algebra; a survey of recent major advances in combinatorial set theory; a tutorial on proof theory and modal logic; and a description of Bernays philosophy of mathematics.

On two problems concerning end extensions

We study problems of Clote and Paris, concerning the existence of end extensions of models of Σn-collection. We continue the study of the notion of ‘Γ-fullness’, begun by Wilkie and Paris (Logic, Methodology and Philosophy of Science VIII (Moscow, 1987). Stud. Logic Found. Math., vol. 126, pp. 143–161. North- Holland, Amsterdam, 1989) and introduce and study a generalization of it, to be used in connection with the existence of Σn-elementary end extensions (instead of plain end extensions). We obtain (a) alternative proofs of results (Adamowicz in Fund. Math. 136, 133–145, 1990) and (Wilkie and Paris in Logic, Methodology and Philosophy of Science VIII (Moscow, 1987). Stud. Logic Found. Math., vol. 126, pp. 143–161. North-Holland, Amsterdam, 1989) related to the problem of Paris and (b) a partial solution to the problem of Clote.

A note on end extensions

Abstract. We provide an alternative proof of a theorem of P. Clote concerning end extensions of models of

On a Problem of J. Paris

\Sigma_n

On end extensions of models of subsystems of peano arithmetic

-induction, for

Logic and theory of algorithms: 4th Conference on Computability in Europe, CiE 2008, Athens, Greece, June 15-20, 2008: Proceedings

n \geq 2

A note on exponentiation

.

Logic Colloquium 2005

We give alternative proofs for results of T. Slaman and N. Thapen, concerning the problem whether or not | Δ1 implies B Σ1. Our proofs isolate structures that could disprove this implication.

3 PAUL BERNAYS’LATER PHILOSOPHY OF MATHEMATICS

Abstract We survey results and problems concerning subsystems of Peano Arithmetic. In particular, we deal with end extensions of models of such theories. First, we discuss the results of Paris and Kirby (Logic Colloquium ’77, North-Holland, Amsterdam, 1978, pp. 199–209) and of Clote (Fund. Math. 127 (1986) 163; Fund. Math. 158 (1998) 301), which generalize the MacDowell and Specker theorem (Proc. Symp. on Foundation of Mathematics, Warsaw, 1959, Pergamon Press, Oxford, 1961, p. 257–263) we also discuss a related problem of Kaufmann (On existence of Σ n end extensions, Lecture Notes in Mathematics, Vol. 859, Springer, Berlin, 1980, pp. 92). Then we sketch an alternative proof of Clotes theorem, using the arithmetized completeness theorem in the spirit of McAloon (Trans. Amer. Math. Soc. 239 (1978) 253) and Paris (Some conservation results for fragments of arithmetic, Lecture Notes in Mathematics, Vol. 890, Springer, Berlin, 1981, p. 251).

Realism vs anti-realism and alternative logics

Motivated by the classical Ramsey for pairs problem in reverse mathematics we investigate the recursion-theoretic complexity of certain assertions which are related to the Erdos-Szekeres theorem. We show that resulting density principles give rise to Ackermannian growth. We then parameterize these assertions with respect to a number-theoretic function f and investigate for which functions f Ackermannian growth is still preserved. We show that this is the case for f (i) = (d)root i but not for f(i) = log (i).Topological dynamics of cellular automata (CA), inherited from classical dynamical systems theory, has been essentially studied in dimension 1. This paper focuses on 2D CA and aims at showing that the situation is different and more complex. The main results are the existence of non sensitive CA without equicontinuous points, the non-recursivity of sensitivity constants and the existence of CA having only non-recursive equicontinuous points. They all show a difference between the 1D and the 2D case. Thanks to these new constructions, we also extend undecidability results concerning topological classification previously obtained in the 1D case.