Existentially Closed Models in the Framework of Arithmetic

We prove that the standard cut is definable in each existentially closed model of IΔ0 + exp by a (parameter free) П1–formula. This definition is optimal with respect to quantifier complexity and allows us to improve some previously known results on existentially closed models of fragments of arithmetic.Ministerio de Educación y Ciencia MTM2011–2684

Bounded Arithmetic in Free Logic

One of the central open questions in bounded arithmetic is whether Buss' hierarchy of theories of bounded arithmetic collapses or not. In this paper, we reformulate Buss' theories using free logic and conjecture that such theories are easier to handle. To show this, we first prove that Buss' theories prove consistencies of induction-free fragments of our theories whose formulae have bounded complexity. Next, we prove that although our theories are based on an apparently weaker logic, we can interpret theories in Buss' hierarchy by our theories using a simple translation. Finally, we investigate finitistic G\"odel sentences in our systems in the hope of proving that a theory in a lower level of Buss' hierarchy cannot prove consistency of induction-free fragments of our theories whose formulae have higher complexity

The Matiyasevich Theorem. Preliminaries

In this article, we prove selected properties of Pell’s equation that are essential to finally prove the Diophantine property of two equations. These equations are explored in the proof of Matiyasevich’s negative solution of Hilbert’s tenth problem.This work has been financed by the resources of the Polish National Science Centre granted by decision no. DEC-2015/19/D/ST6/01473.Institute of Informatics University of Białystok, Białystok, PolandMarcin Acewicz and Karol Pak. Pell’s equation. Formalized Mathematics, 25(3):197-204, 2017. doi: 10.1515/forma-2017-0019.Zofia Adamowicz and Paweł Zbierski. Logic of Mathematics: A Modern Course of Classical Logic. Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts. Wiley-Interscience, 1997.Martin Davis. Hilbert’s tenth problem is unsolvable. The American Mathematical Monthly, Mathematical Association of America, 80(3):233-269, 1973. doi: 10.2307/2318447.Yoshinori Fujisawa and Yasushi Fuwa. The Euler’s function. Formalized Mathematics, 6 (4):549-551, 1997.Xiquan Liang, Li Yan, and Junjie Zhao. Linear congruence relation and complete residue systems. Formalized Mathematics, 15(4):181-187, 2007. doi: 10.2478/v10037-007-0022-7.Robert Milewski. Natural numbers. Formalized Mathematics, 7(1):19-22, 1998.Rafał Ziobro. Fermat’s Little Theorem via divisibility of Newton’s binomial. Formalized Mathematics, 23(3):215-229, 2015. doi: 10.1515/forma-2015-0018.25431532

Ku dobrej szkole skoncentrowanej na uczniach T. 2

Praca recenzowana / peer-reviewed pape

Diagonal reasonings in mathematical logic

First we show a few well known mathematical diagonal reasonings. Then we concentrate on diagonal reasonings typical for mathematical logic