Streszczenie odczytu (Talk summary)

From arithmetic without analysis to the finite combinatorics in set theory.

Peano Arithmetic is an axiomatization of the ``elementary arithmetic''. In this axiom system we can formulate and prove most of the classical results in number theory including those results whose known proofs involve complex analysis. It is widely believed that the Fermat's Last Theorem can be proved in this axiom system while it is not trivial to see that the theorem can be formulated in a single sentence in the language of the Peano Arithmetic. On the other hand there are many ``mathematical'' theorems which can be formulated in the language of Peano Arithmetic but cannot be proved in the axiom system. Paris-Harrington Theorem shows one of such examples. There are even statements which can be formulated in the language of the Peano Arithmetic but whose validity cannot be decided inside the usual mathematics.
In this talk I will present some more details of these situations around the Peano Arithmetic and the ``mathematics''.