Rola aksjomatu komprehensji w rozwoju teorii mnogości

Abstract

The publication of Georg Cantor’s „Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen“ in the 1874 gave birth to the new scientific discipline – set theory. One of the main points of the Cantor’s theory was the axiom of comprehension which states that for any formula there exists a set consisting of only those elements that satisfy this formula. This axiom played a significant role in the development of the set theory in the 19th and 20th centuries. On the one hand it seemed indispensable for a consistence of theory, but on the other it turned out to be a „saboteur” that destroyed the very basis of mathematics. The paper presents the origins and the importance of comprehension axiom: its applications and dangers that it brings. It attempts also to answer the question whether is this axiom indeed indispensable in the set theory or it could (or even should) be replaced with some other formula.