Dva pristupa paradoksima u logici i matematici

Abstract

Uobičajeno je razumevanje paradoksa kao ozbiljnih problema koji sprečavaju dalji razvoj matematičke ili logičke teorije unutar koje su formulisani. Paradoksi navodno pokazuju da je data teorija postavljena na nestabilne, ili čak kontradiktorne osnove, i time ukazuju na neophodnost njihove revizije. Moguć je, međutim, i drugačiji pogled na paradokse. Oni se mogu razumeti i kao da otkrivaju netačnost pojedinačnih pretpostavki o prirodi objekata u vezi kojih se javljaju ili načina na koji se o njima rasuđuje. Paradoksi bi u tom smislu mogli biti shvaćeni kao argumenti koji dovode do novih saznanja o prirodi tih objekata i predstavljaju značajnu motivaciju za unapređenje njihovog razumevanja. U ovom radu, opisaćemo uticaj koji je prvi od navedenih pristupa paradoksima imao na razvoj logike i matematike, pre svega teorije skupova. Pokazaćemo na konkretnim primerima da alternativni pogled na paradokse ili njihove formalne ekvivalente zapravo može voditi do značajnih rezultata u logici i omogućiti zasnivanje nove logičke discipline - teorije pojmova.

It is usual to think of the paradoxes appearing inside a particular logical or mathematical theory as the serious obstacles hindering any further development of that theory. Paradoxes are supposed to show that a theory in question is built on an unstable, or even contradictory foundation, and thus point the need for its complete revision. However, a different view on the paradoxes is also possible. They could instead be understood as the arguments which show that some particular assumptions concerning the objects with respect to which they appear, or the ways of reasoning about them, are wrong. If treated in that way, paradoxes or their solutions could lead to some new insights into the nature of objects they concern. They could thus turn out to make a useful focus in developing the understanding of these objects. In this work, the effect that the first approach towards the paradoxes had on development of logic and mathematics, in particular set theory, will be described. Using some examples, we will try to show that the alternative view on the paradoxes or their formal equivalents actually leads to some important results in logic, and at the same time, opens the door to a new logical theory - the so-called theory of concepts.