Set-Theoretic and Algebraic Preliminaries

Abstract

Set theory is not just one of the main tools in mathematics, it is the very root of mathematics from which all mathematical disciplines stem. Due to a series of his papers, the first of which appeared in 1874, the great German mathematician, Georg Ferdinand Cantor (1845 – 1918), is considered to be the sole founder of set theory. Although the Czech Bernard Bolzano (1781 – 1848) made one of the first attempts to formalize set theory, in particular in his Paradoxien des Unendlichen (of 1851), by considering the one-to-one correspondence between two sets (later on developed by Cantor to what we now know as cardinals), neither he, nor anyone else, was really a predecessor to Cantor's creation. Ernst Zermelo (1871 – 1953) was another German, who among his numerous contributions to set theory, is the author of the first axiom for set theory (of 1908) and undoubtedly one of the primary axioms of the whole mathematics.