Set Theory

Set theory studies well determined collections, called sets, of objects called members or elements, both finite and infinite, with infinite sets its central concern. Georg Cantor founded the field as a distinct discipline in late 1873 when he discovered that the linear continuum, the real number line, is not countable, showing that infinite sets can come in genuinely different sizes. He introduced cardinality, the idea that two sets are the same size when their elements can be placed in one to one correspondence with each other, and in 1878 proposed the continuum hypothesis, that infinite sets of real numbers fall into exactly two size categories, a claim that remains formally undecided within the standard axioms. Ernst Zermelo began axiomatizing set theory in 1908, and the resulting system is now the standard foundation onto which essentially every mathematical object and theorem can be reduced.