Georg Cantor

Overview
“The actual infinite arises in three contexts: first when it is realized in the most complete form, in a fully independent other-worldly being, in Deo, where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto as a mathematical magnitude, number, or order type. I wish to make a sharp distinction between the Absolute and what I call the Transfinite, that is, the actual infinities of the last two sorts, which are clearly limited, subject to further increase, and thus related to the finite.” - Georg Cantor, Foundations of a General Theory of Aggregates, 1883 While the metaphysical conclusions Georg Cantor draws in the above excerpt are contemporary topics of debate, the magnitude of Cantor's contribution to the rigorous study of infinity is not. In the words of infinity scholar Rudy Rucker, "It was Cantor's great achievement to demonstrate that we could, in fact, discuss the infinite in scientific as well as theological terms." That is, prior to Cantor's revolutionary work in Set Theory in the nineteenth century, discourse about infinity was dominated by either classical notions like horror infiniti and the Aristotlean categories of actual infinity and potential infinity, or by purely theological constructs as seen in apophatic religious traditions or the philosophy of Thomas Aquinas.

His work with sets demonstrated, for example, that there are multiple kinds of mathematical infinity--that there are, in fact, infinity kinds of infinity, and that some of these infinities are larger than others. His nomination of the term transfinite numbers gave cardinality to these discrete classes of infinity, thus actualizing them.

Despite the outspoken support from peer mathematicians like David Hilbert, who went so far as to proclaim that "No one shall expel us from the paradise that Cantor has created," Cantor's ideas, new as they were in the late-1800s, were subject to widespread scrutiny during his lifetime. Scholar Robert John Russell speculates that such criticism may have led to Cantor's precipitous decline in health toward the end of his life. Cantor died from a heart attack on January 6, 1918, in a sanatorium.

Major Contributions to Infinity

 * 1865-'72: In a series of letters between Cantor and fellow set theoretical mathematician and longtime friend Richard Dedekind, the two begin developing the idea of one-to-one correspondence.
 * 1873: Proves that the sets of rational numbers and algebraic numbers are countable.
 * 1874: Proves the set of real numbers is uncountable.
 * 1883: Begins working on the Continuum Hypothesis.
 * 1895-'97: Conceives of infinite ordinals, transfinite numbers, and what will later be known as Cantor's Paradox.