Georg Cantor (1845-1918) was a German mathematician who created the field of Set Theory. His motivation was Christian, and he dedicated his last publication to a passage from 1 Corinthians: "The time will come when these things which are now hidden from you will be brought into the light." Cantor was vilified by other mathematicians of his time, such as Henri Poincare, but Cantor's revolutionary work proved to be the most influential of all of them.
Cantor was the first to "put infinity on a firm logical foundation." His invention of set theory helped him achieve this extraordinary goal. He "described a way to do arithmetic with infinite quantities useful to mathematics."
Cantor invented the concept of cardinality and ordered pairs, and how some infinities are larger tan others.
Cantor's best known proof is his technique of Cantorian Diagonalization, a method useful to prove that the real numbers are larger in cardinality than the integers. He is also known for his Cantor Set and Cantor Intersection Theorem.
The set of integers (or their equivalent) is the smallest infinite set. Cantor's "Continuum Hypothesis" is the conjecture that the set of all real numbers is the second smallest infinite set.
Paul Cohen proved in the 1960s that the "Continuum Hypothesis" can neither be proved nor disproved within conventional (axiomatized) set theory. Stated another way, it is possible to develop a non-Cantorian set theory that negates the "Continuum Hypothesis."