From Goldmans's THE GOD OF THE MATHEMATICIANS

THE GOD OF THE MATHEMATICIANS
THE RELIGIOUS BELIEFS THAT GUIDED KURT GÖDEL’S REVOLUTIONARY IDEAS
by David P. Goldman
August 2010

http://www.firstthings.com/article/2010/08/the-god-of-the-mathematicians

Cantor asserted that the infinity of the integers and rational numbers was the first transfinite number, and he named it “Aleph-zero.” What, then, was the second transfinite number, or “Aleph-one”? He had proven that the infinity of the continuum of real numbers was “denser” than that of the integers; unlike that of the rational numbers, it could not be counted. He assumed that if the first transfinite number contained the integers, the second transfinite number would contain the continuum, and that no other transfinite number could be discovered between these two. 

That is Cantor’s “continuum hypothesis,” which attempts to identify a first and second transfinite cardinal number. From there, he believed, all the possible orders of infinity could be counted, the same way the integers count groups of one, two, three, and so forth. He not only recognized, but was driven by, the ontological implications of this assertion: If the continuum hypothesis turned out to be true, Spinoza would be vindicated because God’s infinity could be packaged into a neat series of numbers. Cantor spent the last thirty-five years of his life in a vain effort to prove this. He died in 1918 in a mental hospital. 

It was Gödel and, later, Paul Cohen who demonstrated respectively that Cantor’s continuum hypothesis could be neither proved nor disproved within existing set theory. Indeed, Cantor’s hypothesis remains maddeningly undecidable. Intuition, added Gödel, strongly suggests that Cantor’s hypothesis is wrong: Among the infinite number of transfinite numbers, there are an infinite number of cardinalities between the integers and the points on the continuum line, and mathematical investigation of the infinite will remain infinitely fruitful. God’s infinitude remains safe in heaven. Mathematicians have proven that an infinite number of transfinite numbers exist but cannot tell what they are or in what order they should be arranged.