foundations of mathematics Cantor

In the 19th century, the German mathematician Georg Cantor (1845–1918) returned once more to the notion of infinity and showed that, surprisingly, there is not just one kind of infinity but many kinds. In particular, while the set null of natural numbers and the set of all subsets of null are both infinite, the latter collection is more numerous, in a way that Cantor made precise, than the former. He proved that null, null, and null all have the same size, since it is possible to put them into one-to-one correspondence with one another, but that null is bigger, having the same size as the set of all subsets of null.

However, Cantor was unable to prove the so-called continuum hypothesis, which asserts that there is no set that is larger than null yet smaller than the set of its subsets. It was shown only in the 20th century, by Gödel and the American logician Paul Cohen (b. 1934), that the continuum hypothesis can be neither proved nor disproved from the usual axioms of set theory. Cantor had his detractors, most notably the German mathematician Leopold Kronecker (1823–91), who felt that Cantor’s theory was too metaphysical and that his methods were not sufficiently constructive (see below The quest for rigour: Formal foundations: Nonconstructive arguments).