Cantor began a correspondence with Dedekind to try to understand how to solve the problems but recurring bouts of his mental illness forced him to stop writing to Dedekind in These proved that real numbers are much more numerous than natural numbers. He was finally admitted to a sanatorium.
As Cantor later confided to the mathematician David Hilbertthis statement was inspired by the negative criticism Kronecker had made of set theory and was a call for open-mindedness among mathematicians, especially in dealing with new and novel ideas proposed by younger mathematicians.
Today, the highest award conferred by the German Mathematical Society is a medal honoring its first president, Georg Cantor. During his honeymoon in the Harz mountainsCantor spent much time in mathematical discussions with Richard Dedekindwhom he had met two years earlier while on Swiss holiday.
Given a trigonometric series f x with S as its set of zeros, Cantor had discovered a procedure that produced another trigonometric series that had S1 as its set of zeros, where S1 is the set of limit points of S.
In addition to introducing such concepts as everywhere-dense sets, he showed that whereas everywhere-dense sets were necessarily of the second species, first-species sets could never be everywhere-dense. Cantor starts his second construction with any sequence of real numbers.
By the smallest transfinite cardinal number he meant the cardinal number of any set that can be placed in one-to-one correspondence with the positive integers.