Kurt Gödel became interested in university-level mathematics while he was still at school.
He went on to study in Vienna, where he immersed himself in the study of logic, and was influenced greatly by, and built on, Bertrand Russell’s work in mathematical logic.
What interested Gödel was the foundations of mathematics - what are its basic assumptions and what are its limitations and scope?
Hitherto, it had been assumed that all statements in mathematics were either true or, they were false. If it was true then the search was on for a proof; if it was not true then a simple counterexample would exist.
Fermat’s Last Theorem was just such a statement but despite enormous effort no counterexample had been found and neither had a proof.
Gödel’s great achievement was to consider the possibility that unanswerable questions might exist; he then developed a theory about the nature of such questions in his Incompleteness Theorems.
Think about that: Gödel showed that in any mathematical axiomatic system (a system which is self-evident), there are statements that that cannot be proved or disproved.
Contrary to contemporary thought (and work by people like Bertrand Russell) this meant that mathematics could no longer be viewed as a complete subject.
Life for Gödel changed dramatically in 1933 when Hitler came to power. Another mathematician, Schick, who had greatly inspired Gödel, was murdered by a Nazi student.
This so disturbed Gödel that he had a nervous breakdown. After he recovered, he left Vienna for life in America, taking on a professorship at Princeton. Although he returned to Germany a number of times, Gödel eventually emigrated to America.
According to family and friends, Gödel could be quite stubborn. After suffering from a severe ulcer, he kept to a diet so strict, it slowly led him to lose weight. In the last years of his life, he became convinced that someone was trying to poison him, and began to refuse to eat. He eventually starved himself to death.