Kurt Godel Quotes

Time-travel is possible, but no person will ever manage to kill his past self. Godel laughed his laugh then, and concluded, The a priori is greatly neglected. Logic is very powerful.

Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F.

In 1931, Kurt Godel proved in his famous second incompleteness theorem that there could be no finitary proof of the consistency of arithmetic. He had killed Hilbert's program with a single stroke. So should you be worried that all of mathematics might collapse tomorrow afternoon? For what it's worth, I'm not. I do believe in infinite sets, and I find the proofs of consistency that use infinite sets to be convincing enough to let me sleep at night.

The essence of Hilbert's program was to find a decision process that would operate on symbols in a purely mechanical fashion, without requiring any understanding of their meaning. Since mathematics was reduced to a collection of marks on paper, the decision process should concern itself only with the marks and not with the fallible human intuitions out of which the marks were reduced. In spite of the prolonged efforts of Hilbert and his disciples, the Entscheidungsproblem was never solved. Success was achieved only in highly restricted domains of mathematics, excluding all the deeper and more interesting concepts. Hilbert never gave up hope, but as the years went by his program became an exercise in formal logic having little connection with real mathematics. Finally, when Hilbert was seventy years old, Kurt Godel proved by a brilliant analysis that the Entscheindungsproblem as Hilbert formulated it cannot be solved. Godel proved that in any formulation of mathematics, including the rules of ordinary arithmetic, a formal process for separating statements into true and false cannot exist. He proved the stronger result which is now known as Godel's theorem, that in any formalization of mathematics including the rules of ordinary arithmetic there are meaningful arithmetical statements that cannot be proved true or false. Godel's theorem shows conclusively that in pure mathematics reductionism does not work. To decide whether a mathematical statement is true, it is not sufficient to reduce the statement to marks on paper and to study the behavior of the marks. Except in trivial cases, you can decide the truth of a statement only by studying its meaning and its context in the larger world of mathematical ideas.