User:Boris Tsirelson/Sandbox1

David Hilbert aimed to find axioms sufficient for all mathematics and to prove their consistency from the assumption that the "finitary arithmetic" (a subsystem of the usual arithmetic of the positive integers, chosen to be philosophically uncontroversial) was consistent. A fatal blow was dealt by the second Gödel's incompleteness theorem. Consistency of a theory cannot be proved by a weaker theory, nor by the same theory. It can be proved by a stronger theory, which does not dispel doubts: if the given theory is inconsistent then the stronger theory, being all the more inconsistent, can prove every claim, be it true or false.

Many mathematicians feel that specialized theories, being more reliable than universal theories, are like watertight compartments. If a contradiction will be found in the used universal theory, specialized theories will separate and wait for a better universal theory.

I have always felt that, if one day someone came up with a contradiction in mathematics, I would just say, "Well, those crazy logicians are at it again," and go about my business as I was going the day before.

Casacuberta, C & M Castellet, eds. (1992), Mathematical research today and tomorrow: Viewpoints of seven Fields medalists, Lecture Notes in Mathematics, vol. 1525, Springer-Verlag, ISBN 3-540-56011-4.