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The theorem that in any sufficiently powerful, logically consistent formulation of logic or mathematics there must be true formulas which are neither provable nor disprovable. The theorem entails the corollary that the consistency of a logical system cannot be proved within that system.
- ‘The incompleteness theorem thus raises doubts about any philosophy of mathematics (formalist or otherwise) that requires a single deductive system for all of arithmetic - a single formal method for deriving every arithmetic truth.’
- ‘Gödel's incompleteness theorem tells us that within mathematics there are statements that are unknowable, or undecidable.’
- ‘The fallout, however, from this mathematical bomb was even more perilous than that from the incompleteness theorem.’
- ‘The second incompleteness theorem, which follows pretty straightforwardly from the first, proves that one of the things that you can't prove in a formal system of arithmetic is the consistency of that very system.’
- ‘He used Gödel's incompleteness theorem to argue that our minds' activities exceed what can be programmed into computers.’
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