Discussion and Implications
The incompleteness results affect the philosophy of mathematics, particularly versions of formalism, which use a single system formal logic to define their principles. One can paraphrase the first theorem as saying the following:
- An all-encompassing axiomatic system can never be found that is able to prove all mathematical truths, but no falsehoods.
On the other hand, from a strict formalist perspective this paraphrase would be considered meaningless because it presupposes that mathematical "truth" and "falsehood" are well-defined in an absolute sense, rather than relative to each formal system.
The following rephrasing of the second theorem is even more unsettling to the foundations of mathematics:
- If an axiomatic system can be proven to be consistent from within itself, then it is inconsistent.
Therefore, to establish the consistency of a system S, one needs to use some other system T, but a proof in T is not completely convincing unless T's consistency has already been established without using S.
Theories such as Peano arithmetic, for which any computably enumerable consistent extension is incomplete, are called essentially undecidable or essentially incomplete.
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