0

Given a mathematical statement, can it be always be proven to be correct or incorrect, or is it possible that a proof can not be created? Could all cases that could not be proven, be proven that no proof exists also?

For example, with the Beal Conjecture, is it guaranteed that a proof exists, or could no such proof exist?

I am thinking that some proofs could not exist such as problems that could take longer than the life of the universe to solve. For example, proving that a hashed value equals a given number.

Eric Johnson
  • 115
  • 1
  • 8
  • See here. – Dietrich Burde Aug 16 '20 at 15:53
  • https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems – cr001 Aug 16 '20 at 15:54
  • Whether or not a statement has a proof is not related to the lifetime of the universe or laws of physics. That kind of "practical" issue is not what people mean when they say that certain results (like the continuum hypothesis in ZFC) are not provable. If you think "time" is supposed to be relevant then do you think there are only finitely many integers because you could not write down infinitely many integers during the life of the universe? – KCd Aug 16 '20 at 16:18

1 Answers1

1

A proof has a context, namely the assumptions you take for granted (axioms). Sometimes it is possible to prove that a certain set of assumptions can neither prove nor disprove a certain assertion. For instance Godel proved, the Continuum hypothesis is independent of the set theory ZFC.

Arrow
  • 13,810