Godel proved in 1931 that if [Peano arithmetic] is consistent, then it is incomplete. He constructed a statement that was semantically true but that had no proof, by coding up formulas and proofs as numbers and then creating a formula with code n that asserted that the formula with code n had no proof.

Context

This quote is from my CS 245 course notes. Basically, it’s saying that given this set of axioms, the Peano axioms, it is possible to construct statements that are semantically true, but have no proof of it. That is, something can be true and it will be impossible to prove it. It goes on to say later, that you can axioms to prove that these Peano axioms are consistent and complete, but you can’t prove that those axioms you’ve just added are consistent and complete.

Tags: Christianity

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