A logical system (e.g. standard mathematics) cannot prove its own axioms
It can, and trivially so. Because every statement implies itself, we can just use modus ponens on each axiom A and (A => A) and we get A (if you even need an inference rule).
What the theorem says is that a relevant logical system (not just any logical system) cannot prove that it is not self-contradictory.
It can, and trivially so. Because every statement implies itself, we can just use modus ponens on each axiom A and (A => A) and we get A (if you even need an inference rule).
What the theorem says is that a relevant logical system (not just any logical system) cannot prove that it is not self-contradictory.