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.
Reminds me of Gödel's second incompleteness theorem
A logical system (e.g. standard mathematics) cannot prove its own axioms. Therefore not all problems are solvable using standard mathematics.
However, this restriction only applies to consistent systems. An illogical system (e.g. liberalism) can prove its own axioms.
Legitimate truth nuke🤯
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.