My current understanding:
Math starts with a set of basic (purportedly self-evident) statements that are taken as a given without the need to prove them true, like e.g., a + b = b + a etc. Such initial statements are called axioms.
Any further mathematical statement though is only considered true when shown to be in accordance with the axioms. To the contrary, if in disagreement with at least one of the axioms, a corresponding statement is considered false.
Gödel’s Incompleteness Theorem
However, according to Gödel there are statements like "This sentence is false" which are true despite how they cannot be successfully reduced to the pre-existing axioms, i.e. cannot be proven true.
What confuses me:
What I encounter difficulty with to understand is the precise definition of truth in the context of the incompleteness theorem. First, truth is defined as a state where a statement is demonstrated (“proven”) to be in accordance with the axioms, i.e. truth is established by proof. Then however, truth is claimed to exist even if a statement cannot be proved true given the current set of axioms, suggesting that the knowledge that some statement is true can be reached by “bypassing” any proof.
Doesn’t the above imply that truth can be established without the requirement of any proof? If so, why then the need to prove any statement at all to demonstrate it is true?