Is the induction axiom (postulated in the Peano Axioms or in Dedekinds version using only FOL) a logical truth or a contigent truth (so that it could turn out false)?
IMO the induction axiom would be a contigent truth even in the following case, namely when we would interpret ℕ = {0, 1, 2, …, 1.000.000.000} where we would just additionally postulate in the axioms that 1.000.000.000 is the „last natural number“, i.e. having no successor. Correct?
Because logical truth means true in any interpretation of the symbols and obviously you could interpret some symbols in the induction axiom so that it becomes false.
But even if we focus just on the standard interpretation of the induction axiom in the Peano Axioms it could be false, right? Because we cannot prove it, so how could we be sure that it is true after all? Postulating truth as an axiom has nothing to do with being true, right?
(1) "A logical truth or a contingent truth" aren't exhaustive options. Something can be necessarily true, non-contingent, without being a truth of logic. Arguably the induction principle holds necessarily of the natural numbers, in virtue of the way the natural numbers form an $\omega$-sequence. That doesn't make it a truth of logic (in the normally understood sense).
(2) You might find the following helpful: §9.1(a) and §13.1 of An Introduction to Gödel's Theorems, downloadable from https://www.logicmatters.net/igt .