In answer to the question of how can something be true, but not provable I want to point to the Goldbach conjecture which says "every even number > 2 is the sum of two primes".
If the Goldbach conjecture is false then there's a counterexample which can be checked in finite time (eg. just try adding all pairs of primes less than that number although there are faster ways).
If there isn't a counterexample then the Goldbach conjecture is true.
To be provable however, there would have to exist a proof of the Goldbach conjecture. No such proof is known to exist.
Here "truth" is exactly what it intuitively means. Either a counterexample exists or it doesn't. The "truth" of the Goldbach conjecture won't depend on your choice of axioms. All you need are the primitives necessary to define the natural numbers, primality, and addition (in particular, it won't depend on AC).
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