You are using the word “truth” but it is more correct to use provable/non-provable. What Godel showed is that -limiting the discussion to the natural numbers for simplicity - there are statements that are true in the standard model of the natural numbers that are not provable in the first order Peano Axiomatic system for the natural numbers. What this means is that such a statement will be false in some non-standard model of the natural numbers.

It’s important that we are talking about models of the first order Peano Axioms. One can always find a system of axioms in which all true statements are provable. To do this just take the collection of all true statements in the standard model. Now every true statement is a theorem. It’s easy to have a complete set of axioms. What can’t happen is a recursively enumerable set of axioms that is complete and consistent.

Recursively enumerability is needed so that one can have an effective means of determining if a statement is an axiom. Think computable when I say effective.

When talking about truth we need to be careful because this is tied to a model of an axiomatic system. By the Completeness Theorem a statement that is true in all models of the system is provable.