Yesterday I thought I had found a paradox about undecidable statements. Let's say there is some property P about integers, which is formulated as follows: ∀ n ∈ ℕ, something(n). Where "something" is a algebraic expression about n. Let's also assume that P has been proven to be undecidable.
Now, here is my train of thoughts: let's assume that there exists an integer k ∈ ℕ such that something(k) is false. In this case, it's easy to prove P to be false by just providing this counter-example. So that would mean that P is decidable since it's provably false. That is inconsistent with the assumption that P is undecidable, so it must mean that there isn't any k ∈ ℕ such that something(k) is false, but then we just proved that P is true, which is also inconsistent with the fact that it is undecidable.
Trying to understand this paradox, I found this on Wikipedia:
Undecidability of a statement in a particular deductive system does not, in and of itself, address the question of whether the truth value of the statement is well-defined, or whether it can be determined by other means. Undecidability only implies that the particular deductive system being considered does not prove the truth or falsity of the statement. Whether there exist so-called "absolutely undecidable" statements, whose truth value can never be known or is ill-specified, is a controversial point among various philosophical schools.
Which, in this context, I interpret by the idea that finding a counter-example k ∈ ℕ such that something(k) is false is not inconsistant with the undecidability of P, because the process of finding a counter-example is unrelated to the process of proving the statement true or false by deduction, and the undecidability of P only relates to deduction.
So, assuming this interpretation is correct, my question is : is there any example of such property P which we know is true or false thanks to examples/counter-examples, but which, regardless, has been proven undecidable?