I'm wondering if anyone has found, or can find, a sequence of statements $P(n)$ ($n \in \mathbb{N}$) such that:

Heuristic arguments using probability theory suggest that all the statements $P(n)$ are true.

One can prove in some widely accepted axiom system $X$, preferably by making the heuristic arguments rigorous, that "$P(n)$ holds for infinitely $n$".

One can prove in some widely accepted axiom system $Y$ that "$X$ cannot prove $\forall n P(n)$".

The goal here would be to find statements that are true 'just because they are probably true', not because we can put our finger on a reason why any individual one must be true.

An example of such a sequence of statements might be 'if $p_n$ is the $n$th prime, there are infinitely many repunit primes in base $p_n$'. However while this example meets condition 1 we seem far from having enough understanding of mathematics to prove 2, and 3 seems hopeless. I think we need a less charismatic sequence of statements that are more carefully crafted to the task at hand.