[eru]

There are systems that we can proof the consistency of just fine. See eg https://en.wikipedia.org/wiki/Presburger_arithmetic

Presburger Arithmetic cannot prove its own consistency inside of itself, but that doesn't mean we can't prove its consistency 'at all'.

[mathgradthrow]

What on earth do you mean by prove, If not within a system of proof?

You can prove the consistency of PBA, But you cannot prove that you cannot prove the inconsistency of PBA, because The inability to do both is the definition of consistency in the larger system.

[Scarblac]

But it's not really relevant for anything to be able to prove that you cannot prove the inconsistency.