you cannot prove the consistency of a system of proof within that system, ie at all.
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'.
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.
But proving inconsistency can be done - show that a contradiction follows from the axioms.