3 ms·

you cannot prove the consistency of a system of proof within that system, ie at all.

• eru 1y ago
  There are systems that we can proof the consistency of just fine. See eg https://en.wikipedia.org/wiki/Presburger_arithmetic 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 1y ago
    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 1y ago
      But it's not really relevant for anything to be able to prove that you cannot prove the inconsistency.
• Scarblac 1y ago
  But proving inconsistency can be done - show that a contradiction follows from the axioms.