3 ms·
> In particular, ZF can't prove itself, but it can prove that Peano is consistent. Consistent within Peano, or consistent within ZF? That is to say, could anot
by anonymouskimmer 3y ago
> In particular, ZF can't prove itself, but it can prove that Peano is consistent.
Consistent within Peano, or consistent within ZF? That is to say, could another large system of axioms exist which fully contains the Peano axioms, but in which the Peano axioms are inconsistent (in a non-trivial manner)?
I really don't know much about formal logical consistency, but am mildly curious as I see these sorts of claims mentioned.
- Kranar 3y agoThe technical answer to your question is yes, another axiomatic system can exist that is able to prove that Peano is inconsistent. However, assuming that ZF is consistent, then that other axiomatic system would have to be inconsistent.
- Dylan16807 3y agoConsistent means that it doesn't contradict itself. Peano is consistent within Peano. The system that checks if Peano is consistent doesn't have to be similar at all or share any axioms.