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In addition to weakening the logic (or avoiding the parts that introduce incompleteness, especially the existential quantifier) you also need to weaken your ari
by chithanh 5y ago
In addition to weakening the logic (or avoiding the parts that introduce incompleteness, especially the existential quantifier) you also need to weaken your arithmetic. We know that Primitive Recursion (PR) is too weak to prove its own consistency, and Peano Arithmetic (PA) is too strong. So you may have to look at a strengthening of PR and/or weakening of PA in order to achieve a theory that proves its own consistency. Whether that theory exists is still an open question.
- Sniffnoy 5y agoGetting a theory that proves its own consistency isn't what's being discussed here. What's being discussed here is just weakening a theory to remove a particular inconsistency after an inconsistency is found. Trying to make a usable theory that also proves its own consistency is probably also a futile goal, per Gödel, but it's a separate one, and not one that people would likely go for if an inconsistency were found in Peano arithmetic.