In “What if Current Foundations of Mathematics are Inconsistent?” Voevodsky (2010) suggests that there are three options in light of Goedel’s theorems:
Either:
1. If we “know” arithmetic is consistent, it should be provable, so Goedel’s second incompleteness theorem is false.Or:
2. Admit there can be provably unprovable arithmetic “knowledge”Or:
3. Admit that “knowing” arithmetic is consistent is an illusion, and arithmetic is inconsistent.
But why make any mention of psychological states like “knowing” at all?
Surely, regardless of our intuitions, the only truths (besides the Cogito) that we can “know” to be true, i.e., certain (rather than just probably true on all available evidence) are the truths that we have proved to be necessarily true, on pain of contradiction
Why not the following?—
4. Admit that arithmetic’s consistency is provably unprovable, but that then it may either be (unprovably) true (rather than unprovably “known”) that arithetic is consistent — or it may be false that arithmetic is consistent.
5. If arithmetic’s consistency is true (but unprovable, hence unknowable), then all proven theorems are true (except that their consistency cannot be proven).
6. If arithmetic’s consistency is false, then either an instance of inconsistency will be found (hence inconsistency will be “proven”) or it will not be found, in which case it will never be known whether arithmetic is consistent or inconsistent, hence whether the negations of theorems we have proved are also provable.
“Reliability” does not seem to be a valid substitute for provability-on-pain-of-contradiction. It would make mathematics into something more like inductive empirical science: provisionally true on the available evidence until/unless contradictory evidence is encountered. That is just the conjunction of 5 and 6. It also has some of the flavor of intuitionistic reasoning (insofar as the excluded middle is concerned).
As usual, this uncertainty only besets infinities, not finite constructions.
Or does the notion of “deductive rigor” all reside in the provability of consistency in nonfinite mathematics?
(The problem of possible mistakes in proofs (and the partial solution of computer-aided proofs) concerns another kind of reliability, and again seems to be a solution only for finite mathematics.)