Senses in which ZFC consistency is or isn't provable

Question

From the discussion in this post about ZFC consistency, I've extracted the following hopefully accurate (I think?) summary conclusions:

I) Consistency of the ZFC axioms can never be definitively proven by formal axiomatic system(s) alone. For example, if you use Morse-Kelley set theory to prove Con(ZFC), then you're left with the issue of proving the consistency of the Morse-Kelley axioms. You could prove this within some other formal axiomatic system Foo, but now you have to worry about proving Con(Foo) via yet another formal axiomatic system Bar whose own consistency remains unproven, and so on. Thus at the top of any such "hierarchy" of axiomatic systems, there's always that last one whose consistency remains unproven.

II) However, ZFC axiom consistency can be proven "informally" (i.e. not via a formal axiomatic system) with a level of rigour sufficient to convince most set theorists to not seriously worry about the possibility of ZFC being inconsistent (the top-voted answer implies one such "informal" proof is based on showing that Con(ZFC) is logically equivalent to demonstrating non-existence of positive integer solutions to a fairly straightforward Diophantine equation).

Hence two concrete questions:

Are these summary conclusions (I & II) as I've described them in fact correct?

Does anyone happen to have a reference to a book or paper or something with concrete detail about the concept of reducing proof of Con(ZFC) to solving a Diophantine equation? That sounds intriguing, but what I've found online is relatively vague. Has someone actually worked out the specific polynomial and then solved it?

Answer 1

You mixed the "Confusing Point 2" and "An informal, 'mathematical' proof of the consistency of ZFC".

The reduction of Con(ZFC) to a concrete Diophantine equation is not meant to convince anyone that Con(ZFC) shall hold (This is rather done in other ways, such as "intuitive picture of the cumulative hierarchy" mentioned in the linked posts, just like we believe Con(PA) for our intuition of an indefinitely growing sequence of "natural" numbers). After all, the statement of Fermat's last theorem is not gonna convince anyone of its correctness, and the equation for Con(ZFC) is not gonna convince anyone of anything by itself.

Rather, the "No solution to this Diophantine equation" is Con(ZFC) in ZFC: Con(ZFC) itself is not formulated within ZFC, and we have to use an encoding scheme to formulate Con(ZFC) within the language of ZFC, which through Godel's numbering produces the Diophatine equation. After all, reasoning within ZFC is just printing new sequences of symbols from old ones based on some rules, and these sequences/process can be encoded/simulated by numbers and functions about which ZFC is capable of talking.

From this point of view, Con(ZFC) is not a special proposition at all. It's just that, as someone who lives beyond ZFC, if a solution to the Diaphatine equation exists, we have a way to turn it into a proof of Con(ZFC) which is forbidden by Godel's incompleteness theorem (if ZFC is indeed consistent).

So if you want to read about the reduction, just read about Godel's incompleteness theorems. And further we may add Gentzen's consistency proof of PA to the list of readings.

Edit: Con(PA) (not even to mention the much stronger Con(ZFC)) is not a completely settled issue among all mathematicians. In recent times, at least two prominent mathematicians -- Edward Nelson and Vladimir Voevodsky had cast doubt on whether it holds.