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Is it impossible that Con(ZFC) could be once upon a time proven by a formal system S that is able to prove its own Con(S), i.e. proving Con(ZFC) for good?

S would need to avoid Gödel 2, i.e. needing to be weaker than our arithmetic with natural numbers. But that would mean Cn(S) $\subseteq$ Cn(ZFC) and a weaker system - even if itself consistent - could never prove the consistency of a stronger system because it does not range over all statements of the stronger system, so it could always miss a crucial statement that leads to a contradiction.

Is this last argument correct?

Pippen
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    For good? Why should we trust S that ZFC is consistent simply because it can prove its own consistency? If it were inconsistent, it would prove its own consistency, and ZFC's consistency too. – spaceisdarkgreen Sep 11 '24 at 02:02
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    No, being able to avoid Gödel’s second incompleteness doesn’t mean it’s weaker than ZFC in the sense you wrote down. You’re still thinking that “weaker” and “stronger” is a linear order, which it is not. In fact, there are (presumably) consistent theories that can prove its own consistency and prove Con(ZFC), see https://en.wikipedia.org/wiki/Self-verifying_theories. But as @spaceisdarkgreen already pointed out, that hardly means Con(ZFC) is proved “for good”. (In fact, this system has Con(ZFC) as an axiom and its own consistency possibly depends on Con(ZFC) being true in the first place.) – David Gao Sep 11 '24 at 03:24
  • S proves Con(S) insofar as we just find some x that S cannot prove. In that moment we’d knew that Con(S) is for real, wouldn‘t we? But I see the problem: outside of a very simple, finite S one could always argue that S is nevertheless inconsistent and x provable, we just haven‘t found the contradiction and therefore efq to produce x. But that means all the hype about math as a more certain science is just that: hype. Math is not superior to natural sciences, both produce theorems that are essentially fallible statements with good reasons (proofs). :( – Pippen Sep 11 '24 at 16:14
  • @Pippen I wouldn't claim any kind of blanket superiority to other fields of study, other than the subjective sense that it's the most interesting to me. But as to the "hype" about math being "more certain", "more precise", "more rigorous", etc., I'd direct you to the dictionary definition of the word "more". – spaceisdarkgreen Sep 11 '24 at 18:37

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In practice, "trust" in mathematics is rarely built by logical analysis, but rather by a slow build of intuition, starting from simple concepts and building out from there. The purpose of a proof isn't to be formalized in ZFC, but to convince, and ideally to give a coherent picture of why the result is true (or why it 'follows from the axioms' if we don't believe in that notion).

When the story hangs together, and on iteration produces a beautiful web of interconnected ideas, we gain confidence that it is meaningful.

But, I'd have to say, a formal consistency proof could have, in theory, helped matters during the foundational crisis, when there were lots of untested new methodologies and ideas and Russel's paradox was discovered. But we still would ultimately lean on our intuition for why the thing we proved consistent is in fact consistent.

David Gao points out in the comments some technical problems with your specific argument, and I say why a theory 'proving its own consistency' is not the best criterion. However, what makes a bit more sense is the more ambitious goal that a very weak "more trustworthy" subsystem (e.g. PA or PRA) prove the consistency of some grand theory like ZFC, and this is for practical purposes ruled out by the incompleteness theorem, with the caveat that the word subtheory is carrying a lot of weight (more on that below).

But if we had (hypothetically) accomplished the (now known to be impossible) feat of proving the consistency of a very strong theory like ZFC from a very weak subtheory, we'd still need to trust the weak system.$^*$ And I would argue that trust wouldn't just be built on our perceptions of the weak theory's axioms being more "self-evident" or what-have-you, but by us looking at the consistency proof itself and understanding it.

Gentzen's proof of the consistency of PA is an instructive example here. To me, it's pretty self-evident that PA is consistent... though I confess I've never so much as conceived of an arithmetical property that has 1000 alternating universal and existential quantifiers, I can't see a reason why the simple idea of mathematical induction that works so nicely with properties I do understand shouldn't apply there as well.

But Gentzen's proof gives a new idea for why this system can't generate a contradiction via direct analysis of formal proofs, in quite a different light from my naive intuitions about how natural numbers work, and crucially not hand-waving away the issue of unbounded logical complexity. The elephant in the room is the proof's need for (quantifier-free!) induction up to the ordinal $\epsilon_0$, which of course can't be formalized in PA$^{**}$. But when we have our skeptic hats on, we tend to get preoccupied with what's 'sufficiently finitary' and 'non-circular' rather than appreciating the fact that we have a whole new way to think about the consistency of PA that meaningfully addresses the most salient sources of concern.

We don't yet have a working analysis of this sort for ZFC and we aren't particularly close to having one. We have a confluence of naive set-theoretical intuition, the idea of the cumulative hierarchy, and some other concepts that seem perhaps to justify it, as well as a pile of results that seem to hang together rather nicely. That's good enough for me, personally, but the Gentzen example shows that we shouldn't count out meaningful progress beyond this, despite Godel's theorem.

$^*$Then there's the perennial question/complaint that comes up on MSE about how when we define formal proofs, we (gasp!) talk about sets of formulas, and the like. In other words, how can we even trust our reasoning system for thinking about what the weak system proves? This kind of skepticism makes sense for abstract nonsense, but gets a little silly for matters of finite combinatorics, and once we're down to that level, this cycle could go on forever... at some point, you need to take a stand and believe in something!

$^{**}$As alluded to by David Gao in the comments, there's a crucial "loophole" in the incompleteness theorem being exploited here. The system that proves consistency need not be strictly stronger... it can be incomparable. It can be weaker in a lot of ways, as is the case here where we only use quantifier-free induction and primitive-recursive functions. It just can't be weaker in all ways, hence the need for $\epsilon_0$ induction, which is just out of reach for PA.

spaceisdarkgreen
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