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If you ask a mathematician, particularly a set theorist, about whether ZFC is consistent, they will answer that we can't know for sure because of Gödel's theorems. If you ask what evidence at all is there that ZFC is actually consistent, one reply (and I do know the reply involving V) is that ZFC has been around for a century and no mathematician has ever found a contradiction, which is inductive evidence that ZFC is most likely consistent.

Does it follow, therefore, that the following statement is true?

Even one hour of a mathematician working in or with ZFC and finding no contradiction leads to the probability of "Con(ZFC)" getting closer to 100%.

I do realize that (almost certainly) if there is such an increment, it is unfathomably small (but still finite) and because we are dealing with induction we can never reach 100%.

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    I am not sure that probability of "Con(ZFC)" is a clear enough concept to talk about coherently. That aside, there would have to be a minimal duration threshold for a mathematician to be able to do something consequential, a minute might be too short, and it has to be independent of what others have already done to contribute as new evidence for a Bayesian update. Reprising old arguments around Russell's paradox, etc., would not count. Time elapsed is only a proxy for the amount of independent work done around consistency since ZFC was formulated.
    – Conifold
    Dec 26, 2020 at 10:50
  • @Conifold, leave an answer, dammit! :D Your comments on my questions are always half-answers.
    – Alex
    Dec 26, 2020 at 10:55
  • @Alex I feel you, Conifold tends to do so ;) About the question: perhaps you'd want to widen it to be a question about induction and logic ("can induction be an incremental proof going forward instead of non-disproof"); but I think the root of the question here is: what does it matter? We won't ever reach that 100%, as you admitted, so how is it any different from how we treat induction anyway? Dec 30, 2020 at 8:02
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    I don't understand why this was closed; it seems like a fine philosophy of mathematics question to me. Jan 5, 2021 at 2:23
  • @NoahSchweber, well, weird stuff tends to happen on SE, it's alright...
    – Alex
    Jan 6, 2021 at 2:00

2 Answers 2

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I would respond that the question is moot: it is too difficult (at present) to talk in a meaningful way about the probability of a mathematical statement.

This is because one of the standard interpretations (or classes of interpretations) of probability is as quantifying over "possible states" - so that when we say that flipping a coin results in heads with probability 50%, what we mean is something like "50% of the possible outcomes of flipping a coin result in heads." Exactly what constitutes a "possible state" is a source of intense discussion, but the general theme is certainly a common one.

The problem with this type of interpretation is that it's not clear what "possible state" should mean in the context of mathematics, at least if we assume a "naive Platonist" assumption. A statement like "There is a contradiction in ZFC of length <10^10" is either true, full stop, or false, full stop. And it's exactly this type of sentence for which we need to develop a notion of probability in order to make sense of your question.

The notion of degrees of confidence rears its head now, but ultimately doesn't seem to do much better: this interpretation is tied to the notion of an ideal rational agent, and it's hard to pin down how such an agent can have limited mathematical knowledge.

I think this all becomes clearer when we replace Con(ZFC) with something not philosophically loaded. For example, how does verifying yet another case of Goldbach's conjecture affect (our assessment of) the probability of its truth? I think beyond a certain point it doesn't. Checking "small" cases of a given conjecture may verify that a wide range of ideas for producing counterexamples are not (trivially) successful, but beyond that it doesn't do much, and since ruling out larger counterexamples doesn't rule out additional ideas as far as we can tell, it doesn't yield a meaningful increase in confidence.


So how comfortable should we be about the consistency of ZFC?

Well, I personally would argue: not very! There are many arguments for its consistency, but they're all rather fuzzy (necessarily so). But I think this draws attention exactly to one of the strengths of the ZFC-style framework, namely that it provides not just a single system but a well-understood hierarchy of systems which give us strategies for handling discovered inconsistency. Fundamentally, very little would change in non-set-theoretic mathematics if an inconsistency were discovered in ZFC, because (i) ZFC is overkill for almost everything and (ii) we understand how ZFC is overkill for almost everything.

  • Incidentally, the systems studied in reverse mathematics let us continue that hierarchy further down, that is further in the direction of confident consistency - roughly speaking, I'd say that the weakest of the natural ZFC-style systems is KP, and this is between the reverse mathematical systems ACAo and ATRo in a couple precise senses.

It's worth noting that this hierarchy extends upwards as well, via large cardinal axioms. Basically, ZFC/large-cardinal-style set theories provide both a framework for handling discovered inconsistencies and a framework for understanding results which, whether we realize it or not, are independent of ZFC. While of course this is not perfect (per Godel), it is incredibly rich and successful, and to my mind constitutes the real foundational advantage of ZFC and the large cardinal framework (at the moment ... :P).

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  • A brilliant and technical answer widening my understanding of the foundations. Thank you!
    – Alex
    Dec 26, 2020 at 22:36
  • BTW, you didn't finish the last sentence of your third paragraph.
    – Alex
    Dec 27, 2020 at 12:23
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I agree with Noah Schweber's answer, but I think if we swap out Con(ZFC) for something else, we might still be able to find a plausible example of a mathematical conjecture for which we might say there is evidence for its truth even though we lack a proof.

I have in mind the conjecture from computational complexity theory: P is not identical with NP. In simple terms it states that the class of decision problems that are solvable in polynomial time is not identical (is a proper subset of) the class of decision problems that are solvable in nondeterministic polynomial time. NP problems have solutions that can be verified in polynomial time, but not solved in polynomial time.

Over the last 60 years, thanks to the development of inexpensive computers, a huge amount of work has been done in developing algorithms for solving decision problems. We have become good at finding solutions for problems, demonstrating that some solutions have equivalent complexity to others, and assigning solutions to a number of distinct complexity classes. If it were in fact the case that P = NP, it would mean that for every problem that we know can be solved in NP time, there must exist an algorithm that solves it in polynomial time, we just haven't discovered it yet. Given that there are a lot of NP problems, and a great many clever people have tried to find polynomial solutions for them, it just seems highly implausible to suppose that these solutions are simply eluding us.

So, maybe, we could say that inductively the more we study decision problems and algorithms, and the more we find that there are no polynomial algorithms that solve NP problems, the more plausible it becomes to believe that P is not identical with NP, even in the absence of a proof. Perhaps it would be best not to call this a probability, just a high degree of plausibility.

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    What gives me pause is the obscurity of "degree of plausibility". Is it quantifiable, does it obey Bayesian rules? What exactly do we mean by degree being "high"? Primality was long considered to be hard, and wasn't. We could even say, perhaps, that all our encounters with consistency or NP problems took place (and will always take place) in a "0 measure" subset of the whole, so their contribution to plausibility is exactly 0 in that sense. Or maybe that we did not encounter X in some area D is a good predictor that we will continue not to in the future, but not at all that there is no X in D.
    – Conifold
    Dec 29, 2020 at 3:57

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