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An assumption underlying this earlier question was the existence (and greater expressive strength) of infinite proofs in logics like $\mathcal{L}_{\omega_{1}^{CK}, \omega}$ (based on, for example, the discussion in §2 chap. IX of Mathematical Logic by Ebbinghaus et. al., and answers like this and this)

However the comments in response to the question have led me to question this assumption. For example:

  • Comment by @Z. A. K. stating that: "you can't just talk about 'proof' in the infinitary language $\mathcal{L}_{\omega_{1},\omega}$ since abstract logics don't come equipped with a specific proof theory or even any specific notion of proof" but "there are other ways we can make sense of infinite proofs".
  • Other comments suggesting that infinite formulas are in many cases equivalent to finite formulas in some sufficiently strong first-order theory and that infinitary proofs "don't do anything new."

All this leads me to the following "prequel" question: In what sense do infinite proofs actually exist? Are there formal proof systems which allow for actual infinite proofs? What are, to borrow the languange of the aforementioned comment, the "other ways we can make sense of infinite proofs"?

  • The idea I have in mind of "infinite proof" is one involving either infinitely many steps or statements/formulas of infinite length (but maybe this conception of "infinite proof" is also wrong in some way?)

I realize this might not be feasible to explain in comprehensive detail in the space of a StackExchange answer, but a summary and/or reference to further reading would be quite helpful.

Tankut Beygu
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NikS
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    I have discovered a truly marvelous proof, which this observable universe is too narrow to contain. – not_a_chatbot Dec 07 '24 at 13:43
  • One axiom system's infinite is another axiom system's finite. For example ZFC allows for transfinite induction. From the point of view of PA, those cannot been proofs because they would require infinite proofs. Outside of formal logic, I'm unaware of anything used in the rest of mathematics that can't be described as a finite proof in a sufficiently strong first-order system. (Which usually is ZFC.) – btilly Apr 03 '25 at 20:26
  • @btilly : You included the qualifier “outside formal logic” — does that mean there is a well-known type of formal logic that does formalize a notion of infinite proof? – NikS Apr 04 '25 at 04:44
  • @NikS Logicians have indeed studied that. See https://plato.stanford.edu/entries/logic-infinitary/ (linked in one of your links) for an example. Logicians have also invented the idea of second order logic, which is a concept that can't be properly expressed in first order logic. – btilly Apr 04 '25 at 04:50
  • @btilly : Is it necessarily true that an infinite proof in such an infinitary logic can always be “translated” into a finite proof in some other sufficiently strong first-order theory? (i.e. some other theory in ordinary, finitary first-order logic) – NikS Apr 04 '25 at 05:00
  • Godel's first establishes a statement (for a somewhat arbitrary logic) $P(k)$ where $\forall k ~P(k)$ has no proof but there is a proof for $P(0)$ and a proof for $P(1)$ etc. So in some sense the list of proofs of each $P(k)$ is a truly infinite proof with no finite version for that specific logic. – DanielV Apr 04 '25 at 05:53
  • @NikS I'm sure that some logician has a complicated answer to that. As for me, it is sufficient to note that any proof which can be verified by people can also be verified by some computable system, which in turn can be reduced to arithmetic, which can be modeled in first order logic. I'm willing to accept that as a translation, even though the first order system doing the modeling doesn't mean the same thing as the original proof. – btilly Apr 07 '25 at 21:19

2 Answers2

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Henry Towsner has a preprint Proofs that Modify Proofs which describes a system that works with proof trees which may not be well-founded. This is recent work and I am really only aware of its existence, not any of the particulars. I'm not quite sure if this is the sort of thing you were looking for.

Old Math Logan
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    To add to this answer, there is a (growing) area of proof theory that studies non-wellfounded proofs (and circular proofs). Also, infinite wellfounded proofs feature in ordinal analysis. For example, the $\omega$-rule is an example of an inference rule which has an infinite number of premises. – Alvaro Pintado Dec 07 '24 at 14:36
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If I understand this properly, the easiest way to do this would be to demand that universal quantification be written out to continuous conjunctions over an infinite set in a domain of discourse, and that existential quantification be written out to continuous disjunctions over an infinite set in a domain of discourse.

Then, yes, the proof for a universal theorem would have infinitely many steps. This is precisely why we instead permit arbitrary instantiations in such proofs.

Joshua Harwood
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