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.