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I'm aware of provability logics which have a notation $\square P$ for "$P$ is provable", but I'm not aware of one which is more 'fine grained' and has a notion of proof term size (e.g. with predicates $\mathsf{proof}(x, P)$ for "$x$ is a proof of $P$" and $\mathsf{size}(x, n)$ for "$x$ is a term of size $n$").

Such a system could express the statement "there is no proof term for $\bot$ of size less than $3\widehat{}\widehat{}\widehat{}3$". Perhaps it has a proof of that statement, and the proof term is (much) smaller than $3\widehat{}\widehat{}\widehat{}3$. This leads me to some follow-up questions:

  • Would this result be useful? I assume most mathematicians want their logical systems to be completely consistent, but I could settle for "consistent within the bounds of proof terms I can ever observe".
  • Is there some obvious diagonalization/Gödelian argument which prevents this sort of result?
  • Is this somehow related to paraconsistent arithmetic, in the sense that we don't care about contradictions which are sufficiently 'far away'?
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    $\begingroup$ You may be interested in some work in ultrafinitism, such as Vladimir Yu. Sazonov's "On Feasible Numbers". Here the premise is to show that a system of arithmetic, plus the axiom that "$2^{1000}$ doesn't exist" has a shortest proof of contradiction of a particularly large size (specifically too large to actually write down in the known physical universe). $\endgroup$
    – TomKern
    Aug 2 at 3:03
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It’s easy to express concepts like proof length in any sufficiently complex logical language through Godelization. Once you can express a proof as a number or string, you can define a function as it’s length, so it’s not necessary to add a new symbol for it beyond syntactic sugar.

Complexity is a pretty common concepts in logic and computer science. There are numerous theorems related to it https://en.m.wikipedia.org/wiki/Proof_complexity

Generally speaking, bounding by a fixed constant isn’t that useful since there are infinitely many true things, so you won’t be able to prove everything true within a fixed length. A more useful concept is whether every true statement admits a polynomially bounded proof. The existence of such a proof system is an open problem.

Showing whether any given theorem has a lower bound on proof size is also a generally difficult problem, though we have some proofs in specific cases.

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