At least for fragments of arithmetic, a more explicit dividing line between weak and strong systems has been used (in the cited chapter 2 http://www.math.ucsd.edu/~sbuss/ResearchWeb/handbookII/): “The line between strong and weak fragments is somewhat arbitrarily drawn between those theories which can prove the arithmetized version of the cut-elimination theorem and those which cannot; in practice, this is equivalent to whether the theory can prove that the superexponential function is total.”

According to this dividing line, Friedman’s exponential function arithmetic (EFA) is a weak fragment of arithmetic. But if the question is whether independence results for P != NP can be proved, then EFA feels like a really interesting candidate, precisely because it cannot prove cut-elimination. It would cast an interesting light on the role of higher-order reasoning.

Of course, it is unrealistic to hope to demonstrate that “P!=NP cannot be proven in EFA”. But if you believe P=NP, then also EXP=NEXP, …, 3EXP=N3EXP, … follows. So maybe it is less unrealistic to hope to demonstrate that “3EXP!=N3EXP cannot be proven in EFA”, or at least that there is some K for which it is possible to demonstrate that “KEXP!=NKEXP cannot be proven in EFA”…

]]>otherwise, a general text on proof complexity, I recommend, is Krajicek’s latest book: https://www.karlin.mff.cuni.cz/~krajicek/prf2.pdf

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