What is the standard reference for the provability of the cut-elimination theorem in PRA?

Update: Rasmus Blanck has offered a reference for a system other than Gentzen's $\mathfrak L \mathfrak K$. The bounty is for a reference for the provability of the cut-elimination theorem for the sequent calculus $\mathfrak L \mathfrak K$ in PRA.

  • $\begingroup$ I don't have library resources where I am, so I can't verify the two suggestions by Not Mike and Rasmus before the bounty runs out. I am awarding a bounty to both answers. $\endgroup$ Sep 24, 2017 at 20:53

2 Answers 2


My suggestion is Theorem 5.17 in section V.5.(d) of Hájek and Pudlák's Metamathematics of First-Order Arithmetic, Springer-Verlag, 1993. There, cut-elimination is shown to be provable in the even weaker theory $I\Delta_0 + \text{superexp}$.

  • 3
    $\begingroup$ Hájek and Pudlák appear to formulate the cut-elimination theorem for a system substantially different from Gentzen's $\mathfrak L \mathfrak K$, and adapting their result to derivations in $\mathfrak L \mathfrak K$ looks like a bit of a chore. Can you suggest another source for derivations in $\mathfrak L \mathfrak K$, or a definitive statement linking the two systems? $\endgroup$ Sep 17, 2017 at 6:04
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    $\begingroup$ Perhaps chapter 1 of Handbook of Proof Theory (edited by Samuel Buss, Elsevier, 1998), might be what you are looking for. Cut-elimination is discussed in section 2.4. $\endgroup$
    – user103227
    Sep 23, 2017 at 19:10

I think the chapter "Proof Theory: Some Applications of Cut-Elimination", by Helmut Schwichtenberg, from "Handbook of Mathematical Logic", might be what you are looking for.


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