Proof Theory: Some Applications of Cut-Elimination

Abstract

This chapter discusses the Cut-Elimination Theorem for first-order logic. The proof of Cut-Elimination Theorem is set up in such a way that it can be easily generalized to many other cases where a cut-elimination argument is applied. The chapter also discusses the provability and unprovability of initial cases of transfinite induction for arithmetic Z. The result is well known: given a natural well-ordering < of order type ɛ0, with respect to <, transfinite induction is provable up to any ordinal < ɛ0, but not up to ɛ0 itself. The underivability in Z of transfinite induction up to ɛ0 will also follow from Gödel's Second Incompleteness Theorem, together with the fact that transfinite induction up to ɛ0 suffices to prove the reflection principle for Z and hence the consistency of Z. A direct proof of this underivability result is presented using a cut-elimination argument. Technically, this provides an easy and convincing example of the usefulness of infinite derivations and the strength of the cut-elimination method when applied to infinite derivations.