An Introduction to Proof Theory

Abstract

Proof theory is a central area of mathematical logic of special interest to philosophy. It has its roots in the foundational debate of the 1920s, in particular, in Hilbert’s program in the philosophy of mathematics, which called for a formalization of mathematics, as well as for a proof, using philosophically unproblematic, “finitary” means, that these systems are free from contradiction. Structural proof theory investigates the structure and properties of proofs in different formal deductive systems, including axiomatic derivations, natural deduction, and the sequent calculus. Central results in structural proof theory are the normalization theorem for natural deduction, proved here for both intuitionistic and classical logic, and the cut-elimination theorem for the sequent calculus. In formal systems of number theory formulated in the sequent calculus, the induction rule plays a central role. It can be eliminated from proofs of sequents of a certain elementary form: every proof of an atomic sequent can be transformed into a “simple” proof. This is Hilbert’s central idea for giving finitary consistency proofs. The proof requires a measure of proof complexity called an ordinal notation. The branch of proof theory dealing with mathematical systems such as arithmetic thus has come to be called ordinal proof theory. The theory of ordinal notations is developed here in purely combinatorial terms, and the consistency proof for arithmetic presented in detail.

Takeuti's Well-ordering Proof

“… Akiyoshi and Arana indicate that they intend to further explore Takeuti's philosophical commitments in that regard and we are hopeful that such investigations will contribute to both our understanding of Takeuti's thought, as well as the space of modern constructivist positions more generally.2 It appears that[2] also contains a reconstruction or reformulation of Takeuti's wellordering proof using modern techniques, however that paper is written in Japanese (which we are regrettably unable to read), and approaches the reconstruction from a different perspective than the present paper.3 This section has been adapted from §3 of Darnell & Thomas-Bolduc[6]. For an introduction to these methods and proofs requiring little background in proof theory, see[14].Australasian Journal of Logic (19:1) 2022, Article no. 1…”