Philosophical scope of proof-theoretic methods for securing ideal mathematics

Determine the extent to which proof-theoretic methods and the associated ordinal-notation-based termination arguments for consistency and conservativity proofs—such as the epsilon-substitution method and Gentzen-style analyses relying on induction up to the ordinal ε0—support philosophically robust conclusions about the security of ideal, infinitary mathematics.

Background

The paper explains Hilbert’s plan to justify the use of infinitary mathematics by transforming proofs in an ideal system into proofs in a finitary ‘real’ subsystem, exemplified by the epsilon-substitution method. Such transformations remove problematic infinitary expressions from derivations, but proving that the procedures always terminate typically requires induction on complex ordinal notations, up to ε0 for first-order arithmetic.

Gentzen’s results show that induction up to ε0 suffices to imply the consistency of Peano Arithmetic, while Gödel’s second incompleteness theorem prevents Peano Arithmetic from proving the well-ordering of the corresponding ordinal notations. Consequently, whether these ordinal-based meta-arguments can be justified purely finitarily is contested, raising a broader philosophical issue about what these proof-theoretic achievements establish regarding the ‘security’ of ideal mathematics.

Against this backdrop, the paper explicitly notes that the degree to which such methods and ordinal notation systems warrant philosophical conclusions about the security of infinitary mathematics is not settled, identifying an open question at the philosophy–proof theory interface.