Epsilon substitution for transfinite induction

Epsilon substitution for transfinite induction

We apply Mints’ technique for proving the termination of the epsilon substitution method via cut-elimination to the system of Peano Arithmetic with Transfinite Induction given by Arai.

Epsilon Substitution for ID1

Hilbert’s epsilon substitution method provides a technique for showing that a theory is consistent by producing progressively more accurate computable approximations to the non-computable components of a proof. If it can be shown that this process eventually halts with a sufficiently good approximation, the theory is consistent. Here we produce a new formulation of the method for the theory ID1...

Epsilon substitution method for elementary analysis

Transfinite Induction within Peano Arithmetic

The relative strengths of first-order theories axiomatized by transfinite induction, for ordinals less-than ~0, and formulas restricted in quantifier complexity, is determined. This is done, in part, by describing the provably recursive functions of such theories. Upper bounds for the provably recursive functions are obtained using model-theoretic techniques. A variety of additional results tha...

Arithmetical transfinite induction and hierarchies of functions

We generalize to the case of arithmetical transfinite induction the following three theorems for PA: the Wainer Theorem, the Paris–Harrington Theorem, and a version of the Solovay–Ketonen Theorem. We give uniform proofs using combinatorial constructions.