Ideas in the epsilon substitution method for Π10-FIX

Dedicated to Professor Wolfram Pohlers on the occasion of his 60th birthday
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Abstract

Hilbert proposed the epsilon substitution method as a basis for consistency proofs. Hilbert’s Ansatz for finding a solving substitution for any given finite set of transfinite axioms is, starting with the null substitution S0, to correct false values step by step and thereby generate the process S0,S1,. The problem is to show that the approximating process terminates. After Gentzen’s innovation, Ackermann [W. Ackermann, Zur Widerspruchsfreiheit der Zahlentheorie, Math. Ann. 117 (1940) 162–194] succeeded in proving the termination of the process for the first order arithmetic.

In this note we report recent progress on the subject, and expound basic ideas of the epsilon substitution method à la Ackermann for the theory Π10-FIX of non-monotonic Π10 inductive definitions.

MSC

03F05
03F35koko

Keywords

Epsilon substitution
Termination proof

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