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. 117 162–194] 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 -FIX of non-monotonic inductive definitions.