Epsilon – Substitution Method for the Ramified Language and ∆ 11-Comprehension Rule

We extend to Ramified Analysis the definition and termination proof of Hilbert’s substitution method. This forms a base for future extensions to predicatively reducible subsystems of analysis. First such system treated here is second order arithmetic with ∆1-comprehension rule. Introduction The epsilon substitution method is based on the language introduced by Hilbert [7]. The main non-boolean construction of this language is xF [x], read as “an x satisfying the condition F [x]”. When x is numerical, it is interpreted as the least x satisfying F [x]”. Existential and universal quantifiers become explicitly definable by ∃xF [x] := F [ xF [x]] and ∀xF [x] := F [ x¬F [x]]. The main axioms of the corresponding formalism are critical formulas F [t]→ F [ xF [x]] (1) Hilbert suggested an approach (Ansatz) to transforming arbitrary (non-finitistic) number-theoretic proofs into finitistic (combinatorial) proofs by means of the substitution method using substitution S of numerals for (a finite number of) constant epsilon-terms (cf. [7, 15]). If all critical formulas (1) are true under S, it is called a solving substitution for the system E. Hilbert’s approach is based on the following idea of generating substitutions of numerals for closed epsilon-terms. The initial approximation S0 is identically 0. If substitutions S0, . . . , Si are already generated, and Si is not yet a solving substitution, then Si+1 is found by putting Si+1( xF ) = (the least N ≤ t) (Si(F [n]) = true) for the first formula which is false under Si. The problem stated by Hilbert was to prove termination of the sequence S0, S1, S2, . . . after a finite number of steps for any system of critical formulas (1). After von Neumann’s [17] attack on this (see below) Hilbert [6] stated further problems: