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Published online by Cambridge University Press: 02 December 2009
From Section 10.4 we know that all theories (R) and (R) are distinct. In this chapter we examine specific, more direct independence proofs for theories (R), (R), and(R), and we strengthen Corollary 10.4.3.
Herbrandization of induction axioms
In this section we shall examine the following idea for independence proofs: Take an induction axiom for a (α)-formula. It has the complexity (α). Introduce a new function symbol to obtain a Herbrand form of the axiom, as at the beginning of Section 7.3. But this time we reduce the axiom to an existential formula. This allows us to use a simpler witnessing theorem (Theorem 7.2.3) than the original form of the axiom would require.
Consider first the simplest case (which will turn out to be the only one for which the idea works). Let α(x, y) be a binary predicate. Then the herbrandization of the induction axiom for the formula A(a) ≔ ∃u ≥ a, α(u, a)
is the formula
Denote this formula JNDH(A(a)).
Theorem 11.1.1. The formula INDH(A(a)) is provable in (α, f) but not in (α, f). Hence (α, f) is not (α, f)-conservative over (α, f).
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