Proof of Theorem 4 [Part (ii)]

Abstract

The purpose of this chapter is to prove part (ii) of Theorem 4, which we restate for convenience: Theorem 4 [Part (ii)]. For each nonlexical sequence α = α₀, α₁,…,α_k) ε Ρ, the set C(α) is empty. Remarks. (a) For L_ζ R_ζ -sequences beginning with R ^ℓ_ζ (ℓ ≥ 2) on the right, the proof of the theorem has already been given in the proof of Theorem 5 [see Eqs. (7.6)]. (b) The proof of this theorem draws on Theorems 4 [part (i)], Corollary ₂ (see the Remarks at the end of Chapter 7), the results of the previous chapter summarized in Lemma 17, the notion of a maximal lexical sequence given in Definition 9 (Chapter 7), and properties of lexical and nonlexical sequences developed in Appendices B and D. Accordingly, its proof must be regarded as quite difficult. The proof is given in Lemmas 18–19 below, which assert important properties of the _X-functions of nonlexical sequences.