Elimination of monotone Skolem functions

Let us begin with a

Notational convention: throughout this chapter, when dealing with the systems G _n A^ω,G _n A ^ω _i from chapter 3, we will not include all the arbitrary universal axioms ‘11)’ but only universal sentence that w.r.t. to the canonical embedding of G _n A^ω into $\widehat{\mbox{WE-PA}}^{\omega}|\!\raisebox{1mm}{\ensuremath{\scriptscriptstyle\setminus}}$ (i.e. with the standard primitive recursive definitions of the constants and the relation ≤ from ℒ(G _n A^ω) that are not included as primitive notions in ℒ( $\widehat{\mbox{WE-PA}}^{\omega}|\!\raisebox{1mm}{\ensuremath{\scriptscriptstyle\setminus}}$ )) are provable in $\widehat{\mbox{WE-PA}}^{\omega}|\!\raisebox{1mm}{\ensuremath{\scriptscriptstyle\setminus}}$ . Most notably this yields the schema of quantifier-free induction QF-IA. In this way these systems will (modulo that aforementioned canonical embedding) be subsystems of $\widehat{\mbox{WE-PA}}^{\omega},\widehat{\mbox{WE-HA}}^{\omega}$ . Since we allow in the main results of this chapter arbitrary further axioms of the form Δ (as introduced in theorem 10.21) this, anyway, covers any additional universal axioms we might want to use as well. The reason why we included arbitrary universal axioms when defining the systems G _n A^ω in chapter 3 was that for the cases where n=1,2 it would be quite tedious to verify whether certain basic arithmetical facts are provable. However in this chapter we mainly deal with the case n≥3 or even n=∞ where these things are much easier to verify.