Let us begin with a
Notational convention: throughout this chapter, when dealing with the systems G n Aω,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 Aω 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 Aω) 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 Aω 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.
Mathematics Subject Classification (2000)
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© 2008 Springer-Verlag Berlin Heidelberg
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(2008). Elimination of monotone Skolem functions. In: Applied Proof Theory: Proof Interpretations and Their Use in Mathematics. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77533-1_13
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DOI: https://doi.org/10.1007/978-3-540-77533-1_13
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