First-order Peano Arithmetic
Peter James Smith · Cambridge University Press eBooks · 2013
In the last chapter, we considered the theory IΔ 0 built in the language L A , whose axioms are those of Q, plus (the universal closures of) all instances of the Induction Schema for Δ 0 predicates. Now we lift that restriction on induction, and allow any L A predicate to appear in instances of the Schema. The result is (first-order) Peano Arithmetic. Being generous with induction (a) Given what we said in Section 9.1(a) about the motivation for the induction principle, any instance of the Induction Schema will be intuitively acceptable as an axiom, so long as we replace φ in the Schema by a suitable open wff which expresses a genuine property/relation. We argued at the beginning of the last chapter that Δ 0 wffs are eminently suitable, and we considered the theory you get by adding to Q the instances of the Induction Schema involving such wffs. But why should we be so very restrictive? Take any open wff φ of L A at all. This will be built from no more than the constant term ‘0’, the familiar successor, addition and multiplication functions, plus identity and other logical apparatus. Therefore – you might very well suppose – it ought also to express a perfectly determinate arithmetical property or relation. So why not be generous and allow any open L A wff to be substituted for φ in the Induction Schema? The result of adding to Q (the universal closures of) every instance of the Schema is PA – First-order Peano Arithmetic . (b) …