Conservativeness of PAV

This post continues on the theme of the argument against deflationism, concerning the conservativness of truth theories. I want to discuss the role of induction.

It turns out that this is a very complicated issue, and people have slipped up a bit over it. But there are some points that aren't too complicated, and one concerns the result of extending PA with new vocabulary, and therefore new induction instances, but without new axioms for the new vocabulary. If there are no new axioms, then the result is conservative.

Suppose we let L be the usual first-order language of arithmetic (non-logical vocabulary is {0,,+,×}) and let
V:={P1,}
be some new vocabulary. Each Pi is a predicate symbol (say, of arity ki). Let LV be the extended language. Let PA be the usual system of Peano arithmetic in L, with the induction scheme,
ϕ0xx(ϕϕxx)xϕ
(here ϕ is a formula (possibly with parameters), and ϕtx is the result of substituting the term t for all free occurrences of x, relabelling bound variables inside ϕ if necessary to avoid collisions).

Definition: PAV is the result of extending PA with all instances of induction for the extended language LV.

TheoremPAV conservatively extends PA.

Proof. The "proof idea" is to give a translation that maps every proof (derivation) in the "bigger" theory into a proof in the "smaller" theory. We define a translation
:LVL
as follows. For any L symbol, including variables and terms, the translation is the identity mapping. For compounds, we assume just commutes. For a n-ary predicate symbol PV, we translate an atomic formula P(t1,,tn) as follows:
(P(t1,,tn)):=(P)(t1,,tn).
where (P)(x1,,xn) is any L-formula. This translation is crazy, of course. But we have no axioms constraining the symbol P, so we can translate it any way we like. (If we did have such axioms, say AxP, we would need to try and translate AxP as a theorem of PA.)

The formula (P(t1,,tn)) is then an L-formula. This means that, for any ϕLV, (ϕ) is equivalent to some L-forrnula, say θ. Now consider the induction axiom for any ϕLV:
ϕ0xx(ϕϕxx)xϕ
Its translation under :LVL is equivalent (because commutes with connectives and quantifiers) to
θ0xx(θθxx)xθ
And this is an induction axiom in L. Hence, it is an axiom of PA.
Finally, suppose that
PAVψ
where ψL. Then applying the translation to all the formulas in the derivation of ψ converts the derivation into a derivation of ψ in PA (in L), as each induction instance is translated to an induction instance in L (which is an axiom of PA) and the translation preserves too. So,
PAψ.
QED.

This may be applied to the case where the new vocabulary is a predicate T(x), perhaps to be thought of as a truth predicate, but we do not include any new axioms for T(x) itself. Halbach 2011 calls the corresponding theory PAT. So, PAT conservatively extends PA.

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