Arithmetical conservation results

Abstract

Abstract. In this paper we present a proof of Goodman's Theorem, a classical result in the metamathematics of constructivism, which states that the addition of the axiom of choice to Heyting arithmetic in finite types does not increase the collection of provable arithmetical sentences. Our proof relies on several ideas from earlier proofs by other authors, but adds some new ones as well. In particular, we show how a recent paper by Jaap van Oosten can be used to simplify a key step in the proof. We have also included an interesting corollary for classical systems pointed out to us by Ulrich Kohlenbach.

Extensional Realizability and Choice for Dependent Types in Intuitionistic Set Theory

“…By the foregoing discussion, we have that ( 7), (8), and (11) hold true for some z, y ∈ V ext (A). By the properties of pairing and equality, it follows from ( 8) and ( 11) that rca = rdb a , e A = z, for some closed term r. We may then let ((cc) 1 a) 1 = p(c 10 a) 0 (rca).…”

“…On the other hand, every primitive recursive number-theoretic function and hence relation is canonically definable by a Σ formula in the language of set theory (with just ∈ as primitive). 8 We call such formulas primitive recursive. Kleene's T-predicate T (e, x, z) and the function U (z) are among them.…”

“…These are built up from atomic formulas in the new language by closing under logical connectives and bounded quantifiers of the form ∀x ∈ t and ∃x ∈ t, where t is a term not containing x. Needless to say, as long as we add only constants, since parameters are allowed, bounded separation in the enriched language is equivalent to bounded separation in the primitive language with ∈ and = only. 8 A formula is Σ if it is built up from bounded formulas by means of ∧, ∨, bounded quantifiers ∀x ∈ y and ∃x ∈ y, and unbounded existential quantifiers ∃x.…”

“…Suppose c, d , z ∈ h. By (4), a = (c 0 c) 0 ∼ (d 0 d ) 0 = b, and for some y ∈ V ext (A), (c 0 c) 110 = (d 0 d ) 110 OP( z, a , y). As before, there are z, y ∈ V ext (A) such that ( 7), (8), and (11) hold true of a and b. On account of (6), it follows from ( 7), (8), and ( 11 For the right to left direction, we aim for a closed term r such that for every (vi) We aim for w X W i = W(X , F ).…”

“…As before, there are z, y ∈ V ext (A) such that ( 7), (8), and (11) hold true of a and b. On account of (6), it follows from ( 7), (8), and ( 11 For the right to left direction, we aim for a closed term r such that for every (vi) We aim for w X W i = W(X , F ). Set = W i.…”