On the contrapositive of countable choice

We show that in elementary analysis (EL) the contrapositive of countable choice (CCC) is equivalent to double negation elimination for Σ2-formulas. By also proving a recursive adaptation of this equivalence in Heyting arithmetic (HA), we give an instance of the conservativity of EL over HA with respect to recursive functions and predicates. As a complement, we prove in HA enriched with the (extended) Church thesis that every decidable predicate is recursive. Throughout let x, y, z, z′, e stand for numbers (i.e., nonnegative integers); f for everywhere defined number-number functions; and P , Q for decidable predicates of numbers. The focus is on the contrapositive of countable choice [5, 2, 1]: CCC ∀f∃xP (x, f (x))→ ∃x∀y P (x, y). In [1, Section 2.4, Lemma 5.5] it was proved that CCC follows from double negation elimination for Σ2-formulas: Σ2-DNE ¬¬∃x ∀y P (x, y)→ ∃x∀y P (x, y); and that CCC implies the law of excluded middle for Σ1-formulas: Σ1-LEM ∃xP (x) ∨ ∀x¬P (x). In [1, Footnote 4] it was also conjectured that CCC lies strictly in between these two instances of the law of excluded middle. The objective of the present note is to show that if one works in elementary intuitionistic analysis EL [4, 3.6], and restricts P to quantifier-free predicates, then CCC is actually equivalent to Σ2-DNE. Now EL is conservative over intuitionistic first-order arithmetic HA ([3] and [4, 3.6.2]) whenever the function variables characteristic of EL are interpreted as ranging over all (total) recursive functions, and thus—by Kleene’s normal form ∗Corresponding author. Where Σ2-DNE and Σ 0 1-LEM were called 2-Markov and 1-EM, respectively.