Abstract
We show that in elementary analysis (EL) the contrapositive of countable choice is equivalent to double negation elimination for \({\Sigma_{2}^{0}}\)-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.
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Ishihara, H., Schuster, P. On the contrapositive of countable choice. Arch. Math. Logic 50, 137–143 (2011). https://doi.org/10.1007/s00153-010-0205-8
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DOI: https://doi.org/10.1007/s00153-010-0205-8
Keywords
- Contrapositive of countable choice
- Double negation elimination
- Heyting arithmetic
- Elementary analysis
- Recursive
- Decidable
- Church’s thesis