Logic and the Axiom of Choice

Publisher Summary This chapter focuses on logic and the axiom of choice. The chapter proves that (1) ∀x∃y φ(x,y) →∃f∀x φ(x, fx) is conservative over classic first order logic, (2) ∀x∃y φ(x, y)→∃f∀x φ(x, fx) is conservative over intuitionistic logic without equality, (3) ∀x∃y φ(x, y)→∃f∀x φ(x, fx) is conservative over intuitionistic logic with decidable equality, and (4) ∀x∃y φ (x, y)→ ∃f∀x φ(x, fx) is not conservative over intuitionistic logic. Addition of finitely many instances of the respective schema with all parameters generalized is conservative over any first order theory in the respective logic.