@MISC{Carlström04em+, author = {Jesper Carlström}, title = {EM + Ext − + ACint is equivalent to ACext}, year = {2004} }

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Abstract

It is well known that the extensional axiom of choice (ACext) implies the law of excluded middle (EM). We here prove that the converse holds as well if we have the intensional (‘type-theoretical’) axiom of choice ACint, which is provable in Martin-Löf’s type theory, and a weak extensionality principle (Ext−), which is provable in Martin-Löf’s extensional type theory. In particular, EM ⇔ ACext holds in extensional type theory. The following is the principle ACint of intensional choice: if A, B are sets and R a relation such that (∀x: A)(∃y: B)R(x, y) is true, then there is a function f: A → B such that (∀x: A)R(x, f(x)) is true. It is provable in Martin-Löf’s type theory [8, p. 50]. It follows from ACint that surjective functions have right inverses: If =B is an equivalence relation on B and f: A → B, we say that f is surjective if (∀y: B)(∃x: A)(y =B f(x)) is true. With R(y, x) def = (y =B f(x)), surjectivity