EM + Ext- + ACint is equivalent to ACext

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 is an instance of the premise needed to apply intensional choice (switch A and B). It says that there is a function g : B → A such that (∀y : B)(y =B f(g(y))) is true, that is, a right inverse of f . This, however, does not mean that g is extensional, that is, that it preserves equivalence relations. If =A is an equivalence relation on A and =B is an equivalence relation on B, it might very well happen that f preserves them but g does not. If both A and B are the set of Cauchy sequences of rational numbers, for instance, and =A is pointwise equality, while =B is real number equality, then the identity function is clearly extensional from A/=A to B/=B, but not in the other direction. Thus it has a right inverse (itself) which is not extensional. In fact, we have no reason to expect that we should be able to construct an extensional right inverse in this case. The principle ACext of extensional choice states that there is an extensional function with the property required. To be precise, it states that if R is an extensional relation (that is, it respects the equivalence relations) and (∀x : A)(∃y : B)R(x, y) is true, then there is an extensional function f : A → B such that (∀x : A)R(x, f(x)) is true. As the before-mentioned example with real numbers indicates, one cannot justify ACext constructively. In fact, it implies