Page "Constructivism (mathematics)" Paragraph 22

In intuitionistic theories of type theory ( especially higher-type arithmetic ), many forms of the axiom of choice are permitted.

For example, the axiom AC < sub > 11 </ sub > can be paraphrased to say that for any relation R on the set of real numbers, if you have proved that for each real number x there is a real number y such that R ( x, y ) holds, then there is actually a function F such that R ( x, F ( x )) holds for all real numbers.

Similar choice principles are accepted for all finite types.

The motivation for accepting these seemingly nonconstructive principles is the intuitionistic understanding of the proof that " for each real number x there is a real number y such that R ( x, y ) holds ".

According to the BHK interpretation, this proof itself is essentially the function F that is desired.

The choice principles that intuitionists accept do not imply the law of the excluded middle.