  This paper is about the recent notion of computably probably approximately
correct learning, which lies between the statistical learning theory where
there is no computational requirement on the learner and efficient PAC where
the learner must be polynomially bounded. Examples have recently been given of
hypothesis classes which are PAC learnable but not computably PAC learnable,
but these hypothesis classes are unnatural or non-canonical in the sense that
they depend on a numbering of proofs, formulas, or programs. We use the
on-a-cone machinery from computability theory to prove that, under mild
assumptions such as that the hypothesis class can be computably listable, any
natural hypothesis class which is learnable must be computably learnable. Thus
the counterexamples given previously are necessarily unnatural.
