I would be surprised if the motivation for projective irreps does not also motivate extending higher up the Postnikov tower; you don’t just stop at the universal cover.
As for sets, suppose you went with the definition: any function that takes in a rational number and spits out a boolean. So the Dedekind cut for could be expressed as
sqrt2 = lambda x. (x * x < 2)
I expect that
((serpinski sqrt2) 3/4)
could indeed be -reduced to a boolean, if you choose the right path. So basically, (serpinski S) is also a set if S is a set.
I would be surprised if the motivation for projective irreps does not also motivate extending higher up the Postnikov tower; you don’t just stop at the universal cover.
As for sets, suppose you went with the definition: any function that takes in a rational number and spits out a boolean. So the Dedekind cut for
I expect that
could indeed be
(serpinski S)is also a set ifSis a set.