> To some extend this can be encoded in set theory, but only with ridiculously large (Mahlo) cardinals.
I don't see why the size of Mahlo cardinals is ridiculous. Considering all the strongly inaccessible cardinals that have been discovered, Mahlo cardinals is rather weak. Also, the definition of a Grothendieck universe strongly resembles the definition of a strongly inaccessible cardinal, imho. That they are equivalent are (relatively) straightforward.
Harvey Friedmans work on provable equivalences between large cardinals way larger than Mahlo cardinals and theorems about objects in the domain of the rationals further substantiates this. Very fascinating.
Just to clarify a little bit: For each integer n there is also an infinite amount of rationals (fractions) between n and n+1 and the rationals is also a countable set like the integers (just as 'infinite', so to speak). Quite counter-intuitive, I think.
I don't see why the size of Mahlo cardinals is ridiculous. Considering all the strongly inaccessible cardinals that have been discovered, Mahlo cardinals is rather weak. Also, the definition of a Grothendieck universe strongly resembles the definition of a strongly inaccessible cardinal, imho. That they are equivalent are (relatively) straightforward.
Harvey Friedmans work on provable equivalences between large cardinals way larger than Mahlo cardinals and theorems about objects in the domain of the rationals further substantiates this. Very fascinating.