In classical logic plus ZF, the field of real numbers admits infinitely many isomorphic realizations as a numeral system --- as the radix varies. The intuitionistic status of these systems seems less clear however. First of all, the notion of "field" has several
distinct intuitionistic interpretations (e.g. non-zero implies invertible; non-invertible implies zero; invertible or zero). And even with a way to set up things to make these numeral systems fields in some appropriate sense, one still may lack for the maps between them that make them isomorphic.


If a topos $T$ has a natural number object, surely one can make sense of "the numeral system with radix $n$" at least with $n$ any fixed (real world) natural number > 1. So fixing $T$ one gets an equivalence relation on the real world natural numbers according to the isomorphism of numeral systems of various radix. (Well, depending upon the topos and the interpretation of numeral system, perhaps one might have to discard some radices altogether, where one doesn't have total operations.)


My question: Can one realize any equivalence relation on the natural numbers by choosing some appropriate topos? Failing a full answer, what example topos distinguish some systems but not others?



Radix notation and toposes
No, and actually, you cannot realise any non-trivial equivalence relation this way. If any of the (non-trivial) pairs of radix systems are isomorphic, they all are. It is "well-known" that not every real number has a decimal expansion. This extends to this situation.


Specifically, for any pair n and m (unless one divides the other), the assumption that every radix-n number has a radix-m expansion impies a weak form of the law of the excluded middle, called the limited principle of omniscience (LPO): this says that for every sequence of integers, either it is constant zero, or it has a non-zero element. Given a sequence of integers, you can construct a radix-n number such that the first digit of the corresponding radix-m number decides the answer. (E.g. given base-10 0.3333333... convert it to base-3, it's 0.100000... or 0.022222...)


Conversely if LPO holds, you can show all radix-n systems are equivalent.


It's also been pointed out that these number systems aren't necessarily even rings. The same thing happens - if any of them are rings, then LPO holds. And if LPO holds, then they are all fields (in any sense).


This reasoning goes through in any topos (even Pi-pretopos) with a natural numbers object.


