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?