Thomas Streicher <streicher@mathematik.tu-darmstadt.de> writes:
Recently I was asking myself what is the relation between the arithmetical (Dedekind) reals in a topos and a ring R satisfying the usual SDG axioms (i.e. at least the Kock-Lawvere) axiom.
Perhaps it is worth mentioning that in the ring of smooth reals R the sequence a_k = 2^(-k) is Cauchy but has many "limits" because every infinitesimal dx satisfies the condition "dx is the limit of a_k". This shows that R is not Cauchy complete, not because limits of Cauchy sequences are missing but because there are too many. I once thought the above observation implied there can be no isometric embedding of a Cauchy-complete field (e.g. the Dedekind reals) into R, but now I am not convinced anymore. Andrej Bauer Department of Mathematics and Physics University of Ljubljana http://andrej.com