 The constructions of the natural numbers and the real numbers with which we started go much as usual, except that we get two kinds of natural numbers (the finite von Neumann ordinals in the set universe (nonstandard), and the finite von Neumann set ordinals (standard)). The classical reals can be defined as Dedekind cuts in the standard rationals; these are not sets, but any real can then be approximated by a nonstandard rational. One can proceed to do analysis with some (but not quite all) of the tools of the usual nonstandard analysis.