Questions about hyperreal numbers, as used in non-standard analysis. Please be patient with this long post. This should probably go in linear & abstract algebra forum, but it has ideas from linear algebra, set theory, and calculus. doesn't fit into any one of the forums. 1. From Wiki: "Unlike the reals, the hyperreals do not form a standard metric space..." Why not? From definition of metric space: d: SxS -> R(reals) is a metric on S iff: d(x,y) = 0 iff x=y (infinitely close numbers are the same); d(x,z) ≤ d(x,y) + d(z,y). Is it because the metric on *R is not |x-y|, but st(|x-y|), (st is standard part function; allowing some numbers to be infinitely close to each other)? 2. What are the resp. cardinalities of the hypernaturals *N and the hyperintegers *Z? If *Z is countable, then it will establish countability of *Q (the hyperrationals; set of all a/b such that a, b are in *Z, and b is nonzero). It will mean that not all hyperreal numbers (*R has cardinality of reals = uncountable) can be expressed as *Q (cf. question 4). It will also mean that *Q will be "finer" (has smaller numbers/distances) than R but still has less numbers than R, which is very counterintuitive... 3. Can digits go on beyond the infinite (digits in the infinite H-th place) digits, truly forever, in *R? 4. Are there "hyper-hyperreals", "hyper-hyper-hyperreals", [...] "hyper^n-reals", etc? apologies for some informal abuse of "^n". 5. usage the transfer priniple when defining hyperreal extensions of functions defined in terms of integrals, for example, erf and the gamma function. I think you will need to extend the epsilon-delta limit definition "for all real ε>0, there is a real δ>0..." to "for all HYPERreal ε>0, there is a HYPERreal δ>0...". I can't think of anything else besides introducing "hyper-hyperreals" to have the "standard hyperreal part", and i think the limit is the only way to go.