Our question arises from wondering about the systems of natural numbers in models ZFC + Con(ZFC) and ZFC + $\neg$Con(ZFC). In thinking of the systems of natural numbers of these models, we came to the following remark and questions. If we take the ultrapower of a model of Peano arithmetic we get a model of Peano arithmetic with nonstandard natural numbers like <1, 2, 3, ...> which have infinitely many natural numbers below them. My question is: if we take an ultrapower, $M$, of a model of ZFC, $V$, does $M$ have nonstandard natural numbers?

An ultrapower of a model $M$ of ZFC will have nonstandard natural numbers if and only if either $M$ has nonstandard natural numbers or the ultrafilter used in forming the ultrapower is countably incomplete. The natural numbers of the ultrapower of $M$ with respect to an ultrafilter $U$ are isomorphic to the ultrapower with respect to U of the natural numbers of $M$. (More generally, the operations of $U$ultrapower and taking the subset of a model defined by a particular formula always commute up to canonical isomorphism.) The ultrapower will satisfy Con($PA$) if and only if $M$ does, by Los's theorem. The preceding was written with "ordinary" ultrapowers in mind. You could also form "internal" ultrapowers of a model $M$ of ZFC. You'd use an element $U$ of $M$ that satisfies in $M$ the formula "$U$ is an ultrafilter on a set $I$", and elements of the ultrapower would be represented by functions on $I$ in the sense of $M$ (i.e., elements $f$ of $M$ that satisfy in $M$ the formula "$f$ is a function on $I$"). Everything I wrote in the first paragraph would still be correct provided you interpret "countably complete" in the sense of $M$. Finally, one can, in some situations, form ultrapowers where the ultrafilter is external to the model $M$ but the functions representing ultrapower elements are internal to $M$. Similar (but not identical) results hold for these also. 


In each copy of $V$ there's some set is a model of $PA$. This sequence of sets corresponds to a set in $M$ which, since being a model of $PA$ is a sentence in the language of $ZFC$, is a model of $PA$. Since there is a unique model of $PA$, this is the model of $PA$. This set is an ultrapower of a model of $PA$, so it contains nonstandard natural numbers. (Edit: As Andreas points out, only if the ultrapower is countably incomplete.) 

