A clarification of first-order axiomatizability I have been told that certain classes of structures are not first-order axiomatizable, like the class of finite groups. However, in ZFC set theory, we can define both finiteness and groups, and therefore finite groups, by a very complicated formula. But isn't ZFC itself a first-order theory? I am just a bit confused on this point. I would like a clarification of this little paradox.
 A: Finiteness is not first-order axiomatizable. That can be shown by using the compactness theorem of first-order logic. So there are models of ZFC in which sets that satisfy a definition of finiteness within ZFC are not finite. Just as, for example, there are countable models of ZFC in which the theorem asserting the uncountability of the set of all sets of finite cardinal numbers is true.
Imagine a set $[n]=\{0,1,2,\ldots,n\},$ where $n$ is a finite cardinal number within a model of ZFC. A theorem within ZFC states that for every one-to-one correspondence between that set and some subset of $[n+3] = \{0,1,2,\ldots,n,n+1,n+2,n+3\}$ there are exactly three elements of the latter set that are not within that subset. But if $[n]=\{0,1,2,\ldots,n\}$ is an infinite set, then we know there are one-to-one correspondences between $[n]$ and some of its subsets with infinite complements. But none of those latter correspondences is a member of the model, thus the theorem saying there are always three left over can remain true within the model. That is how an infinite set can satisfying the definition-within-ZFC of finiteness. Similarly, a countable set may satisfy a definition-within-ZFC of uncountability because none of its enumerations is a member of the model.
