The standard model of the reals, the unique field that obeys the second order axioms inside standard models of set theory...
(Emphasis added.)
Are there such things as "standard models of set theory"? This page from a book on model theory says that there is no standard model. The closest things, it says, are something called "natural models". I only glanced at it, but the notion of "natural model" appears to be a second-order concept that depends on the set theory with which you started.
Yes, I'm not sure about this myself. But people do seem to feel that some models of set theory are non-standard (eg countable models), and that there is a standard model of the reals. I get the impression that some models of set theory are "standardler" than others...
With thanks to Paul Christiano
My previous post left one important issue unresolved. Second order logic needed to make use of set theory in order to work its magic, pin down a single copy of the reals and natural numbers, and so on. But set theory is a first order theory, with all the problems that this brings - multiple models, of uncontrollable sizes. How can these two facts be reconciled?
Quite simply, it turns out: for any given model of set theory, the uniqueness proofs still work. Hence the proper statement is:
Often, different models of set theory will have the same model of the reals inside them; but not always. Countable models of set theory, for instance, will have a countable model of the reals. So models of the reals can be divided into three categories:
And similarly for the natural numbers.