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Tyrrell_McAllister comments on Second order logic, in first order set-theory: what gives? - Less Wrong Discussion
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(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...