Models of the Reals of All Cardinalities and Equivalence

[WWGD]

We know that different models of the Reals are elementary equivalent but not isomorphic. In the hyperreals, we have that the Archimedean principle does not hold but it does hold in the standard Reals ( i.e., the hyperreals allow for the existence of indefinitely-small and indefinitely -large numbers, but the standard Reals do not; indefinitely-small numbers must equal zero) . Let's make some set-theoretic assumption about cardinalities: is there a standard second-order property that holds for models of fixed cardinality but does not hold for models of lower cardinality, like in the case of the Archimedean principle for the hyperreals and the Reals?
Thanks.

 

[nomadreid]

I would like to get your question straight before I make a fool of myself by answering a question you didn't ask, and not answering your question.

First, since the Löwenheim–Skolem theorem guarantees that each theory has a model in each infinite cardinality, then one interpretation of your question would give a definite "no". However, perhaps that is not the interpretation you meant?

Perhaps you are looking at it from the other direction: for example, you need an uncountable measurable cardinal so that such-and-such measure would exist... if this is the spirit of your question, then in general the axioms for the large cardinals may be what you need.

Other possible interpretations might need someone better versed in Stability Theory than I am.

 

[WWGD]

Thanks, nomadreid, what I mean is this: models of the Reals of different cardinalities are elementary -equivalent to each other, but not isomorphic. This means that every first-order formula in models of lower cardinality are preserved for models of higher cardinality. But second-order formulas are not preserved. Is there a general characterization of the second order formulas that are
true in models of high cardinality but not so in models of lower cardinality? The only example I know is that of the Archimedean property, that is true for the usual model of the Reals, but not for other models of the Reals of higher cardinality.