Transfinite Cardinals and Expressive Power Consider a language with a sufficiently rich lexicon such that, for every (finite and transfinite) cardinal K, it's possible to express the statement that there exist K-many objects.  Two general sorts of questions:
(1) In general, what sorts of languages (if any) are capable of expressing such statements?  And what do the sentences that express such statements look like?
(2) Let L be a language of this sort, K some cardinal, and S(K) the L-sentence that says that there are K-many things.  How many models does S(K) have?
I'm guessing that the language can't be that of 1st-order logic since no 1st-order sentence has for its models just the infinite structures (and hence can't express that infinitely many objects exist), but my knowledge of logic and set theory runs too thin to know how to proceed any further.
 A: The answer you seek is probably $L_{\infty\infty}$, which is the logic obtained by allowing any set-many quantifiers and disjunctions.
Then an expressing saying that there are $\kappa$ objects of a certain type would be just ${\large\exists}_{\alpha<\kappa}x_\alpha(\bigwedge_{\alpha<\beta<\kappa}x_\alpha\neq x_\beta\land\bigwedge_{\alpha<\kappa}\varphi(x_\alpha))$ where $\varphi$ is the wanted property (if you want to write "exactly" you need to add that for all $y$ satisfying $\varphi$, $\bigvee_{\alpha<\kappa}y=x_\alpha$ as well).
The answer as to how many models can be difficult to answer "just like that" without further information. It could be zero models if the property is flatout inconsistent, or one model if you spell out sophisticated enough properties, or a class-model (which is not really a model in the usual sense of the term, since its universe is not a set), or it could be there are just class-many models of the theory in different cardinalities.
After learning some more set theory and model theory, you might want to check out the relevant parts in:

J. Barwise, S. Feferman, eds., Model-Theoretic Logics (New York: Springer-Verlag, 1985)

