Let's think of cardinals as collections of otherwise featureless points. 

A justly famous theory of cardinals is Lawvere's ETCS. (Note: ETCS is sometimes called a "structural set theory"; but in that nomenclature, a "structural set" is just what I am calling a "cardinal".) 

Whilst there are some nice discussions of ETCS's axioms, I have not seen anyone attempt to present the 'intended model' of ETCS, or indeed of any theory of cardinals. This contrasts with the case of set theories where we are often told that the intended model of ZFC is 'the cumulative hierarchy of pure sets'.

So: in this talk, I will introduce you to a Relationally-presented theory of Stagewise Cardinals (RSC). This axiomatizes the idea that cardinals are found stage-by-stage; it has a nicely intended model. 

We can go further. Osius (1974) showed that an extension of ETCS is equivalent (in a precise sense) to an extension of Zermelo's theory. Using similar proof ideas: RSC is equivalent (in that same precise sense) to a weak but quasi-categorical theory of 'the cumulative hierarchy of pure sets'. 

Should we conclude that theories of (stagewise) cardinals and theories of (cumulative) sets are just "notational variants"? Not so fast! The theories are not (and must not be) bi-interpretable. I therefore suggest that genuine mathematical content is lost in the move from sets to cardinals.