∈ and ≅
The universe of ZFC is inhabited by sets. In fact, the only symbol for the theory of ZFC—besides the usual first-order symbols—is ∈. Everything is built on top of ∈.
In the category Set, isomorphisms are bijections. If two sets have the same cardinality, they are isomorphic.
Given that (very roughly) “isomorphic” is supposed to mean “essentially the same thing”, and given that the only way we can characterize a ‘thing’ in ZFC is with ∈, then one might expect ∈ and ≅ to be highly related.
As it turns out, this is wildly untrue! ∈ is all about knowing in particular which sets are elements of other sets. Contrasting, the truthhood of A≅B does not at all care about specifically which sets A and B contain.
One example of this is as follows. Take some cardinal κ=0. We know that
κ∈/κ
However, for the right choice of κ−≅κ−1, we can let X=κ−∪{κ}; then we have that
κ∈X≅κ
If ∈ and ≅ “respected each other” then we might expect κ∈X≅κ to imply that (or at least be a similar statement to) κ∈κ; contradiction. But as discussed ∈ and ≅ are in some sense antithetical, and so there is no contradiction here.
I hope I have explained myself well.
In fact, ∈ and ≅ are so unrelated that knowing x∈y tells you essentially nothing about the relationship of cd(x) and cd(y), and knowing cd(x) and cd(y) tells you essentially nothing abut the truthhood of x∈y.
To make this explicit, observe that the following table is completely populated with examples