The Von Neumann universe $V$ satisfies ZFC, and there are other models within $V$ that are non-standard and satisfy ZFC. If we look at one of these non-standard models $M$ with a non-standard model of PA $P$, then internally to $M$ we have a notion of 'numbers' in $P$.
Consider a non-standard number $H$ in $P$. I want to understand what a set $A$ (in the non-standard model of ZFC $M$) of cardinality $H$ looks like. What will such a set's external (i.e., w.r.t. $V$) cardinality be? What about a set $B$ of cardinality $H+1$? It seems like you could just take the union of $A$ with some element of $P$ not in $A$ to get a set of cardinality $H+1$.
I am seeing a few ways to resolve this:
Sets of nonstandard cardinalities don't exist within $M$.
$A$ and $B$ exist in $M$; they have different cardinalities within $M$. However, with respect to $V$, they are both infinite and of the same cardinality. The only reason they have different cardinalities in $M$ is that the bijection between $A$ and $B$ is not within $M$.
$A$ and $B$ are infinite and have different external cardinalities, but you can't take the union of $A$ and some element of $P$ to get something of cardinality $H+1$ for some reason.
$A$ and $B$ are infinite and have different external cardinalities, but taking the union of $A$ and some element of $P$ is somehow externally not just adding one thing to $A$, but enough things to change its cardinality.
Which of these, if any, is the right answer?