Is there an absolute notion of the infinite?

Question

Skolem's paradox has been explained by the proposition that the notion of countability is not absolute in first-order logic. Intuitively, that makes sense to me, as a smaller model of ZFC might not be rich enough to talk about a bijection that a larger model understands. I'm wondering, however, if there are models of ZFC that disagree on the notion of being finite/infinite, so that a given set is thought of as infinite in one model, but finite in another? I suspect the answer is no, but I'm not sure how to prove it. Does anyone have any insight?

Answer 1

The finite/infinite distinction is not absolute. Indeed, different models of set theory can think vastly different things about the sizes of a set whose elements they have in common.

For example, there can be models of set theory $M$ ad $N$, both satisfying all the ZFC axioms (assuming that ZFC is consistent), such that a set $x$ is thought to be finite in $M$, but $N$ thinks it has uncountably many elements.

To construct such an example, let $M$ be any model of ZFC, and let $N$ be the model of ZFC that arises from an internal utrapower of $M$ by a nonprincipal ultrafilter on $\mathbb{N}$. Thus, $N$ is a class of $M$, and for any set $x\in N$, the collection of objects that $N$ thinks are elements of $x$ is a set in $M$. So this is a sense in which the set exists in both models. Since the ultrafilter was nonprincipal, it is not difficult to see that $N$ will have a nonstandard collection of integers. So let $n$ be one of the nonstandard integers of $N$, and consider the set in $N$ consisting of the predecessors of $n$ in $N$. Since $N$ thinks that $n$ is a natural number, and doesn't see that it is nonstandard, it follows that $n$ thinks that $n$ has only finitely many predecessors. But $M$, on the other hand, can see that $n$ is nonstandard, and in fact arises from the ultrapower construction, and so $M$ can see that $n$ has continuum many predecessors. So the set of predecessors of $n$ is finite in $N$ and size continuum in $M$.

One can make much worsely behaved examples: for any $M\models$ZFC and any cardinal $\kappa$ of $M$, there is a model $N\models$ZFC constructible inside $M$, such that $N$ has a set that it thinks is finite, but which $M$ thinks has infinite size $\kappa$.

Update. I wrote an extended essay on Skolem's paradox on Infinitely More. What I show there by the end is that there can be a tower of models of set theory, with a set in common that is thought to be finite in the first model, then uncountable, then countable, then uncountable again, then countable, and so forth, in a pattern as long as desired.