How large can a non-standard element of $\omega$ be?

Question

I realized that the definition of a finite set in ZFC is not the same as a finite set in the intuitive sense. A set is said to be finite if it is equipotent to an integer (i.e. an element of $\omega$). So a non-standard element of $\omega$ is considered finite in ZFC, while they have infinitely many elements (because every natural number is in it). My question is, in the intuitive sense, how large can a non-standard integer be? Can it be uncountable? I don't even know if we are can talk about "cardinality" in the intuitive sense.

Answer 1

The external size of the natural numbers can be anything. You can see this from a compactness argument. For any infinite cardinality $\kappa,$ add constants $\{c_i:i\in\kappa\}$ to the language, then to ZFC, add the axioms $c_i\ne c_j$ for all $i\ne j,$ and that $c_i$ is a natural number for each $i$. Assuming there is a model of ZFC, then we can use it to satisfy any finite subtheory, and thus the theory has a model, and this model has a set of natural numbers of external size $\kappa.$

And then to address your specific question, since any natural number has all lesser natural numbers as elements, we can find models with natural numbers of any external size.

Answer 2

If the question is just the cardinalities (in the metatheory) of initial segments of the natural numbers in models of $\mathrm{ZFC}$, a straightforward compactness argument shows that you can produce a model in which you can embed any linear ordering. Let $(I,<)$ be a linear order and $\mathcal{L}=\{\in,c_i\}_{i\in I}$, where the $c_i$ are new constants; let $\Gamma=\{c_i\in c_j\;|\;i,j\in I\wedge i<j\}\cup \{c_i\in\omega\;|\;i\in I\}$. Clearly $\mathrm{ZFC}\cup\Gamma$ is finitely satisfiable (if $\mathrm{ZFC}$ is satisfiable), so $\mathrm{ZFC}\cup\Gamma$ is, too. But a model $M$ of this theory is one in which $(I,<)$ is embedded in what $M$ thinks is $\omega$.

So in such an $M$, an initial segment of $\omega$ can be at least as big as any initial segment of $I$.

(Granted, you don't really need to control things about order types of subsets of $\omega$. It's just fun to do so.)