# Show that the theory of the $Th(\mathbb N)$ in first order logic with the finiteness quantifier is categorical

Suppose that a finiteness quantifier $\mathbf Fx$ is added to first order logic. Its semantics are: $\mathbf Fx\Phi(x)$ is true in a model just in case there a finitely many things in the domain of the model that satisfy $\Phi(x)$

So I have to prove an isomorphism with of $Th(\mathbb N)$ with the finiteness quantifier. But the kicker is that I cannot use any of the metatheory of regular FOL for this. No compactness etc. I really don't know how to proceed.

Does $Th(\mathbb N)$ even look that much different with a finiteness quantifier?

Any help is quite appreciated.

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Show that $\forall x.\mathbf{F}y(y<x)$ is true in $\mathbb N$. Now suppose that $M$ is another model of $Th(\mathbb N)$, it has to be linearly ordered, and every initial segment has to be finite.