TITLE: Is there a least-fixed-point formulation of inaccessible cardinals?
QUESTION [11 upvotes]: The infinity axiom can be formulated by defining a function $S$ as
$$S(N) = \{0\} \cup \{n+1\\ |\\ n \in N\}$$
(FWIW, I'm assuming the von Neumann ordinals.) The axiom is then
$$\exists I . I = S(I)$$
which gives us our first infinite set. Then $\omega$ is the intersection of all the subsets of $I$ that are also fixed points, or the least fixed point of $S$.
I've been curious about this for a while now: can the first (strongly, uncountable) inaccessible cardinal be defined similarly?

REPLY [5 votes]: One can attain something like a positive solution with the observation that the inaccessible cardinals are precisely the regular fixed points of the beth function $\alpha\mapsto\beth_\alpha$, which is monotone and continuous. In particular, the first inaccessible cardinal is the  smallest regular beth fixed point.