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first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, philosophical logic, modal logic, completeness, Gödel incompleteness, decidability, undecidability, theories of truth, truth revision, consistency.

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Is it consistent with ZFC(A) for the Hartogs number of a proper class to be $\aleph_0$?

I'm specifically assuming that we have replacement instead of collection; collection breaks things (because then there is a set that contains a map from $n$ to $C$ for every $n\in\mathbb N$, and you c …
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