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forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set theory, independence, axiom of choice, continuum hypothesis, determinacy, Borel equivalence relations, Boolean-valued models, embeddings, orders, relations, transfinite recursion, set theory as a foundation of mathematics, the philosophy of set theory.

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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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