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

Question

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 can look within that set to get an injection from an infinite subset of $C$ to some infinite set, so $\aleph(C)$ couldn't have been $\aleph_0$.

The construction that seems to work in ZFCA is to take a model of ZFA with infinitely many atoms and then take the direct limit of $L(A_n)$, where $A_n$ is the set of the first $n$ atoms. This seems to be a model of ZFCA, because every set belongs to $L(A_n)$ for some $n$ and therefore inherits a well-ordering from that (as $L(A_n)$ still has a global choice function definable from parameters).

I can't see how replacement fails where collection did, because I can't think any $\varphi$ such that $\forall n\exists!x\varphi(n,x)$ and $\varphi(n,x)$ implies that $x$ is an injection from $n$ into the class of atoms, or anything else that would cause the union of all such $x$s to not fit into any $L(A_n)$.

This of course cannot be extended to a model of GBCA, because the existence of a proper class that is smaller than the universe prevents the existence of a global choice function.

If the construction works and can be extended to ZFC, it does prove that the Hartogs number of classes is interesting in NBG even with the axiom of choice (NBG being conservative over ZFC).

Answer 1

Yes. If you start with infinitely many urelements and then take the sets whose kernel is finite (the kernel of a set is the set of urelements in its transitive closure), the resultant inner model will satisfy Replacement. A more generalized argument is included in my dissertation (https://arxiv.org/abs/2303.14274, Theorem 26).

Let ZFU$_\text{R}$ denote ZF set theory with urelements axiomatized with Replacement. In general, given a model $U$ of ZFU$_R$ an ideal $\mathcal{I}$ of the class of all urelements that contains every urelement singleton, the inner model $U^\mathcal{I}$, which contains all urelements and sets whose kernel is in $\mathcal{I}$, is a model of ZFU$_R$ also. And $U^\mathcal{I}$ satisfies AC if $U$ does. So in the case of ZFCU$_\text{R}$, the Hartogs number of a proper class can be any infinite cardinal.

For ZFCU$_\text{R}$ + Collection, the Hartogs number of a proper class cannot be any limit cardinal by the argument you have given. But this is the only constraint: for every successor infinite cardinal $\kappa^+$, there can be models of ZFCU$_\text{R}$ + Collection where the Hartogs number of the class of urelements is $\kappa^+$. This follows from Lemma 21 and Theorem 26 in my dissertation.

The result for ZFCU$_\text{R}$ can be extended to GBU$_\text{R}$ (in fact, KMU$_\text{R}$) with a global choice function. This is due to Felgner (see Theorem 108 in my dissertation). In class theory with urelements, it is important to distinguish different versions of second-order AC. For example, the existence of a global choice function doesn’t imply the universe can be well-ordered, as shown in Felgner’s model.

However, it is not known, for example, if KMU$_\text{R}$ + Collection + Global Choice (not Global Well-Ordering) is consistent with a proper class with Hartogs number, say, $\aleph_1$.