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Working in $\sf ZF$, regarding an arbitrary Dedekind infinite set $x$, what's the relationship between the cardinality of $\mathcal P(x)$ and $\mathcal H^*_x$? The latter is the set of all sets hereditarily subnumerous to $x$, where subnumerous to means "is injective to" , by "$y$ is to hereditarily subnumerous to $x$" here it specifically means every element of the transitive closure of $\{y\} $ is subnumerous to $x$. Transitive closure defined in the customary way as the minimal transitive superset.

Are there special constraints on how they compare given Choice, GCH, V=L, etc.. Can they be incomparable? I'm interested in cases where $\mathcal H^*_x$ is subnumerous to $\mathcal P(x)$, whether strictly or not, especially outside of choice?

Also how matters would differ if we defined $\mathcal H^*_x$ as the set of all sets whose transitive closure is subnumerous to $x$? Especially as regards it being subnumerous to $\mathcal P(x) $ in absence of choice?

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    $\begingroup$ When $x$ is $\omega$, then $H_\omega$ is countably infinite, but $P(\omega)$ is strictly larger, even in ZF. $\endgroup$
    – Joel David Hamkins
    Commented May 14 at 23:12
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    $\begingroup$ Wouldn't it be more natural to want to compare $H_\kappa$ with $P_\kappa(\kappa)$, which is the set of subsets of $\kappa$ of size less than $\kappa$. In ZFC+GCH, these have the same size for all infinite regular $\kappa$. $\endgroup$
    – Joel David Hamkins
    Commented May 14 at 23:26
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    $\begingroup$ Another issue is that even in ZFC we don't define $H_\kappa$ as the collection of sets that are hereditarily of size less than $\kappa$, if what this means is that they have size less than $\kappa$, and their elements also, and the elements of those sets, etc. This gives the wrong result when $\kappa$ is singular. One normally defines $H_\kappa$ as the collection of sets whose transitive closures have size less than $\kappa$. I'm not sure what "subnumerous" means, but in the ZF context all talk of cardinality is more subtle and complicated, so it might help to say more exactly what is meant. $\endgroup$
    – Joel David Hamkins
    Commented May 14 at 23:29
  • $\begingroup$ Sorry, you don't need GCH in my earlier comment. It is true in ZFC. $\endgroup$
    – Joel David Hamkins
    Commented May 14 at 23:34
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    $\begingroup$ You say "every element of the transitive closure of $\{y\}$ is small", but this is exactly making the mistake I mentioned. The usual definition would say that the transitive closure itself is small. Also, you are using $\leq$ instead of $<$, which means that if ZFC holds, your terminology does not match the established terminology. For example, your notation has no way to refer to the hereditarily finite sets. $\endgroup$
    – Joel David Hamkins
    Commented May 15 at 12:39

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