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The set $H_\kappa$ of sets hereditarily of cardinality less than $\kappa$ is defined as $H_\kappa=\{x||tc(x)|\lt\kappa\}$. What if we define the set $H=H_{Ord}$ of sets hereditarily of cardinality less than $Ord$; $H$ is the class of sets with some ordinal number as there cardinality. Equivalently, $H$ is the class of hereditarily well-ordered sets.

It is immediately obvious $V=H$ is equivalent to $AC$. My questions are as follows:

  1. Is it true that $ZF\vdash H\vDash ZF$? Is it true that $ZF\vdash H\vDash ZFC$?

  2. More generally, does $ZF\vdash H_\kappa\vDash AC$?

  3. What is the consistency strength of the existence of some $j: H\prec H$? Is the existence of such an embedding first order expressible?

(I am obviously working in the context of $ZF$ without choice. I define $|tc(x)|\lt\kappa\leftrightarrow\exists\lambda\lt\kappa(|tc(x)|=\lambda)$)

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No, yes, and not sure.

$H$ (and $H_\kappa$ in general) always satisfies choice, because any family of nonempty sets in $H$ has a well orderable transitive closure, from whence we can define a choice function, which is easily hereditarily well orderable as well.

However, $H\models\sf Power$ if and only if choice holds on $V$. Otherwise, let $\alpha$ be the least ordinal with a power set that cannot be well ordered, then $\mathcal P(\alpha)\subseteq H$ but it is not an element of $H$.

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  • $\begingroup$ Do you have an idea on the consistency strength of some $j: H\prec H$? $\endgroup$
    – Master
    Dec 17 '20 at 19:28
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    $\begingroup$ No, but I'd hazard a guess that it's very high. $\endgroup$
    – Asaf Karagila
    Dec 17 '20 at 19:28
  • $\begingroup$ Thank you for the answer. I will wait a little while to see if anyone has the answer to part 3 before accepting. $\endgroup$
    – Master
    Dec 17 '20 at 19:35
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    $\begingroup$ It's also worth mentioning replacement can fail in $H_{\kappa}.$ For example, in the Feferman-Levy extension of $L,$ $H_{\omega_1}$ fails replacement since the $\omega_n^L$'s are definable in $H_{\omega_1}.$ $\endgroup$ Dec 17 '20 at 20:18
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    $\begingroup$ Even if it is determined $HOD$ is definable in $H$, what is the consistency strength of the existence of some $j: HOD\prec HOD$? $\endgroup$
    – Master
    Dec 17 '20 at 20:48

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