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In the paper "Convex Sets of Cardinals", Truss mentioned a result of Jech:

If $M$ is a countable transitive model of ZFC, and $(P,<)∈M$ is a poset, then there exists a Cohen extension of $M$ such that $(P,<)$ is isomorphic to a set of cardinalities of that model.

The result is referenced to to "On ordering of cardinalities", I found several other mention of this result, also referencing "On incomparable cardinals" by Takahashi. (while this is not my main question, I failed to find both papers, so if someone knows where I can locate them I would love to know).

In his answer here, Asaf said:

We can show that given a model of ZFC, every partial order in that model, and in fact the entire model itself, can be embedded into the cardinals of a larger model

Strengthening the result from above.

Those result led me to think about the internal variation of the question, instead of looking at posets in a model and extending it to a model with enough cardinals, looking at the posets in the universe and asking if the universe has enough cardinals:

  • Is it consistent with ZF that for every partially ordered set $(P,<)$ there exists a set of cardinals that is isomorphic to $(P,<)$?

    • The same question but with the schema statement about definable partially ordered classes as well
    • The same question but in NBG and about partially ordered classes

In other words, is it possible in ZF that the cardinals capture all possible orders?

This also leads to 2 variations of dual questions:

  • Is there a definable class $C$ of partial orders such that if there exists $(P,<)∈C$ that does not embeds to the cardinals, then the axiom of choice holds?

  • Does there exists a minimal definable class $C$ of partial orders such that if non of the orders in $C$ embeds into the cardinals, then the axiom of choice holds?

Both of those variations are unfortunately trivialized by looking $C=\{(\{a,b\}, ∅)\}$

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Yes. This was essentially proved by Honsel and Forti in the 1980s by analysing a model that generalises the Cohen model (essentially, the one Monro used to show it can be consistent for Dedekind finite sets to have large Lindenbaum numbers).

In my preprint Iterated Failures of Choice I provide a different construction for the same result.

The point is that we can essentially embed $V$ with the subset relation into the cardinals.

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  • $\begingroup$ Thanks Asaf, do you know perhaps the name of the paper of Honsel and Forti? $\endgroup$
    – Holo
    Commented Feb 24, 2023 at 12:22
  • $\begingroup$ Looking at your preprint I found their paper "The consistency of the axiom of universality for the ordering of cardinalities" $\endgroup$
    – Holo
    Commented Feb 24, 2023 at 12:31
  • $\begingroup$ Yes, that's the one. $\endgroup$
    – Asaf Karagila
    Commented Feb 24, 2023 at 13:07

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