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 variation 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? (The existence of a maximal such class is trivial by letting $C$ be the class of all non-well-ordered orders)

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 $C=\{(\{a,b\}, ∅)\}$