There are two important numbers that in some meaningful sense describe "how well-orderable" the reals are:

- Hartogs' Number $H(\Bbb R)$, also notated as $\aleph(\Bbb R)$, the least ordinal/well-ordered cardinal that doesn't inject into $\Bbb R$
- The ordinal $\Theta$, also notated as $\aleph^*(\Bbb R)$, the least ordinal/well-ordered cardinal that $\Bbb R$ doesn't surject onto

The first number $H(\Bbb R)$ can be thought of as describing the supremum of the cardinalities of all well-orderable *subsets* of $\Bbb R$, whereas the second number $\Theta$ can be thought of as describing the supremum of the cardinalities of all well-orderable *equivalence classes* of $\Bbb R$.

With choice, these are all equal to $\mathfrak c^+$, the cardinal after $\mathfrak c$.

**My question**: without choice, do we have any results regarding:

- How large each of these numbers can be?
- How small each of these numbers can be?
- Which number is larger or if they can be equal?

I know that $AD$ determines some of these strongly enough to relate them to large cardinals (I believe Woodin cardinals), but I'm also interested in what the possibilities are without something that strong.

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