Given a set $A$, the Hartog number of $A$, denoted with $\aleph(A)$, is defined as the least infinite ordinal which cannot be mapped into $A$ $$\aleph(A) = \min\{\alpha \in \mathsf{Ord}\setminus\omega \mid \alpha\not\le A\}$$ while the Lindenbaum number $\aleph^\star(A)$ is the least infinite ordinal that $A$ cannot be mapped onto$$\aleph^\star(A) = \min\{\alpha \in \mathsf{Ord}\setminus\omega \mid \alpha\not\le^\star A\}$$ Clearly $\forall A, \aleph^\star(A)\ge\aleph(A)$, but to get the equality we need some choice. In particular the statement $\forall A, \aleph^\star(A)=\aleph(A)$ is a direct consequence of the partition principle $\mathsf{PP}$, stating that $\forall A,B$ if $A\le^\star B$ then $A \le B$.
My questions are:
- Is it known whether $\forall A \ (\aleph^\star(A) = \aleph(A)) + \neg \mathsf{PP}$ is consistent?
- Is if known whether $\forall A \ (\aleph^\star(A) = \aleph(A)) + \mathsf{AC}_{WO}+\neg\mathsf{DC}_{\omega_1}$ is consistent?
Thanks!
First of all, the statement $\forall A(\aleph^*(A)=\aleph(A))$ is itself equivalent to $\sf AC_{WO}$. One implication is trivial, if $\sf AC_{WO}$ holds, and $f\colon A\to\kappa$ is a surjection, then $A_\alpha=f^{-1}(\alpha)$ is a family of non-empty set which admits a choice function.
In the other direction this is not so trivial anymore and requires a complicated, but incredibly beautiful argument. The idea is to assume $\forall A(\aleph^*(A)=\aleph(A))$, then prove by induction that $\sf AC_{WO}$ holds. If $\sf AC_{<\lambda}$ holds, and given a family of size $\lambda$ of non-empty sets, we construct a set of partial choice functions (with a twist), so that the set maps onto $\lambda$ and any injection in the inverse direction must also provide us with means to define a choice function from the original family.
This happens to be the same as $\sf PP_{WO}$, namely, any partition into well-orderable sets has an injection into the original set, but that is almost immediately equivalent to the statement $\aleph^*(A)=\aleph(A)$.
Finally, since in all known models of $\sf ZF+\lnot AC$ we also have $\lnot\sf PP$, the first question is easily answered if we can show that $\sf AC_{WO}+\lnot AC$ is consistent. But this is well-known to hold in $L(\Bbb R)$ after adding $\omega_1$ Cohen reals. If my memory serves me right, then $\sf DC_{\omega_1}$ should fail in that model as well, but if not, then we also have models for that. For example, the models from my work on Krivine's result in "Realizing Realizability Results with Classical Constructions".