# Can a stage of the cumulative hierarchy violate the partition principle?

If we violate the partition principle and add to $$\sf ZF$$ the axiom that there exists a set $$X$$ that has a partition on it that is greater in cardinality than the set of singleton subsets of $$X$$.

Can $$X$$ be a standard stage $$V_\alpha$$ of the cumulative hierarchy?

Can $$X$$ be a non-standard stage $$V_{\alpha'}$$ of the cumulative hierarchy?

About the second question what I mean is there is a non-well founded model of $$\sf ZF$$ such that $$\alpha'$$ is seen as an ordinal internally but externally it is a transitive set of transitive sets but has subsets of it that are infinite descending membership chains.

• Partitioning a set into "more" is actually violating the weak partition principle. It is a stronger claim. Commented Jun 10, 2022 at 9:58
• You can't have a transitive set of transitive sets which has a descending membership chain. If you are a transitive set, you are well-founded. Full stop. Non-standard ordinals/models/etc. will always have a membership relationship that is not the real $\in$. Commented Jun 10, 2022 at 10:20
• Assuming ZF, a transitive set is always well-founded. All sets are. Non-standard models have a different membership relation, which means it is irrelevant what is the underlying set. Commented Jun 10, 2022 at 10:31
• I'm speaking about non-well founded models, the background theory for models here is ZF-Reg. So here we can have those transitive and not well founded. Commented Jun 10, 2022 at 10:33
• We had discussed this over some of your previous questions. There is no reasonable notion for "non-standard ordinals" the way you thinking about it; there is even less of a way to define $V_\alpha$ in those cases. Commented Jun 10, 2022 at 10:35

## 1 Answer

If you are asking whether or not a $$V_\alpha$$ could violate the partition principle, the answer is easily yes.

As we all know, it is always the case that $$\Bbb R$$ can be partitioned into $$\aleph_1$$ parts; but it is consistent that there is no injection from $$\omega_1$$ into the reals. Next, observe that $$V_{\omega+1}$$ is exactly the same cardinality as $$\Bbb R$$.

• What about the non-standard stages? Commented Jun 10, 2022 at 10:17
• That's an internal statement. I'm not sure what standard/non-standard have to do with it. Commented Jun 10, 2022 at 10:18
• I mean can we consistently say that we have a non-well founded model of $\sf ZF$, where there is a set $V_{\alpha'}$ in that model where $\alpha'$ is externally a non-standard ordinal (as described above) and such that this model satisfies that $V_{\alpha'}$ violates the weak PP Commented Jun 10, 2022 at 10:58
• @ZuhairAl-Johar Take $\alpha'$ to be the $\omega+1$ of any model of "ZF + PP fails at $V_{\omega+1}$" which is not an $\omega$-model. Commented Jun 11, 2022 at 14:43