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 $X$.



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> Can $X$ be a standard stage $V\_\alpha$ of the cumulative hierarchy?
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> Can $X$ be a non-standard stage $V\_{\alpha'}$ of the cumulative hierarchy?
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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.



Can a stage of the cumulative hierarchy violate the partition principle?
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 $\Bbb R$.


