One more point to make about the paradoxical decomposition to more parts than elements [...] currently we do not know of any model of ZF+$\lnot$AC where such decomposition does not exist. Namely, as far as we know, in all models where choice fails there is some set which can be partitioned into more parts than elements.
(Just for clarification: From the context of the above references the comparison "more parts than" is obviously meant in terms of the cardinality of the suitable partition being strictly larger than the cardinality of the suitable initial set itself.)
Now, I find the negation or denial of such "paradoxial partitioning" interesting; i.e. the statement (proposition, "$\lnot$PP"):
"Each partition of each given set has cardinality less than, or at most equal to, the cardinality of the given set."
And I like to further explore how this suggested "proposition $\lnot$PP" (being considered along with ZF, of course) relates to "the standard set theory including Axiom of Choice", ZFC. Therefore
Are there any models of ZFC known (or could there be any such models, in principle) in which there is some set which can be partitioned into more parts than elements ?
Are there any models of ZF known (or could there be any such models, in principle) in which there is no set which can be partitioned into more parts than elements, and which is not also a model of ZFC ?
(And just for reference:
Is there a conventional or concise way of expressing the suggested "proposition $\lnot$PP" in terms of standard notation, such as used in the sources linked above? Has it perhaps been discussed already, by some other name? ...)