The Morris model

Abstract:

Douglass B. Morris announced in 1970 that it is consistent with $\mathsf {ZF}$ that “For every $\alpha$, there exists a set $A_\alpha$ which is the countable union of countable sets, and $\mathcal P(A_\alpha )$ can be partitioned into $\aleph _\alpha$ non-empty sets”. The result was never published in a journal (it was proved in full in Morris’ dissertation) and seems to have been lost, save a mention in Jech’s “Axiom of Choice”. We provide a proof using modern tools derived from recent work of the author. We also prove a new preservation theorem for general products of symmetric systems, which we use to obtain the consistency of Dependent Choice with the above statement (replacing “countable union of countable sets” by “union of $\kappa$ sets of size $\kappa$”).