Set theory without choice: Not everything on cofinality is possible (Q1354318)

\textit{A. Apter} and \textit{M. Magidor} [``Instances of dependent choice and the measurability of \(\aleph_{\omega+1}\)'', Ann. Pure Appl. Logic 74, No. 3, 203-219 (1995; Zbl 0842.03038)]\ proved the consistency of \(\text{ZF}+\text{DC}_{\aleph_\omega}+ {|{\mathcal H}(\aleph_\omega)|=\aleph_\omega}+ \text{``}\aleph_{\omega+1}\) is measurable''. In contrast to this result the author proves in \(\text{ZF}+\text{DC}\) that if \(\mu\) is a singular cardinal of uncountable cofinality and \({|{\mathcal H}(\mu)|=\mu}\), then \(\mu^+\) is regular and non-measurable. To prove this and other results on cardinal arithmetic and cofinality in set theory with a limited axiom of choice, the author starts by discussing what part of pcf theory [\textit{S. Shelah}, Cardinal arithmetic (1994; Zbl 0848.03025)]\ survives. Let \(D\) be a filter on a set \(A\). The author defines a least upper bound and an exact upper bound in \(({}^A\text{Ord},{<_D})\). He considers \(<_D\)-increasing sequences in \(^A\text{Ord}\) indexed by (ordinals or by) a directed partial order \(I\) and gives some sufficient conditions for the existence of least upper bounds and exact upper bounds assuming sufficiently large cofinality and sufficiently strong directedness of \(I\) under some weak versions of the axiom of choice. In a particular case he assumes DC and the filter is \(\aleph_1\)-complete. To measure the size of the set \(\prod_{a\in A}f(a)/D\) he introduces a variety of ordinals (\(hpp_\Gamma\), \(ehpp_\Gamma\), \(shpp_\Gamma\), \(pp_\Gamma\), \(epp_\Gamma\), where \(\Gamma\) is a set of filters and \(pp\) stands for pseudo power, \(h\) for hereditary, \(s\) for smooth, \(e\) for element) as supremum of the length of a \(<_D\)-increasing sequence (\(D\in\Gamma\)) of sets or elements in \(\prod_{a\in A}f(a)\). Then the existence of \(<_D\)-exact upper bounds gives results on these ordinals. Roughly speaking a family \(E\) of filters \(D\) on \(A\) is nice if it is possible to define ordinal rank functions \(\text{rk}^2_D(f)\), \(\text{rk}^3_D(f)\) for \(f\in{}^A\text{Ord}\) and \(D\in E\). It is proved that under some reasonable assumptions to measure the size of \(\prod_{a\in A}f(a)/D\) using these ranks is the same as to measure them by cardinalities of factorizations \(F/e\) for equivalences \(e\) on sets \(F\subseteq\prod_{a\in A}f(a)\) with \(e{\restriction}F\) containing \({=_D}{\restriction}F\). \(\otimes_{\alpha,R}\) means that there exists a sequence \(\langle e(\delta):\delta<\alpha\) a limit ordinal\(\rangle\) where \(e(\delta)\) is an unbounded subset of \(\delta\) of order type which belongs to \(R\). Assuming DC it is proved that if \(\aleph_0<\text{cf}(\mu)<\mu= |\bigcup_{\alpha<\mu}{\mathcal P}(\alpha)|\) (i.e.\ \(\mu\) is a strong limit cardinal of a certain kind) and \(E\) is a nice family of filters on \(\text{cf}(\mu)\), then \(2^\mu\) is an aleph, \(\mu^+\) is regular, \(\otimes_{2^\mu,\{\delta<2^\mu:\text{cf}(\delta)>\mu\}}\) holds, and if \(\mu<\lambda\leq2^\mu\) then \(\lambda\) is not measurable. The author proves that under some assumptions if \(\langle\mu_i:i<\kappa\rangle\) is an increasing continuous sequence of alephs, then there is \(\lambda\) such that the set \(\{i<\kappa:\text{cf}(\mu_i^+)=\lambda\}\) is stationary. Many theorems in the paper have as a major assumption the existence of nice \(E\). The author shows similarly as in the case of set theory with axiom of choice that if there is no nice \(E\), then the universe is sufficiently similar to some inner model suitable to answer the questions on the exponentiation and cofinality.