On Martin's axiom and forms of choice (Q2813673)

In what follows, \(\mathbf{ZF}\) is Zermelo-Fraenkel set theory (i.e., choiceless set theory) and \(\mathbf{ZFA}\) is the set theory obtained by weakening the axiom of extensionality in order to allow the existence of atoms. \(\mathbf{AC}\) denotes the axiom of choice and \(\mathbf{ZFC} = \mathbf{ZF} + \mathbf{AC}\). \(\mathbf{AC}_{\mathrm{WO}}\) denotes the restriction of the axiom of choice to families of well-orderable non-empty sets. \(\mathbf{AC}^{\aleph_0}\) denotes the axiom of countable choice (which is the restriction of \(\mathbf{AC}\) to countable families of non-empty sets) and \(\mathbf{AC}^{\aleph_0}_{\mathrm{fin}}\) denotes the axiom of countable choice for countable families of non-empty finite sets. \(\mathbf{BPI}\) denotes the Boolean prime ideal theorem. \(\mathbf{CUT}\) denotes the countable union theorem (which states that the union of a countable family of countable sets is a countable set) and \(\mathbf{DF = F}\) denotes the statement ``Every Dedekind-finite set is finite''\, (or, equivalently, ``Every infinite set has an infinite countable subset''). \(\mathbf{DC}\) is the principle of dependent choices, which is the statement: ``If \(R\) is a binary relation on a non-empty set \(A\) such that \((\forall x \in A)(\exists y \in A)[xRy]\), then there is a sequence \(\langle x_n: n < \omega\rangle\) of elements of \(A\) such that \(x_n R x_{n + 1}\) for all \(n < \omega\)''. Notice that \(\mathbf{DC}\) clearly implies \(\mathbf{AC}^{\aleph_0}\).NEWLINENEWLINEWe assume the reader is familiar with the usual terminology needed to define and state the the widely known combinatorial principle \(\mathbf{MA}\) (Martin's axiom). However, as the paper under review works within the choiceless context, we have to be aware of certain details -- so, let us give some definitions. For any well-ordered cardinal \(\kappa\), \(\mathbf{MA}(\kappa)\) is the following statement: ``Whenever \(\mathbb{P}\) is a non-empty c.c.c pre-order and \(\mathcal{D}\) is a family of \(\leqslant \kappa\) dense subsets of \(\mathbb{P}\), there is a filter \(F \subseteq \mathbb{P}\) which is \(\mathcal{D}\)-generic, meaning that \(F \cap D \neq \emptyset\) for all \(D \in \mathcal{D}\)''. \(\mathbf{MA}\) denotes the statement ``For every \(\kappa < 2^{\aleph_0}\), \(\mathbf{MA}(\kappa)\) holds''\,\,(where \(\kappa\) ranges over well-ordered cardinal numbers). Now, let \(\mathfrak{n}\) be a cardinal in the absence of the axiom of choice -- that is, \(\mathfrak{n}\) is not necessarily a well-ordered cardinal. Let \(\mathbf{MA}_{\mathfrak{n}}\) denote the principle ``Whenever \(\mathbb{P}\) is a non-empty c.c.c pre-order with \(|\mathbb{P}| \leqslant \mathfrak{n}\) and \(\mathcal{D}\) is a family of \(\leqslant \mathfrak{n}\) dense subsets of \(\mathbb{P}\), there is a filter \(F \subseteq \mathbb{P}\) which is \(\mathcal{D}\)-generic''. Let \(\mathbf{MA}^*\) denote the statement ``For every \(\mathfrak{n} < 2^{\aleph_0}\), \(\mathbf{MA}_{\mathfrak n}\) holds''\,\,(where \(\mathfrak{n}\) ranges over arbitrary, i.e., not necessarily well-ordered cardinal numbers). It is known that, in \(\mathbf{ZFC}\), \(\mathbf{MA}\) and \(\mathbf{MA}^*\) are equivalent. \(\mathbf{MA}_{\aleph_0}\) is provable without invoking any form of choice, but \(\mathbf{MA}_{\aleph_1}\) is not provable even in \(\mathbf{ZFC}\); on the other hand, it is known that \(\mathbf{DC}\) implies \(\mathbf{MA}(\aleph_0)\) which in turn implies that every compact c.c.c Hausdorff topological space is Baire. It is still unknown whether \(\mathbf{AC}^{\aleph_0}\) implies \(\mathbf{MA}(\aleph_0)\) or if \(\mathbf{MA}(\aleph_0)\) implies \(\mathbf{AC}^{\aleph_0}\).NEWLINENEWLINEIn the paper under review, the author investigates the deductive strength of \(\mathbf{MA}(\aleph_0)\) in the choiceless context, aiming to positionate such statement in the hierarchy of weak choice principles. The question of whether the equivalence between \(\mathbf{MA}\) and \(\mathbf{MA}^*\) still holds in the absence of the axiom of choice is also investigated. Typical results of the paper are: {\parindent=0.8cm \begin{itemize}\item[(i)] ``Every compact c.c.c Hausdorff topological space is Baire''\, + \(\mathbf{BPI}\) implies \(\mathbf{MA}(\aleph_0)\) restricted to complete Boolean algebras. \item[(ii)] \(\mathbf{BPI}\) + \(\mathbf{CUT}\) does not imply \(\mathbf{MA}(\aleph_0)\) in \(\mathbf{ZF}\). \item [(iii)] \(\mathbf{AC}_{\mathrm{WO}}\) does not imply \(\mathbf{MA}(\aleph_0)\) in \(\mathbf{ZFA}\). \item [(iv)] \(\mathbf{MA}(\aleph_0)\) is false in the basic Fraenkel model of \(\mathbf{ZFA}\). \item [(v)] \(\mathbf{MA}(\aleph_0)\) restricted to complete Boolean algebras is false in the second Fraenkel model of \(\mathbf{ZFA}\). \item [(vi)] If \(\mathbf{ZF}\) is consistent, then so is \(\mathbf{ZF} + \mathbf{MA}^* + \neg \mathbf{AC}^{\aleph_0}\). \item [(vii)] If \(\mathbf{ZFA}\) is consistent, then so is \(\mathbf{ZFA} + \mathbf{MA}^* + \neg \mathbf{MA}(\aleph_0) + \mathbf{DF = F} + \mathbf{CUT}\). In particular, \(\mathbf{MA}^* + \neg \mathbf{MA}\) is relatively consistent with \(\mathbf{ZFA}\). NEWLINENEWLINE\end{itemize}} The paper finishes with a number of problems, the first of them asking whether \(\mathbf{AC}^{\aleph_0}\) implies \(\mathbf{MA}(\aleph_0)\) or if \(\mathbf{MA}(\aleph_0)\) implies \(\mathbf{AC}^{\aleph_0}\). The author also asks about the relative consistency (with \(\mathbf{ZF}\) or \(\mathbf{ZFA}\)) of certain statements -- for instance, the negation of \(\mathbf{MA}^*\), or the conjuntion of the negation of \(\mathbf{MA}(\aleph_0)\) with its restriction to complete Boolean algebras.