Abstract In set theory without the Axiom of Choice ( $\mathsf {AC}$ ), we investigate the open problem of the deductive strength of statements which concern the existence of almost disjoint and maximal almost disjoint (MAD) families of infinite-dimensional subspaces of a given infinite-dimensional vector space, as well as the extension of almost disjoint families in infinite-dimensional vector spaces to MAD families.

Abstract Ellis’ Theorem (i.e., “every compact Hausdorff right topological semigroup has an idempotent element”) is known to be proved only under the assumption of the full Axiom of Choice (AC); AC is used in the proof in the disguise of Zorn’s Lemma. In this article, we prove that in ZF, Ellis’ Theorem follows from the Boolean Prime Ideal Theorem (BPI), and hence is strictly weaker than AC in ZF. In fact, we establish that BPI implies the formally stronger (than Ellis’ Theorem) statement “for every family ${\cal A} = \{ ({S_i},{ \cdot _i},{{\cal T}_i}):i \in I\}$ of nontrivial compact Hausdorff right topological semigroups, there exists a function f with domain I such that $f\left( i \right)$ is an idempotent of ${S_i}$ , for all $i \in I$ ”, which in turn implies ACfin (i.e., AC for sets of nonempty finite sets). Furthermore, we prove that in ZFA, the Axiom of Multiple Choice (MC) implies Ellis’ Theorem for abelian semigroups (i.e., “every compact Hausdorff right topological abelian semigroup has an idempotent element”) and that the strictly weaker than MC (in ZFA) principle LW (i.e., “every linearly ordered set can be well-ordered”) implies Ellis’ Theorem for linearly orderable semigroups (i.e., “every compact Hausdorff right topological linearly orderable semigroup has an idempotent element”); thus the latter formally weaker versions of Ellis’ Theorem are strictly weaker than BPI in ZFA. Yet, it is shown that no choice is required in order to prove Ellis’ Theorem for well-orderable semigroups. We also show that each one of the (strictly weaker than AC) statements “the Tychonoff product $2^{\Cal R} $ is compact and Loeb” and $BPI_{\Cal R}$ (BPI for filters on ${\Cal R}$ ) implies “there exists a free idempotent ultrafilter on ω” (which in turn is not provable in ZF). Moreover, we prove that the latter statement does not imply $BP{I_\omega }$ (BPI for filters on ω) in ZF, hence it does not imply any of $AC_{\Cal R} $ (AC for sets of nonempty sets of reals) and $BPI_{\Cal R} $ in ZF, either. In addition, we prove that the statements “there exists a free ultrafilter on ω”, “there exists a free ultrafilter on ω which is not idempotent”, and “for every IP set $A \subseteq \omega$ , there exists a free ultrafilter ${\cal F}$ on ω such that $A \in {\cal F}$ ” are pairwise equivalent in ZF.

Abstract Ramsey’s Theorem is naturally connected to the statement “every infinite partially ordered set has either an infinite chain or an infinite anti-chain”. Indeed, it is a well-known result that Ramsey’s Theorem implies the latter principle. In the book “Consequences of the Axiom of Choice” by P. Howard and J. E. Rubin, it is stated as unknown whether the above implication is reversible, that is whether the principle “every infinite partially ordered set has either an infinite chain or an infinite anti-chain” implies Ramsey’s Theorem. The purpose of this paper is to settle the aforementioned open problem. In particular, we construct a suitable Fraenkel–Mostowski permutation model ${\cal N}$ for ZFA and prove that the above principle for infinite partially ordered sets is true in ${\cal N}$ , whereas Ramsey’s Theorem is false in ${\cal N}$ . Then, based on the existence of ${\cal N}$ and on results of D. Pincus, we show that there is a model of ZF which satisfies “every infinite partially ordered set has either an infinite chain or an infinite anti-chain” and the negation of Ramsey’s Theorem. In addition, we prove that Ramsey’s Theorem (hence, the above principle for infinite partially ordered sets) is true in Mostowski’s linearly ordered model, filling the gap of information in the book “Consequences of the Axiom of Choice”.

Abstract We establish the following results: 1. In ZF (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC), for every set I and for every ordinal number α ≥ ω, the following statements are equivalent: (a) The Tychonoff product of ∣α∣ many non-empty finite discrete subsets of I is compact. (b) The union of ∣α∣ many non-empty finite subsets of I is well orderable. 2. The statement: For every infinite set I, every closed subset of the Tychonoff product [0, 1]Iwhich consists offunctions with finite support is compact, is not provable in ZF set theory. 3. The statement: For every set I, the principle of dependent choices relativised to I implies the Tychonoff product of countably many non-empty finite discrete subsets of I is compact, is not provable in ZF0 (i.e., ZF minus the Axiom of Regularity). 4. The statement: For every set I, every ℵ0-sized family of non-empty finite subsets of I has a choice function implies the Tychonoff product of ℵ0many non-empty finite discrete subsets of I is compact, is not provable in ZF0.