On variants of the principle of consistent choices, the minimal cover property and the 2-compactness of generalized Cantor cubes

Abstract

In set theory without the Axiom of Choice (AC), we study the deductive strength of variants of the Principle of Consistent Choices (PCC) and their relationship with the minimal cover property, the 2-compactness of generalized Cantor cubes, and with certain weak choice principles. (Complete definitions are given in Section “Notation and terminology”.) Among other results, we establish the following:1.“For every infinite set X, the generalized Cantor cube 2X has the minimal cover property” (MCP) implies “PCC restricted to families of 2-element sets” (F2), which in turn implies “for every infinite set X, 2X is 2-compact” (Q(2)). Moreover, ‘MCP implies F2’ is not reversible in ZFA (i.e., ZF, the Zermelo–Fraenkel set theory minus AC, with the Axiom of Extensionality weakened in order to permit the existence of atoms).The above results strengthen related results in Howard–Tachtsis “On the minimal cover property in ZF” and “On the set-theoretic strength of the n-compactness of generalized Cantor cubes”.2.“Every Dedekind-finite set is finite” (DF=F) implies the Principle of Partial Consistent Choices (PPCC) – the latter principle being introduced here – which in turn implies ACfinℵ0 (i.e., the axiom of choice for denumerable families of non-empty finite sets). None of the previous implications is reversible in ZF.3.The Principle of Countable Consistent Choices (PCCℵ0), which is introduced here, is equivalent to ACfinℵ0.4.Rado's Lemma (RL) + “every infinite set has an infinite linearly orderable subset” implies PPCC. In addition, RL does not imply PPCC in ZFA, PPCC does not imply RL in ZF, and PPCC does not imply “every infinite set has an infinite linearly orderable subset” in ZFA.5.PPCC does not imply “there are no amorphous sets” in ZFA.6.F2 implies “there are no amorphous sets” and the implication is not reversible in ZF. This clarifies the relationship between the latter two statements, whose status is mentioned as unknown in Howard–Rubin “Consequences of the Axiom of Choice”.7.“Every infinite partially ordered set has either an infinite chain or an infinite antichain” (CAC) does not imply X, where X∈{PPCC,F2}, in ZF. In addition, F2 does not imply CAC in ZFA.