ON FREE ULTRAFILTERS ON $\omega $ WITH WELL-ORDERABLE BASES IN $\mathsf {ZF}$

Abstract In $\mathsf {ZF}$ (i.e., Zermelo–Fraenkel set theory minus the axiom of choice ( $\mathsf {AC}$ )), we investigate the open problem of the deductive strength of the principle UF wob (ω): “There exists a free ultrafilter on ω with a well-orderable base”, which was introduced by Herzberg, Kanovei, Katz, and Lyubetsky [(2018), Journal of Symbolic Logic , 83(1), 385–391]. Typical results are: (1) “ $\aleph _{1}\leq 2^{\aleph _{0}}$ ” is strictly weaker than $\mathsf {UF_{wob}}(\omega )$ in $\mathsf {ZF}$ . (2) “There exists a free ultrafilter on $\omega $ ” does not imply “ $\aleph _{1}\leq 2^{\aleph _{0}}$ ” in $\mathsf {ZF}$ , and thus (by (1)) neither does it imply $\mathsf {UF_{wob}}(\omega )$ in $\mathsf {ZF}$ . This fills the gap in information in Howard and Rubin [Mathematical Surveys and Monographs, American Mathematical Society, 1998], as well as in Herzberg et al. (2018). (3) Martin’s Axiom ( $\mathsf {MA}$ ) implies “no free ultrafilter on $\omega $ has a well-orderable base of cardinality $&lt;2^{\aleph _{0}}$ ”, and the latter principle is not implied by $\aleph _{0}$ -Martin’s Axiom ( $\mathsf {MA}(\aleph _{0})$ ) in $\mathsf {ZF}$ . (4) $\mathsf {MA} + \mathsf {UF_{wob}}(\omega )$ implies $\mathsf {AC}(\mathbb {R})$ (the axiom of choice for non-empty sets of reals), which in turn implies $\mathsf {UF_{wob}}(\omega )$ . Furthermore, $\mathsf {MA}$ and $\mathsf {UF_{wob}}(\omega )$ are mutually independent in $\mathsf {ZF}$ . (5) For any infinite linearly orderable set X , each of “every filter base on X can be well ordered” and “every filter on X has a well-orderable base” is equivalent to “ $\wp (X)$ can be well ordered”. This yields novel characterizations of the principle “every linearly ordered set can be well ordered” in $\mathsf {ZFA}$ (i.e., Zermelo–Fraenkel set theory with atoms), and of $\mathsf {AC}$ in $\mathsf {ZF}$ . (6)</ja