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Status of PS0(C) in ZF

Determine whether the principle PS0(C)—asserting the existence of certain choice functions for countable partitions witnessing local finite hitting of uncountable families in [X]^{<ω}—holds in ZF or is independent.

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Background

The paper introduces several variants of PS0, notably PS0(II), PS0(C), and PS0(D). Independence results are obtained for PS0(II) and PS0(D), showing neither is provable in ZF. However, the authors point out that the status of PS0(C) in ZF remains unresolved.

References

The independence results of Theorems \ref{s7:t10} and \ref{s7:t12}(2) supply further information about the deductive strength and the relationships between $\mathbf{PS_0}(\mathbf{II})$, $\mathbf{PS}_0(\mathbf{C})$ and $\mathbf{PS}_0(\mathbf{D})$; none of $\mathbf{PS_0}(\mathbf{II})$ and $\mathbf{PS_0(D)}$ is provable in $\mathbf{ZF}$, whereas the status of $\mathbf{PS_0(C)}$ in $\mathbf{ZF}$ is still an open problem.

— Constructing crowded Hausdorff $P$-spaces in set theory without the axiom of choice (2510.11935 - Tachtsis et al., 13 Oct 2025) in Section 7, discussion after Theorem s7:t10