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For $\mathbb{A},\mathbb{B}\subseteq\mathcal{P}(\omega^\omega)$, say that $\mathbb{A}$ spreads onto $\mathbb{B}$ iff there is some $F:\omega^\omega\rightarrow\omega^\omega$ such that $F[A]\in\mathbb{B}$ whenever $A\in\mathbb{A}$. Write "$\mathbb{A}\trianglerighteq\mathbb{B}$" for "$\mathbb{A}$ spreads onto $\mathbb{B}$."

Each cardinal characteristic in Cichon's diagram $\mathfrak{x}$ is defined as "The smallest cardinality of an element of $\mathbb{X}_\mathfrak{x}$" for some $\mathbb{X}_\mathfrak{x}\subseteq\mathcal{P}(\omega^\omega)$. I'm curious how the $\mathsf{ZFC}$-provable cardinality relationships between the $\mathfrak{x}$s compare to the $\mathsf{ZF+DC+AD}$-provable spreading relationships between the $\mathbb{X}_\mathfrak{x}$s. Specifically:

  • Is it the case that, for all $\mathfrak{x},\mathfrak{y}$ in Cichon's diagram, $\mathsf{ZFC}\vdash\mathfrak{x}\le\mathfrak{y}$ implies $\mathsf{ZF+DC+AD}\vdash \mathbb{X}_\mathfrak{x}\trianglelefteq \mathbb{X}_\mathfrak{y}$?

  • Is it the case that, for all $\mathfrak{x},\mathfrak{y}$ in Cichon's diagram, $\mathsf{ZF+DC+AD}\vdash \mathbb{X}_\mathfrak{x}\trianglelefteq \mathbb{X}_\mathfrak{y}$ implies $\mathsf{ZFC}\vdash \mathfrak{x}\le\mathfrak{y}$?

As positive evidence, Paul Larson showed that $\mathsf{ZF+DC+AD}\vdash\mathbb{X}_\mathfrak{b}\triangleleft \mathbb{X}_\mathfrak{d}$, so the "spreading-under-determinacy" picture agrees with (and in fact strengthens) the "cardinality-under-choice" picture in this instance.

I'm particularly interested in the behavior of either of the triples $\mathsf{add}(\mathcal{B}),\mathsf{cov}(\mathcal{B}),\mathfrak{b}$ or $\mathsf{cof}(\mathcal{B}),\mathsf{non}(\mathcal{B}),\mathfrak{d}$, where $\mathcal{B}$ is the $\sigma$-ideal of meager sets, since there is a $\mathsf{ZFC}$-theorem about each triple which isn't a consequence of any of the provable "simple inequalities" - I don't know how (if at all) I'd expect this more complicated relationship to be reflected on the "spreading"-side.

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