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Consider the following variant of Freiling's Axiom of Symmetry, $\mathsf{AS}$, which will be denoted $A_{< 2^{\aleph_0}}$:

  • given any function $f$ from $\mathbb{R}$ into the families of of subsets of $\mathbb{R}$ of size $< 2^{\aleph_0}$ there are $x_1,x_2 \in \mathbb{R}$ such that $x_1 \notin f(x_2)$ and $x_2 \notin f(x_1)$.

Does $\mathsf{ZF} + A_{<2^{\aleph_0}} + \mathsf{LM}$ imply $\neg \mathsf{WCH}$? If not, what is the current state of research in discovering models of $\mathsf{ZF}$ in which $\mathsf{LM} +\lnot \mathsf{WCH}$ holds?

Here

  • $\mathsf{LM}$ is the statement that every set of reals is Lebesgue measurable,
  • $\mathsf{WCH}$ is the statement that every uncountable subset of $\mathbb R$ can be put into 1-1 correspondence with $\mathbb R$.

For background to this question, in his 3 August 2011 FOM post to Timothy Chow, Ali Enayat stated that

$\mathsf{ZF} + \mathsf{AS} + \mathsf{LM} + \mathsf{WCH}$ holds in Solovay's model since in Solovay's model every uncountable subset of $\mathbb{R}$ has a perfect subset.

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Assuming $AD$, a version of the continuum hypothesis holds: every set of reals is either countable or of size continuum (this is a consequence of the perfect set property). So assuming $AD$, your $A_{<2^{\aleph_0}}$ and Freiling's $AS$ are equivalent. In particular, both can hold in Solovay's model, so the answer to your question is "no."


Your second question seems much more interesting. Off the top of my head, I don't see a reason why we can't have a model of ZF+DC+LM+$\neg$WCH (except of course for the small fact that I don't know how to build one).

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  • $\begingroup$ Just to clarify what you wrote: Are you assuming that $\mathsf{AD}$ holds and $L(\mathbb{R})$ is a Solovay model? It seems that you are saying that $\mathsf{AD}$ alone implies that $L(\mathbb{R})$ is a Solovay model. (Well, you do not mention $L(\mathbb{R})$ but if this is not what you meant, I am not sure I understand the "In particular".) $\endgroup$ – Andrés E. Caicedo Sep 29 '15 at 12:08
  • $\begingroup$ @Noahschweber: Do you know of any research going on which might be useful in constructing a model of $ZF$+$DC$+$LM$+$\lnot$$WCH$? $\endgroup$ – Thomas Benjamin Sep 29 '15 at 13:49
  • $\begingroup$ @AndrésCaicedo quite right, that was a silly slip on my part; I've clarified. $\endgroup$ – Noah Schweber Oct 2 '15 at 2:58

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