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Let $LM$ denote "all subsets of $\Bbb{R}$ are Lebesgue measurable", and

$WCH$ (weak continuum hypothesis) denote "every uncountable subset of $\Bbb{R}$ can be be put into 1-1 correspondence with $\Bbb{R}$".

[Warning: in other contexts, weak CH means something totally different , i.e., it sometimes means $2^{\aleph_{0}} < 2^{\aleph_{1}}$].

We know that $LM$ and $WCH$ both hold in Solovay models. By forcing a (Ramsey) ultrafilter over a Solovay model one can also arrange $WCH+\neg LM$ (due to joint work of Di Prisco and Todorčević, who showed that the perfect set property holds in the generic extension).

This prompts my question ($DC$ below is dependent choice).

Question: Is it known, relative to appropriate large cardinal axioms, whether there is a model of $ZF+LM+DC+\neg WCH$?

My question arose from an FOM-question of Tim Chow, and my answer to it; see also Chow's response.

Rahman. M
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Ali Enayat
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  • What's your definition of uncountable: $\aleph_0 < |S|$ or $|S| \not\leq \aleph_0$ ? –  Aug 03 '11 at 23:27
  • I don't think there's a difference in this case - any non-finite set $S\subseteq\mathbb{R}$ is also Dedekind non-finite. This is because $S$ comes with a natural linear ordering, the restriction of the ordering on $\mathbb{R}$. Let $L_S=\lbrace x\in S:$ There are more than finitely many $y\in S$ with $y>x$ as real numbers$\rbrace$, and $R_S=\lbrace x\in S:$ There are more than finitely many $y\in S$ with $y<x$ as real numbers$\rbrace$. Then either $L_S$ has no greatest member (in which case we can get an embedding of $\omega$ into $L_S$), or $R_S$ has no least member (cont'd.) – Noah Schweber Aug 03 '11 at 23:52
  • (in which case we can get an embedding of $\omega^*$ into $L_S$). Either way, we wind up with a subset of $S$ of cardinality $\aleph_0$. So if $S\subseteq \mathbb{R}$ is such that $\vert S\vert\not\leq\aleph_0$, then $\aleph_0<\vert S\vert$. – Noah Schweber Aug 03 '11 at 23:53
  • Assuming $L_S$ has no greatest member, it does not follow that there is an embedding of $\omega$ into $L_S\hspace{.05 in}$. See http://consequences.emich.edu/CONSEQ.HTM, form 13. –  Aug 04 '11 at 00:47
  • @Ricky and Noah: first of all, thanks for your comments. Next: I have added $DC$, since talk of $LM$ without it is rather silly (and of course with $DC$ the two notions of uncountability that Ricky asked about collapse to one). – Ali Enayat Aug 04 '11 at 01:12
  • How is $LM$ at all silly with just $CC$ instead of $DC\hspace{.01 in}$?$\hspace{.04 in}$ (That also make uncountability unambiguous.) –  Aug 04 '11 at 01:19
  • @Ricky: I know that CC is sufficient for the removal of ambiguity, but real analysis is so much smoother with $DC$, so I opted for it even though I realize it is not the most frugal option. – Ali Enayat Aug 04 '11 at 03:51
  • I've heard that Radon-Nikodym uses Dependent Choice; is that basically why? I don't know of any other reason for real analysis to prefer DC over CC. –  Aug 04 '11 at 04:06
  • @Noah: What you are saying is not quite correct, we can have infinite Dedekind finite subsets of the reals. In fact, Arnie Miller proved recently that it is consistent to have a Borel such set. Of course, these worries go away under DC. – Andrés E. Caicedo Aug 04 '11 at 08:58
  • Arnie Miller proved that it is consistent to have a $F_{\sigma \delta}$ such set. –  Aug 04 '11 at 17:19
  • @Ricky: $DC$ comes handy in the proof of Baire Category Thm [indeed $DC$ is equivalent to Baire Category Thm for complete metric spaces (over $ZF$)]; and in Fubini's Thm (among other places). I found the following survey useful: www.drmaciver.com/docs/essay.pdf – Ali Enayat Aug 04 '11 at 17:51
  • Is that for complete defined in the 'right' way (with nets/filters), or with sequences? –  Aug 04 '11 at 21:15
  • @Ricky: sequences, as far as I remember. – Ali Enayat Aug 05 '11 at 04:38
  • I am pretty sure that the desired theory holds in the improved MAD family model from Section 5 of this paper : http://www.users.miamioh.edu/larsonpb/lru5.pdf – Paul Larson May 02 '17 at 17:30
  • Horrowitz and Shelah have a paper which proves similar (better) results from a weaker large cardinal hypothesis. I'll have to ask them if they get the theory above. Both models add a generic MAD family to a Solovay model in such a way that all sets of reals are Lebesgue measurable. The remaining question is whether you can embed the reals into the MAD family in these models, and in our case I'm pretty sure that you can't. I'll have to write up the details, of course. The Horrowitz-Shelah paper can be found here: https://arxiv.org/pdf/1704.08327.pdf – Paul Larson May 02 '17 at 17:34

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This is an expansion of my comments above. In the paper with Zapletal that I reference, we assume a proper class of Woodin cardinals and force over $L(\mathbb{R})$ with a partial order of countable approximations to a certain kind of MAD family (which Jindra named an "improved" MAD family). Although I have yet to write out the details, I believe that the resulting model satisfies LM (it clearly satisfies DC), and that, in this model, $\mathbb{R}$ cannot be injected into the generic MAD family. The arguments I have in mind are straightforward applications of the arguments given in the paper.

The paper of Horowitz and Shelah referenced in my second comment works from the assumption of a strongly inaccessible cardinal and, as I understand it, adds the construction of a generic MAD family to Solovay's argument. As shown in their paper, DC + LM hold in the resulting model. I wrote to Haim and asked if $\mathbb{R}$ can be injected into the generic MAD family in this model, and he said no. He says he'll update their paper to include a proof of this (so they'll probably have a proof out before we do).

Asaf Karagila
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Paul Larson
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