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In his answer to a Math Stack Exchange question of Katlus, Asaf Karagila wrote the following:

"It is a theorem that from $ZF+DC+$"$\aleph_1$$\le$$|$$\mathbb R$$|$" we can prove that there is an unmeasurable set of real numbers, so in Solovay's model we have that there are no sets of real numbers which have size $\aleph_1$. In fact, in Solovay's model every uncountable set of reals is of size continuum, and in some sense the continuum hypothesis holds."

Three questions regarding this statement:

i) Since "it is a theorem that from $ZF+DC+$"$\aleph_1$$\le$$|$$\mathbb R$$|$" we can prove that there is an unmeasurable set of real numbers" could someone provide an explicit construction of this set in the aforementioned system?

ii) Are there models of $ZF$$+$"Every set of reals is Lebesgue measurable"$+$$\lnot$$CH$?

iii) In the models mentioned in ii), does $DC$ fail?

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    $\begingroup$ The result i) is due to Raisonnier link.springer.com/article/10.1007%2FBF02760523 . $\endgroup$ Jan 8, 2015 at 16:26
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    $\begingroup$ @ThomasBenjamin the main issue with your link was simply that the URL was wrong. There was a missing dot between math and stackexchange. $\endgroup$
    – user9072
    Jan 8, 2015 at 16:28
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    $\begingroup$ @quid: Oh... Thanks for fixing the URL. $\endgroup$ Jan 8, 2015 at 16:29
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    $\begingroup$ Quick question: what do you mean by "$\neg CH$?" That there is an uncountable set of reals of size strictly less than that of $\mathbb{R}$, or just that $\vert\mathbb{R}\vert\not=\aleph_1$? $\endgroup$ Jan 8, 2015 at 17:20
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    $\begingroup$ Note that in 1991 Foreman and Wehrung proved from ZF + Hahn Banach theorem that there is a Lebesgue non-measurable set. The paper is interesting also because it develops Lebesgue measure without AC. Here is the reference: hal.archives-ouvertes.fr/hal-00004713/document $\endgroup$
    – Avshalom
    Jan 8, 2015 at 19:38

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For (i) see here. For (ii), (iii) see this.

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