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I was wandering if there was a book, thesis or some notes where Shelah's argument for

  1. $\mathtt{ZF}+\mathtt{DC}+$"All sets of reals are Lebesgue measurable" is equiconsistent with $\mathtt{ZFC} + \exists \kappa$ inaccessible
  2. $\mathtt{ZF}+\mathtt{DC}+$"All sets of reals have the Baire property" is equiconsistent with $\mathtt{ZFC}$

contained in Can you take Solovay's inaccessible away? is explained in a newer and/or "more digestible" way. Is there?

Thanks!

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    $\begingroup$ Raisonnier's follow-up paper addressing 1 may be of interest. $\endgroup$ Apr 22, 2022 at 12:29
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    $\begingroup$ There's a chapter titled The Raisonnier Filter in Ralf Schindler's textbook (Set Theory: Exploring Independence and Truth). It presents the Lebesgue measure result in a learner-friendly manner, I think. $\endgroup$ Apr 23, 2022 at 3:54
  • $\begingroup$ You may look at Ralf Schindler's book on set theory. $\endgroup$ Apr 23, 2022 at 13:39

1 Answer 1

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Chapter 9.5 of the book by Bartoszynski-Judah presents Raisonnier's proof (answering your question 1):

Assume that $\aleph_1$ is not inaccessible in $L$, hence a successor in $L$. So there is a real $x$ which knows that the $L$-predecessor is countable, hence $\aleph_1^{L[x]}=\aleph_1$, so $X:=\mathbb R \cap L[x]$ is uncountable.

Now assume that all $\Sigma^1_2(x)$-sets are Lebesgue measurable. Then (this needs some work) the "Raisonier filter" $F_X$ built from $X$, which is a $\bf \Sigma^1_3$-set, is a rapid filter and therefore (again some work, known earlier) not measurable.

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