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?](http://dx.doi.org/10.1007/BF02760522) is explained in a newer and/or "more digestible" way. Is there?


Thanks!



Shelah's "Can you take Solovay's inaccessible away?"
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.


