Specker proved in 1957 that if in $V$ every set of real numbers has the perfect set property, than in $L$, $\omega\_1^V$ is actually a limit cardinal.


The original proof is in German, and I've been looking for an English account of the proof. I couldn't find it in all the usual places (usually just a reference). I do remember it not being *that* complicated when it was given in a course I attended a couple of years ago, so it seems strange that I couldn't find it in the books.


Can anyone help me find such English version of the proof (or give it here if it is short enough)?



Perfect set property implies $\omega_1$ is a limit cardinal in $L$
The usual proof (as in Kanamori's book, section 11) is as follows: Work in $\mathsf{ZF}$. Note first, with Bernstein, that if $\omega\_1\le\mathfrak c$, then there is a set of reals without the perfect set property: Either $\omega\_1=\mathfrak c$, so $\mathbb R$ can be well-ordered, and we can build Bernstein sets using the usual transfinite recursion, or else $\omega\_1<\mathfrak c$, and any set of reals of size $\omega\_1$ lacks the perfect set property (that there are exactly continuum many perfect sets, and that each perfect set contains a copy of the Cantor set and therefore has size $\mathfrak c$ are provable in $\mathsf{ZF}$).


Now, under the assumption that all sets of reals have the perfect set property, we argue that $\omega\_1$ is a limit cardinal in $L[r]$ for all reals $r$: Suppose otherwise, so for some real $r$ and some $\kappa$, $\omega\_1=\kappa^+$ in $L[r]$. Let $s$ be a real coding $r$ and a well-ordering of $\omega$ in type $\kappa$. In $L[s]$, we have that $\omega\_1$ is computed correctly. But now we see that $\omega\_1\le\mathfrak c$, as witnessed by $\mathbb R^{L[s]}$. 


As Asaf points out in the comments, if $\omega\_1$ is regular in $V$, this gives us that it is inaccessible in $L[r]$ for all reals $r$, but it is equiconsistent with $\mathsf{ZF}$ that $\omega\_1$ is singular and yet the perfect set property holds, see



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> John Truss. [*Models of set theory containing many perfect sets*](http://www.sciencedirect.com/science/article/pii/0003484374900151). Ann. Math. Logic, **7**, (1974), 197–219. [MR0369068 (51 #5304)](http://www.ams.org/mathscinet-getitem?mr=369068). 
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