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I am wondering who proved the following fact:

($\ast$) If $\omega_1$ is not inaccessible in $L$, then there is an uncountable coanalytic set of reals without a perfect subset.

I have been unable to track down a reference for it. All I found so far is:

  1. In the 1930s, Gödel proved that if $V = L$, then there is an uncountable coanalytic set of reals without a perfect subset.

  2. In 1957, Specker proved in $\mathsf{ZF}$ that if $\omega_1$ is regular in $V$ but not inaccessible in $L$, then there is an uncountable set of reals without a perfect subset.

Someone recently said to me that ($\ast$) should be credited to Gödel, but on the other hand the fact that the much later paper of Specker is cited for the weaker result (2) suggests to me that ($\ast$) should not be considered to follow immediately from Gödel's work. I don't read German, so I can't tell if Specker actually proved ($\ast$).

Edit: When I say that (2) is weaker than ($\ast$), I am thinking of ($\ast$) as being proved in the theory $\mathsf{ZF} + {}$"$\omega_1$ is regular in $V$." (It seems to me that any reasonable proof in $\mathsf{ZFC}$ would also work in this context, because $\mathsf{AC}$ does not seem relevant for constructing coanalytic sets.)

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    $\begingroup$ Although I do not know Godel specific example of a thin $\Pi_1^1$ set; however, it probably involved some coding of $L_\alpha$ for $\alpha < \omega_1^L$ by reals. Surely this can be relativized to coding $L_\alpha[x]$ for $\alpha < \omega_1^{L[x]}$ and yield a thin uncountable $\Pi_1^1(x)$ set assuming $\omega_1^{L[x]} = \omega_1^V$. It is likely Godel method could relative to prove (*); however, whether he was aware of enough of these notions for credit is another issue. $\endgroup$
    – William
    Commented Jul 11, 2017 at 19:53
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    $\begingroup$ Nowadays, we think of relativization as a trivial step; if you prove a theorem about $L$ and I point out that it also holds for $L[x]$ for any real $x$, I'm not really making any progress. But things were different in the old days; relativization could be real progress. For example, I vaguely recall reading somewhere that the relative consistency result "if ZFC + $V\neq L$ is consistent then it remains consistent if you adjoin CH" was not explicitly proved before 1950. $\endgroup$
    – Andreas Blass
    Commented Jul 12, 2017 at 2:18
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    $\begingroup$ The result is not literally Gödel's, although of course nowadays we see it as essentially the same proof as item 1. The reason is that the $L[x]$ models had not been considered until Lévy's work, from 1957, and item 1 itself did not see a proof in print until Novikov's work from 1951 (in Russian) and Addison's in 1959. After 1959 the result would have been noticed pretty quickly, but I do not know who mentioned it first. Maybe asking Addison may clarify this point. $\endgroup$
    – Andrés E. Caicedo
    Commented Jul 12, 2017 at 4:37
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    $\begingroup$ I vaguely recall some relevant (or semi-relevant) results in Ioanna Dimitriou's masters thesis. $\endgroup$
    – Asaf Karagila ♦
    Commented Jul 12, 2017 at 10:50
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    $\begingroup$ Trevor, you are misstating Specker's result: that there are uncountable sets without a perfect subset was proved back in 1908, by Bernstein. What Specker did was to prove in $\mathsf{ZF}$ that if $\aleph_1$ is regular in ${\mathrm V}$ but not inaccessible in $L$, then there are uncountable sets of reals without a perfect subset. The point, of course, is that choice is not involved in the argument. (That $\aleph_1$ is regular also ends up being important, but that's a different story.) $\endgroup$
    – Andrés E. Caicedo
    Commented Jul 14, 2017 at 17:16

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