I'm looking for the proof that $\mathtt{PSP}$, the statement that every uncountable subset of the the Baire space $\mathbb{N}^\mathbb{N}$ contains an homeomorphic copy of the Cantor space $2^\mathbb{N}$, implies the consistency of inaccessible cardinals.


I was able to track down Mycielski's paper *On the axiom of determinateness* in which, building on Specker's *Zur Axiomatik der Mengenlehre*, this result is proven (read also this [question](https://mathoverflow.net/questions/275216/inaccessible-cardinals-and-the-perfect-set-property-for-coanalytic-sets) on this account), but I don't read german and Mycielski's article does not report Specker's full argument (and it's also pretty hard to read due to painful notation).


Is there a "newer" account on this proof? Is there a book or thesis where it is the full proof is reported?


Thanks!



$\mathtt{PSP}$ implies the consistency of inaccessible cardinals
See Proposition 11.5 (and the discussion leading up to it) in Kanamori's book **The Higher Infinite**. Note that Kanamori states the result a bit more optimally: if $M\models\mathsf{ZF}$ + "$\omega\_1$ is regular" + $\mathsf{PCP}$ then $\omega\_1^M$ is inaccessible in $L^M$ (and in fact $M\models$ "$\omega\_1$ is inaccessible to reals," that is, $\omega\_1^M$ is inaccessible in $L[a]^M$ for every $a\in\mathbb{R}^M$).




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Since it's pretty short, I'll sketch the proof here for completeness:


Suppose $M$ satisfies the necessary conditions; from now on, all statements are made within $M$.


First, we prove an auxiliary result: that $\omega\_1^{L[a]}$ being countable for each real $a$ implies that $\omega\_1$ is inaccessible to reals. We prove the contrapositive. Suppose $\omega\_1$ is not inaccessible in $L[a]$ for some real $a$. By the usual condensation argument (relativized) we have $L[a]\models\mathsf{GCH}$, so we must have $\omega\_1=\rho^{+^{L[a]}}$ for some $L[a]$-cardinal $\rho$. Now $\rho$ is countable in reality, hence coded by a real; let $b$ be a real coding a well-ordering of $\omega$ isomorphic to $\rho$. Then $a\oplus b$ is a real with $\omega\_1^{L[a\oplus b]}\ge\omega\_1$.


Now suppose $\omega\_1$ is not inaccessible to reals. By the above, this means that $\omega\_1^{L[a]}=\omega\_1$ for some real $a$. But per Godel we know that $L[a]\models\omega\_1\le 2^{\aleph\_0}$, or put another way there is an injection from $\omega\_1^{L[a]}$ into the reals. Sicne $\omega\_1^{L[a]}=\omega\_1$, this means there is an injection from $\omega\_1$ into the reals, whence (by Bernstein) $\mathsf{PSP}$ fails.


