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 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?



1 Answer 1


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 $a\in\mathbb{R}^M$).

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.


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