Set Theory – Reference for Undecidability in ZFC

decidabilityreference-requestset-theory

Note: This is a question and thus doesn't need the usual type of context.

According to this:

It is a surprising fact that the statement
$$A > B \implies \mathcal{P}(A) > \mathcal{P}(B)$$ is undecidable in $\mathrm{ZFC}$.

. . . and . . .

we define $A > B$ to mean: there exists an injection $B\to A$ but no bijection between these two sets.

Please may I have a reference for this?

Context:

It is indeed surprising. I can't say that I would understand a proof of it, but having a reference would go a long way to convincing whoever that it's true; it's the kind of thing I would share with fellow students. If the reference includes some prerequisites for the proof, that would be ideal.

Best Answer

Cohen's original work would be enough.

Cohen, Paul Joseph, The independence of the continuum hypothesis. I, II, Proc. Natl. Acad. Sci. USA 50, 1143-1148 (1963); 51, 105-110 (1964). ZBL0192.04401.

The second paper, Lemma 21 and Lemma 22 give us exactly that if we are adding $\kappa$ many Cohen reals, and $\kappa^{\omega_1}=\kappa$ in the ground model, then $2^{\aleph_1}=\kappa$ in the extension, and therefore $2^{\aleph_0}=2^{\aleph_1}$ in that case.

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