Let $\kappa$ be a cardinal, does $2^\kappa<2^{\kappa^+}$ always hold? It clearly holds if one assumes generalized continuum hypothesis, but does it also hold if one assumes otherwise?
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If I remember correctly, this is independent of ZFC. – Mark Dec 05 '20 at 17:37
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when $\kappa=\aleph_0$ this statement is the negation of Luzin's hypothesis – Atticus Stonestrom Dec 05 '20 at 18:31
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Possible relevant – Hanul Jeon Dec 05 '20 at 19:25
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This is independent of $\mathsf{ZFC}$. It consistently holds, because it holds under $\mathsf{GCH}$, and it consistently doesn't hold, because a two step iteration of Cohen forcing over a model of $\mathsf{GCH}$ can be used to force $2^\omega=2^{\omega_1}$.
Alessandro Codenotti
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@Jonathan First you force with $\mathrm{Fn}(\kappa\times\omega_1,2)$, then with $\mathrm{Fn}(\kappa\times\omega,2)$, where $\kappa$ is the result you want for both exponentiations – Alessandro Codenotti Dec 07 '20 at 13:47
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The two forcings are equivalent and one already suffices (of course assuming $\operatorname{cf}(\kappa) > \omega_1$). I would call this the forcing adding $\kappa$ many Cohen reals or the finite support iteration of Cohen forcing. In any case two-step iteration is confusing to me, since even what you propose is just product forcing. – Jonathan Schilhan Dec 07 '20 at 22:16