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I know we have the consistency of $\mathsf{ZF} + \mathsf{AC} + \mathsf{GCH}$, $\mathsf{ZF} + \neg \mathsf{AC}$, and $\mathsf{ZF} + \mathsf{AC} + \neg \mathsf{GCH}$.

What about $\mathsf{ZF} + \neg \mathsf{AC} + \mathsf{CH}$ and $\mathsf{ZF} + \neg\mathsf{AC} + \neg\mathsf{CH}$?

Trevor Wilson
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user235390
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  • Welcome to Math.SE! Please clarify your question by at least spelling out what all these abbreviations mean. – Hrodelbert Apr 29 '15 at 09:00
  • Are you just asking whether the Continuum Hypothesis is independent of $\mathsf{ZF}{+}{\neg}\mathsf{AC}$? – user642796 Apr 29 '15 at 09:09
  • Here is a related question on mathoverflow. Joel David Hamkins' answer seems to imply that ZF + $\lnot$AC + CH is consistent. – TonyK Apr 29 '15 at 09:34
  • @Arthur: And if $\sf AC$ is independent of $\sf ZF+CH$, as it seems. – Asaf Karagila Apr 29 '15 at 11:12
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    I second Asaf's remark that it is important to define $\mathsf{CH}$ carefully in the absence of $\mathsf{AC}$. It is possible to have a surjection from $\mathbb{R}$ to $\omega_2$ but no injection from $\omega_1$ to $\mathbb{R}$, for example (this follows from the Axiom of Determinacy.) – Trevor Wilson Apr 29 '15 at 18:03
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    @Trevor: Maybe it's also worth pointing out that in models of $\sf ZF+AD$ you also get the perfect set property for $\Bbb R$ so there are no intermediate cardinals, despite the fact that $\Bbb R$ can be mapped onto many different cardinals. – Asaf Karagila Apr 30 '15 at 08:21

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There's really nothing much to it.

We can easily use symmetric forcing the axiom of choice to fail only above rank $\omega+\omega$. Then if $\sf CH$ was true in the ground model, it will be true while $\sf AC$ is false; and if $\sf CH$ was false in the ground model, it will be false in the symmetric extension.

Do note, however, that $\sf CH$ has several formulations which end up non-equivalent when the axiom of choice fails. See How to formulate continuum hypothesis without the axiom of choice? for more details. And while we're handing out links, The Continuum Hypothesis & The Axiom of Choice seems relevant as well.

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Asaf Karagila
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