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I was wondering whether it is consistent to have c=ℵc where c=2ℵ0 is the cardinality of the reals (over ZFC). If so, what interesting consequences of this statement are known (besides ¬CH)? I was curious about this because in some sense c is the largest possible number of cardinals below c, and this is partly motivated by the idea that c may be so large as to be 'unreachable' via approximation by fewer smaller cardinals, which seems similar in nature to an opinion of Cohen on CH.

From what I have read, I think that it is consistent (relative to ZFC) for c to be the ω_1-th fixed-point of \aleph, which would be one possibility satisfying \frak{c} = \aleph_{\frak{c}}. But can \frak{c} be the \frak{c}-th fixed-point of \aleph, and does this yield even more interesting consequences?

Martin Sleziak
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user21820
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    All the things you ask for are consistent by Easton's theorem. For the last part, continuum can be the \omega_1-th cardinal \alpha such that \aleph_\alpha=\alpha. – Wojowu Jun 24 '19 at 10:14
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    @Wojowu: Why Easton? Might as well argue that these things are consistent because starting with a supercompact cardinal, if you add that many Cohen reals, you get the wanted result. (The point I am trying to make is that Cohen and Solovay already prove this result, there's no reason involving class forcing over all regular cardinals here.) – Asaf Karagila Jun 24 '19 at 10:17
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    @AsafKaragila That's a good point. I appeal to Easton because that's one theorem of this sort which I know by the name :P – Wojowu Jun 24 '19 at 12:49
  • @Wojowu: Letting k be the ω_1-th aleph-fixed-point, k cannot be the k-th aleph-fixed-point, so it doesn't answer my last question, but Asaf said my comment works. – user21820 Jul 01 '19 at 14:20
  • Robert Solovay, "2^{\aleph_0} can be anything it ought to be", The theory of models. Proceedings of the 1963 International Symposium at Berkeley. Zbl 0202.30701 – Goldstern Aug 28 '19 at 09:51
  • I cited Solovay's result (independently also proved by Cohen, as Solovay states in his announcement in the AMS Notices, October 1963, p.595) only in order to give you a quotable reference; if you are interested in the proof, you can find it e.g. in Jech's book (ch.15). – Goldstern Aug 30 '19 at 15:18
  • @Goldstern: Okay thanks a lot! – user21820 Aug 30 '19 at 15:25

1 Answers1

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Yes. Start with a model of \sf CH, then take the least fixed point with uncountable cofinality. Call that \kappa. Now add \kappa Cohen reals.

Since fixed points form a club of ordinals, you can iterate the fixed points enumeration. Repeat that \omega_1 times, then take the least one of cofinality \omega_1 in that club. Now call that \kappa, and add that many Cohen reals.

Assuming no large cardinals get involved, that means that \frak c is singular. This by itself implies that Martin's Axiom fails, and that Cichon's diagram is not trivial, since some of the cardinal characteristics are provably regular.

Other than that, I don't believe we can say a lot more without adding more assumptions on the universe.

Asaf Karagila
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  • Thanks, but what about my last question? And I was hoping ZFC alone can prove some interesting things given that \frak{c} is an \aleph-fixed-point. Anyway what can you say under extra assumptions? – user21820 Jun 24 '19 at 11:57
  • Am I right that since we can define (over ZFC) the class function F that maps k to the k-th aleph-fixed-point, we can define an ordinal m that is a fixed-point of F, so that m would be the m-th aleph-fixed-point, and then by Easton's theorem gives consistency of \frak{c} being the \frak{c}-th aleph-fixed-point relative to ZFC? – user21820 Jul 01 '19 at 08:07
  • Yeah, that's about right. – Asaf Karagila Jul 01 '19 at 08:16
  • Okay then thanks, my question is answered! =) – user21820 Jul 01 '19 at 08:29