Hello everybody, little question for logicians: Considering $ZFC + \neg CH$, is it possible to construct a model $V \vDash ZFC + \neg CH$ such that there exists a finite number of cardinal between $\aleph_0$ and $2^{\aleph_0}$ ? Same question with exactly one cardinal between $\aleph_0$ and $2^{\aleph_0}$ ?


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    $\begingroup$ Yes, $2^{\aleph_0}=\aleph_n$ is relatively consistent with ZFC for any positive integer $n$. $\endgroup$ – Emil Jeřábek Feb 29 '12 at 13:40

Yes! Almost anything is possible. You can force over a model of ZFC + CH to create a new model where $2^{\aleph_0}$ is $\aleph_2$, for example, so that there is one cardinal between $\aleph_0$ and the continuum. The idea is to create new binary sequences, new real numbers, with a partial order (a notion of forcing) and allow a generic filter to make it coherent and take you to the new universe where the continuum is a new size. You could have 3 or 4 or any finite number of cardinals between $\aleph_0$ and $2^{\aleph_0}$ by adding new subsets of natural numbers.

See the discussion below about how the answer to the question "What size can the continuum be?" is due to Cohen, Solovay, and Easton. Also, see in the comments how the continuum could reach as far up as $\aleph_{2^{\aleph_0}}$, so there are continuum many cardinals between omega and $2^{\aleph_0}$. Hamkins' paper on the Multiverse shows that the ability to force to create models which have a variety of sizes of the continuum settles the continuum hypothesis. You can read all about how to add new reals to create a new model in Thomas Jech's Set Theory or Kenneth Kunen's book on the same subject.

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    $\begingroup$ "Almost anything is possible": indeed, as Ed Dean pointed out, Easton's theorem gives a precise expression of this fact. $\endgroup$ – Todd Trimble Feb 29 '12 at 14:20
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    $\begingroup$ More precisely, the Cohen-Solovay theorem. Solovay wrote a short note "$2^{\aleph_0}$ can be anything it ought to be" in the 1965 proceedings volume edited by Addison, Henkin and Tarski. MR0195680 (33 #3878). $\endgroup$ – Goldstern Feb 29 '12 at 14:58
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    $\begingroup$ Since the spirit of the question seems to be to look for strange values of the continuum, it might be worth mentioning that it is even possible that there are $2^{\aleph_0}$ cardinals between $\aleph_0$ and $2^{\aleph_0}$. $\endgroup$ – Juris Steprans Feb 29 '12 at 16:20
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    $\begingroup$ Joel, can this be made into a very scary power set operation by Easton's theorem: for every regular $\kappa$ we have that $2^\kappa = \aleph_{2^\kappa}$? $\endgroup$ – Asaf Karagila Feb 29 '12 at 23:26
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    $\begingroup$ Asaf, yes, it does seem that we can achieve your scary situation: just let $E(\kappa)$ be a suitable aleph-fixed point for each regular cardinal, starting from a model of GCH, and then appeal to Easton's theorem. I like it! $\endgroup$ – Joel David Hamkins Mar 1 '12 at 0:27

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