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Let $\kappa$ be an uncountable cardinal of a c.t.m $M$ of $ZFC$.

Is there a generic extension of $M$ like $M[G]$ such that all uncountable cardinals of ground model collapse except $\kappa$?

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  • $\begingroup$ $\aleph_0$ never collapses. $\endgroup$
    – Goldstern
    Commented Dec 31, 2013 at 13:51
  • $\begingroup$ Do you mean a forcing extension? Class forcing? $\endgroup$
    – Goldstern
    Commented Dec 31, 2013 at 13:53
  • $\begingroup$ @Goldstern: Obviously I mean collapse of uncountable cardinals using any possible kind of forcings. $\endgroup$
    – user44891
    Commented Dec 31, 2013 at 14:02

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Obviously we cannot do this and preserve ZFC in the extension, since $M[G]$ will have only one uncountable cardinal. But if you give up power set in the extension, then for many cardinals $\kappa$ you can do this with class forcing. For example, $\text{Coll}(\omega_1,\lt\text{Ord})$ is the forcing having as conditions the countable partial functions from $\omega_1$ to the ordinals, ordered by extension. The generic filter will be a surjection $\omega_1\to\text{Ord}$, which will collapse all cardinals above $\omega_1$ to $\omega_1$ in $M[G]$. This model will not satisfy the power set axiom for uncountable sets, although it will for countable sets, and indeed the CH will hold in this model, as well as the rest of ZFC except for power set.

For any regular $\kappa$ that could become $\omega_1$ in a forcing extension, we could first do that and then do the above forcing, so as to achieve the result with $\kappa$. Singular $\kappa$ are problematic, and if one wants AC in $M[G]$ this will be impossible, since $\kappa$ will become $\omega_1$ in the extension.

It is interesting to consider the similar model $M[H]$ obtained by collapsing all sets to be countable, thereby producing a model of ZFC without power set in which every set is countable.

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  • $\begingroup$ Is there a good source on that $M[H]$? $\endgroup$
    – Noah Schweber
    Commented Dec 31, 2013 at 17:55
  • $\begingroup$ Noah, it comes up in a variety of contexts, but I don't know an authorative account of it. It has come up before on MO at mathoverflow.net/questions/139797/…. $\endgroup$
    – Joel David Hamkins
    Commented Dec 31, 2013 at 19:09
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    $\begingroup$ @Noah If I were to think about $M[H]$, I'd pretend that some more powerful set theorists have continued the cumulative hierarchy beyond the hierarchy that I see, so that what I think is Ord is actually just some inaccessible cardinal $\kappa$. Then what I call $M[H]$ is, for people in that bigger universe, the result of Lévy collapsing below $\kappa$ and then truncating the universe at $\kappa$. That Lévy model is pretty well understood, since it occurs along the way in Solovay's construction of his Lebesgue measure model. $\endgroup$
    – Andreas Blass
    Commented Jan 1, 2014 at 1:30
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    $\begingroup$ @Noah $M[H]$ shows up in a result of Jensen quoted by Mathias as Metatheorem 5.11 in "Happy Families" [Ann. Math. Logic 12 (1977) 59-111]. Jensen's result is the equiconsistency of "$M[H]$ is an elementary extension of the set of hereditarily countable sets (of $M$)" and "ZFC + every closed unbounded class of ordinals contains a regular cardinal (the "Mahlo schema")." $\endgroup$
    – Andreas Blass
    Commented Jan 1, 2014 at 1:44

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