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Recall that Easton forcing was introduced to show that the continuum function at regular cardinals could be anything subject to 'the obvious constraints' (monotonicity etc). However, it is a handy method if one wants to add a proper class of sets. My question is why do would we now restrict to using forcing conditions only using regular cardinals (edit: if we wanted only to add class-many sets)? I've had a read through Friedman's Class forcing, and all the (nontrivial) examples given there are variants on Easton forcing, only playing with things like supports and stationarity.

I'm not interested in preserving AC, though I suspect that we lose tameness at some point, and hence axioms like powerset may fail to hold.

Hmm, let me state my actual question, which was in fact rather implicit (and everyone's comments/answers have helped me figure out how to phrase it, so thank you all).

If I try to add $F(\kappa)$ generic subsets to each cardinal $\kappa$ (by some simple class function $\kappa \mapsto F(\kappa)$, such as the identity, or constant at some given infinite cardinal), will I get a model of ZF(C)? Or is the restriction to adding subsets to only regular cardinals, as in Easton forcing, necessary?

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  • $\begingroup$ Presumably, you still want to add a proper class of sets with a purpose? $\endgroup$
    – François G. Dorais
    Commented Nov 18, 2012 at 5:28
  • $\begingroup$ Yes, it's to do a class-of-atoms version of Jech-Sochor. All I need in the end is a model of ZF. $\endgroup$
    – David Roberts ♦
    Commented Nov 18, 2012 at 7:14
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    $\begingroup$ David, given your purpose and since there are a proper class of regular cardinals, why does it matter to do something at singular cardinals? For example, you can view Easton's iteration as adding a subset to $\kappa^+$ for every infinite cardinal $\kappa$, including singular $\kappa$, since all $\kappa^+$ are regular. $\endgroup$
    – Joel David Hamkins
    Commented Nov 18, 2012 at 10:48
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    $\begingroup$ A Jech-Sochor theorem for a proper class of atoms will need some additional nontrivial structure on the atoms. The point is that the pure sets that play, in the ZF model, the role of the atoms can't all have the same rank. So the set of these surrogate atoms comes with a certain structure, namely (at least) a partition according to rank. If the original set of atoms didn't have this structure, then you won't get the sort of transfer theorem that Jech and Sochor got. $\endgroup$
    – Andreas Blass
    Commented Nov 18, 2012 at 13:33
  • $\begingroup$ @JDH not necessarily in ZF :-) @Andreas that's good to know. I was planning to add sets at all ranks, so that's one objection out of the way. $\endgroup$
    – David Roberts ♦
    Commented Nov 18, 2012 at 19:57

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To increase the power set of a regular cardinal $\kappa$, Easton used forcing conditions that are partial functions of size $<\kappa$. So the forcing is $\kappa$-closed and therefore adds no new subsets of any cardinals below $\kappa$. It therefore doesn't interfere with whatever he was trying to do with the power sets of those smaller cardinals. If he did the same thing with a singular $\kappa$, the forcing would be only cf$(\kappa)$-closed, not $\kappa$-closed. For example, if $\kappa=\aleph_\omega$, then the union of a countable chain of conditions (each of size $<\aleph_\omega$) could have size $\aleph_\omega$ and thus fail to be a condition. As a result, new subsets would be added at cardinals below $\kappa$ (but $\geq$ cf$(\kappa)$), thereby messing up whatever was supposed to happen with the power sets of those cardinals.

A decade later, Silver discovered that not only does Easton's method not work for singular cardinals (which Easton already knew), but there are non-trivial constraints on $2^\kappa$ for singular $\kappa$. In particular, a singular cardinal of uncountable cofinality cannot be the first place where GCH fails. Later, it was shown (I believe first by Magidor) that a singular cardinal of countable cofinality can be the first place where GCH fails, but a large cardinal was needed for the proof and, by a result of Jensen, large cardinals are unavoidable here. Work of Gitik has pinned down the exact large-cardinal strength of the negation of the singular cardinal hypothesis.

The bottom line here is that, in order to get anything like Easton's results for singular cardinals, one must use large cardinals, one must use considerably fancier forcing notions than Easton used, and even then, some manipulations of power sets of singular cardinals are outright impossible.

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    $\begingroup$ Shortly before Silver announced his breakthrough result, Frank Drake wrote a paper explaining why not only Easton's method but also variations that people had been trying didn't work. The paper is "On failures at singular cardinals in generic extensions," J. London Math. Soc., series 2, vol. 9 (1974/75) pp. 219-228. I believe the motivation for Drake's paper was that, because attempts to extend Easton's result to singular cardinals were unsuccessful, they weren't being published, so people were independently inventing the same methods and discovering that they fail. $\endgroup$
    – Andreas Blass
    Commented Nov 18, 2012 at 1:59
  • $\begingroup$ I should clarify I that my main interest is not Easton's results, but in adding proper classes of sets. Cardinal inequalities are unimportant for my purposes. $\endgroup$
    – David Roberts ♦
    Commented Nov 18, 2012 at 3:00
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What Andreas Blass fails to mention is that the forcing used to add subsets to a cardinal $\kappa$ will not only add subsets to $cf(\kappa)$ but even collapse $\kappa$ to $cf(\kappa)$ if $\kappa$ is singular.

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