What goes wrong in Easton forcing if we don't just use regular cardinals?

2012-11-18T01:25:33Z

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?

Answer by Andreas Blass for What goes wrong in Easton forcing if we don't just use regular cardinals?

2012-11-18T01:51:52Z

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.

What goes wrong in Easton forcing if we don't just use regular cardinals? - MathOverflow

What goes wrong in Easton forcing if we don't just use regular cardinals?

Answer by Andreas Blass for What goes wrong in Easton forcing if we don't just use regular cardinals?