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I would like to know examples of forcing arguments where in order to make the cardinal arithmetic go through one needs to assume something about the size of the continuum or other power sets.

The examples that I know of are forcing to add $\aleph_2$ many Cohen reals (which requires the assumption of CH in the ground model to show that the continuum in the extension is exactly $\aleph_2$), and the forcing in Easton's Theorem (which requires the assumption of GCH in the ground model).

Are there other examples of forcing arguments where one needs to know (if only to make the arguments go more smoothly) or needs to assume the sizes of the power sets in the ground model in order to produce the desired result in the extension? I am especially interested in examples where the assumption is more exotic than CH or GCH and also examples where the forced result in the extension is about something other than the size of the continuum or the size of other power sets.

Also, I was looking at whether or not one needs to assume anything about the sizes of the power sets to show that if a partial order P satisfies the $\kappa +$-chain condition then it preserves cardinals above $\kappa$. I did not see any place where an assumption on the sizes of the power sets is required, but I wanted to check if I had overlooked some hidden detail. Along with this, I'm also interested if an assumption on the sizes of power sets is needed (or helpful) for common forcing arguments like this one.

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You don't need CH to add a Cohen real, but perhaps you mean to refer to the argument that one needs to know how big $2^\omega$ is in the ground model in order to know that adding $\aleph_2$ many Cohen reals forces $2^\omega=\aleph_2$. – Joel David Hamkins Nov 15 '11 at 1:04
Yes, of course. Thank you, I will make the correction. – Erin Carmody Nov 15 '11 at 1:35
up vote 7 down vote accepted

Something that happens rather frequently (in my particular field of interest) is that some assumption like CH is needed for the forcing notion to exist at all.
An example is the consistency proof of Todorcevic's Open Coloring Axiom, where the appropriate forcing notion for a single step of the forcing construction requires CH and the bookkeeping requires $\lozenge_{\omega_2}$ in the ground model. The diamond is in fact just another restriction on the size of power sets in the ground model, by Shelah's result that the higher diamonds correspond to instances of GCH.

Another example is oracle forcing, where you start with a ground model satisfying $\lozenge_{\omega_1}$. This is used to construct models where you have control over automorphisms of $\mathcal P(\omega)/fin$ and similar things.

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thank you Stefan, this is exactly what I was looking for. Out of curiousity, what do you call your particular field of interest? – Erin Carmody Nov 15 '11 at 13:43
If anyone else has more examples to add, I'll like to read. – Erin Carmody Nov 15 '11 at 13:43
Erin, I am glad I could be of help. I am interested in set theory of the reals, in particular things with a combinatorial flavour like OCA and automorphisms of $\mathcal P(\omega)/fin$. – Stefan Geschke Nov 15 '11 at 14:00

There are some odd examples coming from PCF theory, where one begins with assumptions about pcf structures that are not known to be consistent (even if one assumes consistency of large cardinals), and then do some forcing in order to make things happen.

For example, consider the following conjecture of Shelah:

"If $\mathfrak{a}$ is a set of regular cardinals greater than $|\mathfrak{a}|$, then ${\rm pcf}(\mathfrak{a})$ cannot have a weakly inaccessible accumulation point."

An affirmative answer to the above conjecture implies \begin{equation} {\rm cf}(\prod{\rm pcf}\mathfrak{a}, <)={\rm cf}(\prod\mathfrak{a},<) \end{equation} for every set of regular cardinals $\mathfrak{a}$ with $|\mathfrak{a}|<\min\mathfrak{a}$. On the other hand, if the above conjecture fails for some $\mathfrak{a}$, one can do some mild forcing and get the consistency of \begin{equation} {\rm cf}(\prod{\rm pcf}\mathfrak{a}, <)\neq{\rm cf}(\prod\mathfrak{a}, <). \end{equation}

Shelah has many arguments of this sort: he's got a whole family of pcf conjectures that are "linearly ordered" (none of which are known to be consistent) and he uses forcing to show that many other natural questions are (essentially) equivalent to one of his basic conjectures.

For a more substantial example, consider the question of whether every compact Hausdorff space can be partitioned into two pieces, neither of which contains a copy of the Cantor set. Assuming a supercompact cardinal, Shelah is able to force the existence of counterexample, but the resulting model satisfies CH. He is able to prove, however, that the existence of such a counterexample in a model where $2^{\aleph_0}=\aleph_2$ implies some weird pcf behavior that is not known to be consistent. Moreover (and this part speaks to your original question), if one assumes there is a model where this weird pcf behavior holds, then one can force the existence of a counterexample together with the continuum being $\aleph_2$. So in this case, the topological question is essentially equivalent to a question about pcf. (This is from his paper [Sh:682].)

Edit: I know your original question asked about assumptions on sizes of power sets, but I interpreted things a little broader to include assumptions about cardinal arithmetic.

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