February 11, 2009
2. Silver’s theorem.
From the results of the previous lectures, we know that any power can be computed from the cofinality and gimel functions (see the Remark at the end of lecture II.2). What we can say about the numbers varies greatly depending on whether is regular or not. If is regular, then As mentioned on lecture II.2, forcing provides us with a great deal of freedom to manipulate the exponential function at least for regular. In fact, the following holds:
Theorem 1. (Easton). If holds, then for any definable function from the class of infinite cardinals to itself such that:
- whenever and
- for all
there is a class forcing that preserves cofinalities and such that in the extension by it holds that for all regular cardinals here, is the function as computed prior to the forcing extension.
For example, it is consistent that for all regular cardinals (as mentioned last lecture, the same result is consistent for all cardinals, as shown by Foreman and Woodin, although their argument is significantly more elaborate that Easton’s). There is almost no limit to the combinations that the theorem allows: We could have whenever is regular and is an even ordinal, and whenever for some odd ordinal Or, if there is a proper class of weakly inaccessible cardinals (regular cardinals such that ) then we could have the third weakly inaccessible strictly larger than for all regular cardinals etc.
Morally, Easton’s theorem says that there is nothing else to say about the gimel function on regular cardinals, and all that is left to be explored is the behavior of for singular In this section we begin this exploration. However, it is perhaps sobering to point out that there are several weaknesses in Easton’s result.
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October 24, 2008
I concluded my series of talks by showing the following theorem of Viale:
Theorem (Viale). Assume and let be an inner model where is regular and such that . Then .
This allows us to conclude, via the results shown last time, that if holds in and computes cardinals correctly, then it also computes correctly ordinals of cofinality .
An elaboration of this argument is expected to show that, at least if we strengthen the assumption of to , then computes correctly ordinals of cofinality .
Under an additional assumption, Viale has shown this: If holds in , is a strong limit cardinal, , and in we have that is regular, then in , the cofinality of cannot be . The new assumption on allows us to use a result of Dzamonja and Shelah, On squares, outside guessing of clubs and , Fund. Math. 148 (1995), 165-198, in place of the structure imposed by . It is still open if the corresponding covering statement follows from , which would eliminate the need for this the strong limit requirement.
- Go to the intermezzo for a discussion of consistency strengths.
October 17, 2008
I presented a sketch of a nice proof due to Todorcevic that implies the P-ideal dichotomy . I then introduced Viale’s covering property and showed that it follows from . Next time I will indicate how it can be used to provide a proof of part 1 of the following theorem:
Theorem (Viale). Assume is an inner model.
- If holds in and computes cardinals correctly, then it also computes correctly ordinals of cofinality .
- If holds in , is a strong limit cardinal, , and in we have that is regular, then in , the cofinality of cannot be .
It follows from this result and the last theorem from last time that if is a model of and a forcing extension of an inner model by a cardinal preserving forcing, then .
In fact, the argument from last time shows that we can weaken the assumption that is a forcing extension to the assumption that for all there is a regular cardinal such that in we have a partition where each is stationary in .
It is possible that this assumption actually follows from in . However, something is required for it: In Gitik, Neeman, Sinapova, A cardinal preserving extension making the set of points of countable cofinality nonstationary, Archive for Mathematical Logic, vol. 46 (2007), 451-456, it is shown that (assuming large cardinals) one can find a (proper class) forcing extension of that preserves cardinals, does not add reals, and (for some cardinal ) the set of points of countable -cofinality in is nonstationary for every regular . Obviously, this situation is incompatible with in , by Viale’s result.
October 1, 2008
(At Randall’s request, this entry will be more detailed than usual.)
Remark 1. is club in , so any is stationary as a subset of iff it is stationary as a subset of . It follows that proper forcing preserves stationary subsets of .
Remark 2. Proper forcing extensions satisfy the countable covering property with respect to , namely, if is proper, then any countable set of ordinals in is contained in a countable set of ordinals in . We won’t prove this for now, but the argument for Axiom A posets resembles the general case reasonably:
Given a name for a countable set of ordinals in the extension, find an appropriate regular and consider a countable elementary containing , , and any other relevant parameters. One can then produce a sequence such that
- Each is in .
- , where enumerates the dense subsets of in .
Let for all . Then , so is a countable set of ordinals in containing in . A density argument completes the proof.
Woodin calls a poset weakly proper if the countable covering property holds between and . Not every weakly proper forcing is proper, but the countable covering property is a very useful consequence of properness. On the other hand, standard examples of stationary set preserving posets that are not proper, like Prikry forcing (changing the cofinality of a measurable cardinal to without adding bounded subsets of ) or Namba forcing (changing the cofinality of to while preserving are not weakly proper, and account for some of the usefulness of over .
The following is obvious:
Fact. Assume is weakly proper. Then either adds no new -sequences of ordinals, or else it adds a real.
The relation between the reals and the -sequences of ordinals in the presence of strong forcing axioms like is a common theme I am exploring through these talks.
September 30, 2008
This posting complements a series of talks given at the Set Theory Seminar at BSU from September 12 to October 24, 2008. Here is a list of links to the talks in this series:
- First talk, September 12, 2008.
- Second talk, September 19, 2008.
- Third talk, September 26, 2008.
- Fourth talk, October 3, 2008.
- Fifth talk, October 10, 2008.
- Sixth talk, October 17, 2008.
- Seventh talk, October 24, 2008.
[Version of October 31.]
I’ll use this post to provide some notes about consistency strength of the different natural hierarchies that forcing axioms and their bounded versions suggest. This entry will be updated with some frequency until I more or less feel I don’t have more to add. Feel free to email me additions, suggestions and corrections, or to post them in the comments. In fact, please do.
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September 12, 2008
Today I started a series of talks on “Forcing axioms and inner models” in the Set Theory Seminar. The goal is to discuss a few results about strong forcing axioms and to see how these axioms impose a certain kind of `rigidity’ to the universe.
I motivated forcing axioms as trying to capture the intuition that the universe is `wide’ or `saturated’ in some sense, the next natural step after the formalization via large cardinal axioms of the intuition that the universe is `tall.’
The extensions of obtained via large cardinals and those obtained via forcing axioms share a few common features that seem to indicate their adoption is not arbitrary. They provide us with reflection principles (typically, at the level of the large cardinals themselves, or at small cardinals, respectively), with regularity properties (and determinacy) for many pointclasses of reals, and with generic absoluteness principles.
The specific format I’m concentrating on is of axioms of the form for a class of posets, stating that any admits filters meeting any given collection of many dense subsets of . The proper forcing axiom is of this kind, with being the class of proper posets. The strongest axiom to fall under this setting is Martin’s maximum , that has as the class of all posets preserving stationary subsets of .
Of particular interest is the `bounded’ version of these axioms, which, if posets in preserve , was shown by Bagaria to correspond precisely to an absoluteness statement, namely that for any .
In the next meeting I will review the notion of properness, and discuss some consequences of .
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