Luminy – Hugh Woodin: Ultimate L (II)

October 21, 2010

For the first lecture, see here.

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Luminy – Hugh Woodin: Ultimate L (I)

October 19, 2010

The XI International Workshop on Set Theory took place October 4-8, 2010. It was hosted by the CIRM, in Luminy, France. I am very glad I was invited, since it was a great experience: The Workshop has a tradition of excellence, and this time was no exception, with several very nice talks. I had the chance to give a talk (available here) and to interact with the other participants. There were two mini-courses, one by Ben Miller and one by Hugh Woodin. Ben has made the slides of his series available at his website.

What follows are my notes on Hugh’s talks. Needless to say, any mistakes are mine. Hugh’s talks took place on October 6, 7, and 8. Though the title of his mini-course was “Long extenders, iteration hypotheses, and ultimate L”, I think that “Ultimate L” reflects most closely the content. The talks were based on a tiny portion of a manuscript Hugh has been writing during the last few years, originally titled “Suitable extender sequences” and more recently, “Suitable extender models” which, unfortunately, is not currently publicly available.

The general theme is that appropriate extender models for supercompactness should provably be an ultimate version of the constructible universe L. The results discussed during the talks aim at supporting this idea.

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580 -Partition calculus (4)

April 9, 2009

 

1. Colorings of pairs. I

 

There are several possible ways in which one can try to generalize Ramsey’s theorem to larger cardinalities. We will discuss some of these generalizations in upcoming lectures. For now, let’s highlight some obstacles.

Theorem 1 ({\mbox{Erd\H os}}-Kakutani) {\omega_1\not\rightarrow(3)^2_\omega.} In fact, {2^\kappa\not\rightarrow(3)^2_\kappa.}

 

Proof: Let {S={}^\kappa\{0,1\}.} Let {F:[S]^2\rightarrow\kappa} be given by

\displaystyle  F(\{f,g\})=\mbox{least }\alpha<\kappa\mbox{ such that }f(\alpha)\ne g(\alpha).

Then, if {f,g,h} are distinct, it is impossible that {F(\{f,g\})=F(\{f,h\})=F(\{g,h\}).} \Box

Theorem 2 (Sierpi\’nski) {\omega_1\not\rightarrow(\omega_1)^2.} In fact, {2^\kappa\not\rightarrow(\kappa^+)^2.}

 

Proof: With {S} as above, let {F:[S]^2\rightarrow2} be given as follows: Let {<} be a well-order of {S} in order type {2^\kappa.} Let {<_{lex}} be the lexicographic ordering on {S.} Set

\displaystyle  F(\{f,g\})=1\mbox{ iff }<_{lex}\mbox{ and }<\mbox{ coincide on }\{f,g\}.

Lemma 3 There is no {<_{lex}}-increasing or decreasing {\kappa^+}-sequence of elements of {S.}

 

Proof: Let {W=\{f_\alpha\colon\alpha<\kappa^+\}} be a counterexample. Let {\gamma\le\kappa} be least such that {\{f_\alpha\upharpoonright\gamma\colon\alpha<\kappa^+\}} has size {\kappa^+,} and let {Z\in[W]^{\kappa^+}} be such that if {f,g\in Z} then {f\upharpoonright\gamma\ne g\upharpoonright\gamma.} To simplify notation, we will identify {Z} and {W.} For {\alpha<\kappa^+} let {\xi_\alpha<\gamma} be such that {f_\alpha\upharpoonright\xi_\alpha=f_{\alpha+1} \upharpoonright\xi_\alpha} but {f_\alpha(\xi_\alpha)=1-f_{\alpha+1}(\xi_\alpha).} By regularity of {\kappa^+,} there is {\xi<\gamma} such that {\xi=\xi_\alpha} for {\kappa^+} many {\alpha.}

But if {\xi=\xi_\alpha=\xi_\beta} and {f_\alpha\upharpoonright\xi=f_\beta\upharpoonright\xi,} then {f_\beta<_{lex} f_{\alpha+1}} iff {f_\alpha<_{lex} f_{\beta+1},} so {f_\alpha=f_\beta.} It follows that {\{f_\alpha\upharpoonright\xi\colon\alpha<\kappa^+\}} has size {\kappa^+,} contradicting the minimality of {\gamma.} \Box

The lemma implies the result: If {H\subseteq S} has size {\kappa^+} and is {F}-homogeneous, then {H} contradicts Lemma 3. \Box

Now I want to present some significant strengthenings of the results above. The results from last lecture exploit the fact that a great deal of coding can be carried out with infinitely many coordinates. Perhaps surprisingly, strong anti-Ramsey results are possible, even if we restrict ourselves to colorings of pairs.

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580 -Cardinal arithmetic (4)

February 11, 2009

2. Silver’s theorem.

From the results of the previous lectures, we know that any power \kappa^\lambda 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 \gimel(\lambda) varies greatly depending on whether \lambda is regular or not. If \lambda is regular, then \gimel(\lambda)=2^\lambda. As mentioned on lecture II.2, forcing provides us with a great deal of freedom to manipulate the exponential function \kappa\mapsto 2^\kappa, at  least for \kappa regular. In fact, the following holds:

Theorem 1. (Easton). If {\sf GCH} holds, then for any definable function F from the class of infinite cardinals to itself such that:

  1. F(\kappa)\le F(\lambda) whenever \kappa\le\lambda, and
  2. \kappa<{\rm cf}(F(\kappa)) for all \kappa,

there is a class forcing {\mathbb P} that preserves cofinalities and such that in the extension by {\mathbb P} it holds that 2^\kappa=F^V(\kappa) for all regular cardinals \kappa; here, F^V is the function F as computed prior to the forcing extension. \Box

For example, it is consistent that 2^\kappa=\kappa^{++} for all regular cardinals \kappa (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 2^\kappa=\kappa^{+16} whenever \kappa=\aleph_\tau is regular and \tau is an even ordinal, and 2^\kappa=\kappa^{+17} whenever \kappa=\aleph_\tau for some odd ordinal \tau. Or, if there is a proper class of weakly inaccessible cardinals (regular cardinals \kappa such that \kappa=\aleph_\kappa) then we could have 2^\kappa= the third weakly inaccessible strictly larger than \kappa, for all regular cardinals \kappa, 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 \gimel(\lambda) for singular \lambda. 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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