Luminy – hugh woodin: Ultimate L (I)
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 superpactness should provably be an ultimate version of the constructible universe L. the results discussed during the talks aim at supporting this idea.
Ultimate L
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I
Let δ be superpact. the basic problem that concerns us is whether there is an L-like inner model N\\subseteq V with δ superpact in N.
of course， the shape of the answer depends on what we mean by “L-like”. there are several possible ways of making this nontrivial. here， we only adopt the very general requirement that the superpactness of δ in N should “directly trace back” to its superpactness in V.
Recall:
we use p_δ(x) to denote the set \\{a\\subseteq x\\mid |a|amp;amp;amp;amp;lt;δ\\}.
An ultrafilter (or measure)U on p_δ(λ) is fine iff for all \\alphaamp;amp;amp;amp;lt;λ we have \\{a\\inp_δ(λ)\\mid \\alpha\\in a\\}\\inU.
the ultrafilter U is normal iff it is δ-plete and for all F:p_δ(λ)oλ， if F is regressive U-ae (i.e.， if \\{a\\mid F(a)\\in a\\}\\inU) then F is constant U-ae， i.e.， there is an \\alphaamp;amp;amp;amp;lt;λ such that \\{a\\mid F(a)=\\alpha\\}\\inU.
δ is superpact iff for all λ there is a normal fine measure U on p_δ(λ).
It is a standard result that δ is superpact iff for all λ there is an elementary embedding j:Vo m with {\\rm cp}(j)=δ， j(δ)amp;amp;amp;amp;gt;λ， and j''λ\\in m (or， equivalently，{}^λ m\\subseteq m).
In fact， given such an embedding j， we can define a normal fine U on p_δ(λ) by
A\\inU iff j''λ\\in j(A).
conversely， given a normal fine ultrafilter U on p_δ(λ)， the ultrapower embedding generated by U is an example of such an embedding j. moreover， if U_j is the ultrafilter on p_δ(λ) derived from j as explained above， then U_j=U.
Another characterization of superpactness was found by magidor， and it will play a key role in these lectures; in this reformulation， rather than the critical point，δ appears as the image of the critical points of the embeddings under consideration. this version seems ideally designed to be used as a guide in the construction of extender models for superpactness， although recent results suggest that this is， in fact， a red herring.
the key notion we will be studying is the following:
definition. N\\subseteq V is a weak extender model for `δ is superpact’ iff for all λamp;amp;amp;amp;gt;δ there is a normal fine U on p_δ(λ) such that:
p_δ(λ)\\cap N\\in U， and
U\\cap N\\in N.
this definition couples the superpactness of δ in N directly with its superpactness in V. In the manuscript， that N is a weak extender model for `δ is superpact’ is denoted by o^N_{\\rm long}(δ)=\\infty. Note that this is a weak notion indeed， in that we are not requiring that N=L[\\vec E] for some (long) sequence \\vec E of extenders. the idea is to study basic properties of N that follow from this notion， in the hopes of better understanding how such an L[\\vec E] model can actually be constructed.
For example， fineness of U already implies that N satisfies a version of covering: If A\\subseteqλ and |A|amp;amp;amp;amp;lt;δ， then there is a b\\inp_{δ}(λ)\\cap N with A\\subseteq b. but in fact a significantly stronger version of covering holds. to prove it， we first need to recall a nice result due to Solovay， who used it to show that {\\sf Sch} holds above a superpact.
Solovay’s Lemma. Let λamp;amp;amp;amp;gt;δ be regular. then there is a set x with the property that the function f:a\\mapsto\\sup(a) is injective on x and， for any normal fine measure U on p_δ(λ)， x\\inU.
It follows from Solovay’s lemma that any such U is equivalent to a measure on ordinals.
proof. Let \\vec S=\\leftamp;amp;amp;amp;lt; S_\\alpha\\mid\\alphaamp;amp;amp;amp;lt;λ\\rightamp;amp;amp;amp;gt; be a partition of S^λ_\\omega into stationary sets.
(we could just as well use S^λ_{\\le\\gamma} for any fixed \\gammaamp;amp;amp;amp;lt;δ. Recall that
S^λ_{\\le\\gamma}=\\{\\alphaamp;amp;amp;amp;lt;λ\\mid{\\rm cf}(\\alpha)\\le\\gamma\\}
and similarly for S^λ_\\gamma=S^λ_{=\\gamma} and S^λ_{amp;amp;amp;amp;lt;\\gamma}.)
It is a well-known result of Solovay that such partitions exist.
hugh actually gave a quick sketch of a crazy proof of this fact: otherwise， attempting to produce such a partition ought to fail， and we can therefore obtain an easily definable λ-plete ultrafilter {\\mathcal V} on λ. the definability in fact ensures that {\\mathcal V}\\in V^λ\/{\\mathcal V}， contradiction. we will encounter a similar definable splitting argument in the third lecture.
Let x consist of those a\\inp_δ(λ) such that， letting \\beta=\\sup(a)， we have {\\rm cf}(\\beta)amp;amp;amp;amp;gt;\\omega， and
a=\\{\\alphaamp;amp;amp;amp;lt;\\beta\\mid S_\\alpha\\cap\\beta is stationary in \\beta\\}.
then f is 1-1 on x since， by definition， any a\\in x can be reconstructed from \\vec S and \\sup(a). All that needs arguing is that x\\inU for any normal fine measure U on p_δ(λ).(this shows that to define U-measure 1 sets， we only need a partition \\vec S of S^λ_\\omega into stationary sets.)
Let j:Vo m be the ultrapower embedding generated by U， so
U=\\{A\\inp_δ(λ)\\mid j''λ\\in j(A)\\}.
we need to verify that j''λ\\in j(x). First， note that j''λ\\in m. Letting au=\\sup(j''λ)， we then have that m\\models{\\rm cf}(au)=λ. Since
m\\models j(λ)\\geau is regular，
it follows that auamp;amp;amp;amp;lt;j(λ). Let \\leftamp;amp;amp;amp;lt;t_\\beta\\mid\\betaamp;amp;amp;amp;lt;j(λ)\\rightamp;amp;amp;amp;gt;=j(\\leftamp;amp;amp;amp;lt;S_\\alpha\\mid\\alphaamp;amp;amp;amp;lt;λ\\rightamp;amp;amp;amp;gt;). In m， the t_\\beta partition S^{j(λ)}_\\omega into stationary sets. Let
A=\\{\\beta
δ是规则的。然后有一个集合x具有函数f:a\\mapsto\\sup(a)在x上是单射的性质，并且，对于任何正常的精细测度U上pδ(λ)， x ∈ U。
从索洛维引理可以得出，任何这样的 U都等价于序数上的测度。
证明。设\\vec S=〈 S_a|aλ〉是S^λ_≤γ的一个分划成平稳集。
(我们也可以使用S^\\λ_≤γ来表示任何固定的γ<δ。回想一下，
S ^λ_γ={a<λ| cf(a)≤γ}
同样的，S^λ_γ=S^λ_=γ和S^λ_<γ)
这种分区的存在是Solovay的一个众所周知的结果。
hugh实际上给出了对这个事实的一个疯狂的证明:否则，试图产生这样一个划分应该会失败，因此我们可以得到一个容易定义的完整超滤器 V on λ。可定义性实际上确保了 V在V^λ V中，矛盾。在第三节课中，我们会遇到一个类似的可定义的分裂论证。
让x由∈pδ(λ)中的一个组成，这样，让β=sup(a)，我们有cf(β)>w，和
a=\\{a<β| S_anβ固定在β}中。
那么f在x上是1-1，因为根据定义，x中的任何a都可以由\\vec S和\\sup(a)重构。所有需要讨论的是x在U中对于U上p_δ(λ)的任何正常精细测量U。(这表明要定义U-测度1集，我们只需要将S^λ_w划分为平稳集。)