求道九州 Ultimate L
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Luminy – hugh woodin: Ultimate L (I)
the xI Iional workshop oheory took place october 4-8， 2010. It was hosted by the cIRm， in Luminy， France. I am very glad I was invited， si was a great experiehe workshop has a tradition of excellence， and this time was no exception， with several very alks. I had the ce to give a talk (available here) and to i with the other partits. 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 pla october 6， 7， and 8. though the title of his mini-course was “Loenders， iteration hypotheses， and ultimate L”， I think that “Ultimate L” reflects most closely the tent. the talks were based on a tiny portion of a manuscript hugh has been writing during the last few years， inally titled “Suitable extender sequences” and more retly，“Suitable extender models” which， unfortunately， is not currently publicly available.
the general theme is that appropriate extender models for superpaess should provably be an ultimate version of the structible universe L. the results discussed during the talks aim at supp this idea.
Ultimate L
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I
Let δ be superpact. the basi that s us is whether there is an L-like inner model N\\subseteq V with δ superpa 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 superpaess of δ in N should “directly trace back” to its superpaess in V.
Recall:
we use p_δ(x) to dehe 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 stant 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 aary 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 su embedding j， we define a normal fine U on p_δ(λ) by
A\\inU iff j''''λ\\in j(A).
versely， given a normal firafilter U on p_δ(λ)， the ultrapower embedding geed by U is an example of su 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 superpaess 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 sideration. this version seems ideally desigo be used as a guide in the stru of extender models for superpaess， although ret 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)\\\\in U， and
U\\\\in N.
this definition couples the superpaess of δ in N directly with its superpaess 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. hat 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 uanding how su L[\\vec E] model actually be structed.
For example， fineness of U already implies that N satisfies a version of c: If A\\subseteqλ and |A|amp;amp;amp;amp;lt;δ， then there is a b\\inp_{δ}(λ)\\ with A\\subseteq b. but in fact a signifitly stronger version of c holds. to prove it， we first o 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 fun f:a\\mapsto\\sup(a) is iive 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-know of Solovay that such partitio.
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 therefore obtain an easily definable λ-plete ultrafilter {\\mathcal V} on λ. the definability in fasures that {\\mathcal V}\\in V^λ\/{\\mathcal V}， tradi. we will enter a similar definable splitting argument ihird lecture.
Let x sist 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 be restructed 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 geed by U， so
U=\\{A\\inp_δ(λ)\\mid j''''λ\\in j(A)\\}.
we o verify that j''''λ\\in j(x). First， hat 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。)
the xI Iional workshop oheory took place october 4-8， 2010. It was hosted by the cIRm， in Luminy， France. I am very glad I was invited， si was a great experiehe workshop has a tradition of excellence， and this time was no exception， with several very alks. I had the ce to give a talk (available here) and to i with the other partits. 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 pla october 6， 7， and 8. though the title of his mini-course was “Loenders， iteration hypotheses， and ultimate L”， I think that “Ultimate L” reflects most closely the tent. the talks were based on a tiny portion of a manuscript hugh has been writing during the last few years， inally titled “Suitable extender sequences” and more retly，“Suitable extender models” which， unfortunately， is not currently publicly available.
the general theme is that appropriate extender models for superpaess should provably be an ultimate version of the structible universe L. the results discussed during the talks aim at supp this idea.
Ultimate L
Advertisements
REpoRt thIS Ad
I
Let δ be superpact. the basi that s us is whether there is an L-like inner model N\\subseteq V with δ superpa 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 superpaess of δ in N should “directly trace back” to its superpaess in V.
Recall:
we use p_δ(x) to dehe 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 stant 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 aary 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 su embedding j， we define a normal fine U on p_δ(λ) by
A\\inU iff j''''λ\\in j(A).
versely， given a normal firafilter U on p_δ(λ)， the ultrapower embedding geed by U is an example of su 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 superpaess 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 sideration. this version seems ideally desigo be used as a guide in the stru of extender models for superpaess， although ret 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)\\\\in U， and
U\\\\in N.
this definition couples the superpaess of δ in N directly with its superpaess 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. hat 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 uanding how su L[\\vec E] model actually be structed.
For example， fineness of U already implies that N satisfies a version of c: If A\\subseteqλ and |A|amp;amp;amp;amp;lt;δ， then there is a b\\inp_{δ}(λ)\\ with A\\subseteq b. but in fact a signifitly stronger version of c holds. to prove it， we first o 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 fun f:a\\mapsto\\sup(a) is iive 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-know of Solovay that such partitio.
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 therefore obtain an easily definable λ-plete ultrafilter {\\mathcal V} on λ. the definability in fasures that {\\mathcal V}\\in V^λ\/{\\mathcal V}， tradi. we will enter a similar definable splitting argument ihird lecture.
Let x sist 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 be restructed 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 geed by U， so
U=\\{A\\inp_δ(λ)\\mid j''''λ\\in j(A)\\}.
we o verify that j''''λ\\in j(x). First， hat 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。)