\(\newcommand\seq[1]{\langle#1\rangle}\newcommand\gkeq{\leftrightarrow} \newcommand{\chang}{\mathbb C} \newcommand\changsharp{\chang^\sharp} \newcommand\changcore{K(\reals)^\chang} \newcommand\reals{\mathcal R} \newcommand\ords{\Omega} \DeclareMathOperator\ult{Ult} \DeclareMathOperator\supp{supp} \DeclareMathOperator\cof{cf} \newcommand\cut{\vert} \newcommand\ecut{\vert} \DeclareMathOperator\crit{crit} \newcommand\set[1]{\{#1\}} \newcommand\ps{\mathcal P} \newcommand\chang{\mathbb C} \DeclareMathOperator\add{add} \newcommand\changone{\chang_{\omega_1}} \newcommand\changsharpone{\changone^{\#}} \DeclareMathOperator\domain{domain} \DeclareMathOperator\range{range}\)

The \(\Omega_1\)-Chang Model — a False Conjecture

The \(\omega_1\)-Chang model is defined in the same way as the \(\chang\)-model, except that it contains all \(\omega_1\) sequences of ordinals. Thus its presumptive sharp, \(\changsharpone\), would be a mouse \(M\) over \(\ps(\omega_1)\). I’m conjecturing at the moment that an extender of length \(\kappa^{+\omega_2}\) implies the existence of \(\changsharpone\), and that this can be proved by substantially the same argument as in my paper [1]changpaper for \(\chang^\sharp\).

Failure of the Use of Gitik’s Reconstruction for a Lower Bound

Originally I had thought that Gitik’s argument for reconstructing extenders from the threads given by iterations could be used to show that \(\chang^\sharp_{\omega_1}\) would be substantially stronger than \(\chang^\sharp\) — perhaps as strong as or stronger than a Woodin cardinal. Ralf Schindler pointed out the flaw in my thinking: Gitik’s reconstruction for uncountable cofinality requires the covering lemma, which makes the argument break down at an extender overlapping an extender of length \(\kappa^{+ +}\). Still, this is enough that I “knew” that that my proof of the existence of \(\chang^\sharp\) would not work for \(\changsharpone\) — but I didn’t know where it broke down. I realized this Oberwolfach, and it kept me awake for a nightwith some anxiety that if I was missing it for \(\changone\) maybe it was something I was also missing for \(\chang\) itself. Finally I realized what was happening: the failure of the covering lemma comes in much earlier than I’d thought: Gitik’s reconstruction for extenders from iterations of length \(\omega_1\) requires covering by sets of size \(\omega_1\), but the appropriate core model for \(\changone\) is \(K(\ps(\reals))\), so every mouse has size at least \(|\ps(\reals)|\), which cannot be smaller than \(\omega_2\).[2]

Woodin’s view

Woodin (private conversation) has said that \(\changsharpone\) is impossible, at least without Determinacy. Obviously I’m disagreeing, though with some trepidation. Here are some thoughts.

UPDATE (3/7/14): He is (of course) correct: \(\kappa\to(\omega_1)^{\omega_1}\) implies that every \(\omega_1\)-Souslin set is determined, by Martin’s proof that \(0^\sharp\) implies \(\Pi_1^1\) determinacy. Thus a sharp for the \(\omega_1\) Chang model implies that \(L(R)\) satisfies \(AD\), but not, so far as I can see, anything beyond that. Is it plausible to conjecture that \(AD\) in \(L(R)\) together with an extender of length \(\omega_2\) implies that there is a sharp for the \(\omega_1\) Chang model?

• As I understand it, Woodin’s reason for believing there cannot be a sharp for the \(\omega_1\)-Chang model is that it would imply infinitary partition properties, which in turn would imply AD. I don’t see that \(\changsharpone\) implies any partition property significantly stronger than \(\kappa\to (\omega_1)^{\omega_1}\), where \(\kappa\) is a member of the class \(I\) of indiscernibles, and is thus a large cardinal in \(\changone\); in particular \(\kappa\) is greater than \(\omega_2\), which we can assume is \(|{2^\omega}|\) in \(V\). I don’t see that it implies that \(\omega_1\to(\omega_1)^{\omega_1}\), since \(\kappa\) is not definably singular in \(\changone\). It’s been a long time since I’ve studied infinitary partition relations, but I don’t see how this would present a problem. In a model of AD there are many cardinals below \(\theta\) satisfying the strong partition property, \(\kappa\to(\kappa)^\kappa\). Here everything is happening above \(\theta\).

• Woodin’s sharp is actually for a stronger model than \(\chang\), which he calls \(\chang^+\) . As I understand it this is the smallest model which not only contains all countable sequences, but is correct about stationarity for all of the sets \(\ps_{\omega_1}(\lambda)\) for regular \(\lambda\). He says that this sharp induces a sharp for \(\chang\), which I certainly believe, and that his proof of impossibility of a sharp for its analog \(\chang^+_{\omega_1}\) also applies down to \(\changsharpone\); this last I find easier to doubt.

• If I’m wrong so far, there are two weaknesses in the current version [1]changpaper of my \(\chang^\sharp\) model which may be relevant. The first is the use of restricted formulas, which do not appear in his sharp for the same model. This could possibly affect the argument he has in mind. However I’m reasonably confident at this point that a better choice of terms will eliminate the need for restricted formulas.

• Woodin said something about a need for patterns as being a possible (though unlikely) out. In [1]changpaper, there is something that may apply: in order to be deemed equivalent, two sequences \(B\) and \(B'\) need to agree not only on their order type, but also on whether any corresponding pairs of consecutive elements of \(B\) and \(B'\) are also consecutive in \(I\), and, if not, whether the larger is a successor or a limit point in \(I\). However I now think it likely that the improvement of the class of terms referred to in the previous item will also eliminate this restriction[3]betterterms.

How to Prove It?

I haven’t tried to put anything on paper (other than this note), but I have been thinking about how to construct \(\changsharpone\). I’ve only found one issue which I’ve recognized as problematic: namely the equivalence relation \(\gkeq\). In [1]changpaper I use (following Gitik), in the tableau \(z\) of a level \(s(\zeta)\) of a condition, functions \(a_{\gamma,\nu}\colon [\bar\kappa_\gamma,\bar\kappa_\gamma^+) \to \supp(E_\nu)\) for \(\nu\lt\gamma\lt\zeta\lt \omega_1\).[4] The definition says that \(z\gkeq z'\) if for some sequence \(\vec n=\seq{n_\nu\mid\nu\in\zeta}\) with the property that \(\forall m\in\omega\,\set{\nu\mid n_\nu\lt m}\) is finite, we have that \(a_{\nu+1,\nu}\gkeq_{n_\nu} a'_{\nu+1,\nu}\) for all \(\nu\in \zeta\).[5] This works for countable \(\zeta\), but for the \(\omega_1\)-Chang model I’d need to deal with \(\zeta\lt\omega_2\).

I think I my may be able to modify the construction by replacing each function \(a_{\nu}\)[6] with a countable sequence \(a_{\nu,n}\) of such functions. The idea would be that \(a_{\nu}\gkeq a'_{\nu}\) if \(a_{\nu,i}\gkeq_{\nu,n_i} a'_{\nu,i}\) for a sequence \(\seq{n_i\mid i\in \omega}\) with \(\forall m\in\omega\forall^\infty i\in\omega\; n_i\ge m\), and then defining \(z\gkeq z'\) if \(a^z_{\nu} \gkeq a^{z'}_\nu\) for all \(\nu\in\zeta\).

This would break the trick I took from Gitik of dealing with the finitely many cases in which \(n_\nu=0\) by using \(\add(s,\vec w)\) where \(\vec w\) adds \(\set{\nu\mid n_\nu=0}\) to \(\domain(s)\). This raises a problem with the proof of the Lemma asserting that if \(s'\le s\gkeq t\) then there are \(s''\le s'\) and \(t''\le t\) such that \(s''\gkeq t''\).

Perhaps this could be dealt with by extending \(\gkeq\) by adding \(a\gkeq a'\) if \(\set{n\mid a_n\not=a'_n}\) is finite. This would interfere with the need to establish a standard name for every generator, since a condition containing \(a_{\nu,n}(\xi)=\beta\) could always be overridden and hence would not establish \(h_{\zeta,\nu,n}(\xi)\) as a name for \(i_\nu\circ i^{-1}_\zeta(\beta)\). Here I could modify the definition of \(\add(s,z)\) by defining, for \(\vec\xi\colon\omega\to[\bar\kappa_\zeta,\bar\kappa_\zeta^+)\), the function \(f_{a,a'}(\vec\xi)=\alpha\) if \(f_{a_n,a'_n}(\xi_n)=\alpha\) (using the definition of \(f_{a,a'}\) from [1]changpaper) for all but finitely many \(n\in\omega\).