\[$\newcommand\seq[1]{\langle#1\rangle} \newcommand{\chang}{\mathbb C} \newcommand\changsharp{\chang^\sharp} \newcommand\changcore{K(\reals)^\chang} \newcommand\reals{\mathcal R} \newcommand\ords{\Omega} \DeclareMathOperator\ult{Ult} \DeclareMathOperator\supp{supp} \DeclareMathOperator\cof{cf} \newcommand\cut{\vert} \newcommand\ecut{\vert} \DeclareMathOperator\crit{crit} \newcommand\set[1]{\{#1\}} \newcommand\belongsto{\sim} \DeclareMathOperator{\genterm}{\mathcal G} \DeclareMathOperator{\lastgen}{\mathcal{Base}} \DeclareMathOperator{\nextgen}{\mathcal{N\!extB}} \DeclareMathOperator{\accpt}{\mathcal{Accpt}} \DeclareMathOperator{\Mfunc}{\mathcal{Mterm}} \]
In this note, I’m trying to see whether it is possible to eliminate the restricted formulas from the characterization of \(\chang^\sharp\) in[1]. As defined in that paper, the class \(I\) is easily definable as the domain of one of the terms in the language, namely the term which, for a \(\lambda=\kappa_\nu\in I\) and \(\beta\in(\kappa^{+\omega_1})^M\), gives the generator \(i_\nu(\beta)\) for \(i_\nu(E)\). The other terms used in the paper are the functions \(i_\ords(f)\) for \(f\colon\kappa\to M\) in \(M\) and the formulas for the inductive definition of \(\chang\); I think that there is no need to change them.[2]
My scenario for reforming the terms assumes that the lower bound, as obtained in the main Theorem 1.4(1) of[1] by using Gitik’s argument for the cofinality \(\omega\) case, is the actual upper (and hence exact) bound. In this case I should be able to take the terms from the full iteration from \(M\) to the core model of \(\chang\). To be specific, I’m assuming that \(K(\reals)^\chang\) is the initial segment below \(\ords\) of an iterate of \(M\), and that \(M\) is the minimal \(K(\reals)\)-mouse such that \(M\) has a final extender \(E\) of length \((\kappa^{\omega_1})^M\).[3] I conjecture that this assumption is true, but don’t have a proof.
I will write \(i_\nu\colon M\to M_\nu=\ult_\nu(M,E)\) for the iteration defined in[1]. I will write \(k_\xi\colon M=\bar M_0\to \bar M_\xi\) for the full ultrapower, i.e., the iteration so that \(K(\reals)^\chang=\bar M_\ords\cut\ords\).
I will write \(\kappa_\nu=i_\nu(\kappa)\), as in the paper, and \(\bar\kappa_\nu\) for the least ordinal \(k_\xi(\kappa)\ge\kappa_\nu\) such that \(k_{\xi,\xi+1}\) is the ultrapower by \(k_\xi(E)\). Equivalently, if we write \(\bar k\colon M_\ords\to \bar M_\ords\) for the map such that \(k_\ords=\bar k\circ i_\ords\), then \(\bar \kappa_\nu=\bar k(\kappa_\nu)\). I will write \(F_\nu\) for the extender such that \(\bar M_{\nu+1}=\ult(]bar M_\nu, F_\nu)\).
Now a few observations:
In \(K(\reals)^\chang\), we have \(o(\kappa_\nu)=0\) except when \(\cof(\nu)=\omega\), in which case \(\kappa_\nu=\bar\kappa_\nu\). The “except when” clause follows from Gitik’s reconstruction construction for defining extenders from their countable threads. The forward direction remains to be proved.
\(\lambda=k_{\lambda}(\kappa)\) for all \(\nu\) whenever \(\lambda=\kappa_\nu\) or \(\lambda=\bar\kappa_\nu\).
For all \(\nu\), \(\kappa_{\nu+1}=k_{\bar\kappa_\nu,\bar\kappa_{\nu}+1}(\bar\kappa_{\nu})=k_{\bar\kappa_\nu,\kappa_{\nu+1}}(\bar\kappa_\nu)\). The iteration \(k_{\bar\kappa_\nu+1,\kappa_{\nu+1}}\) is dealing with extenders on cardinals between \(\bar\kappa_{\nu}\) and \(\kappa_{\nu+1}\).
The terms include the following
\(\nextgen_{f,g} (\alpha,\gamma)\), for \(\alpha \lt\gamma\) and \(f,g\in M\), denotes the least ordinal \(\beta\gt\gamma\) such that for some \(\nu\), \(F_\nu=k_{\nu}(f)(\alpha)\) and \(\beta=(k_\nu(g)(\alpha))\). Thus, in \(\bar M_\omega\), \(\beta\) is an indiscernible for \(k_\ords(f)(\alpha)\) belonging to the extender \(k_\nu(f)(\alpha)\).[4]
\(\lastgen(\gamma)\) denotes the largest \(\lambda\lt\gamma\) (if it exists) such that for some \(\nu\lt\nu'\), \(\lambda=\crit(k_{\nu,\nu+1})\) and \(\gamma=k_{\nu,\nu'}(\lambda)=\crit(k_{\nu',\ords})\).
\(\accpt_{f}(\xi, \gamma)\) is the least ordinal \(\lambda=\crit(k_{\nu,\ords})>\gamma\ge\xi\) such that, writing \(F=k_\nu(f)(\xi)\), for each \(F'\triangleleft F\) in \(\bar M_\nu\) there are cofinally many \(\nu'\lt\nu\) such that \(F'=k_{\nu',\nu}(F_{\nu'})\), or, as a special case, \(k_{\nu}(f)(\xi)=\lambda\) and every extender \(F'\) on \(\lambda\) in \bar M_\nu$ has cofinally many such \(\nu'\).
\(\Mfunc_{f}(\xi)=k_{\ords}(\xi)\) is in[1], but here I use \(k\) instead of \(i\) to define it.
The term \(\lastgen(\gamma)\) is used to obtain \(\bar\kappa_\nu=\lastgen(\kappa_{\nu+1})\) from \(\kappa_{\nu+1}\). Note that its domain is the set of critical points of the the iteration \(k\) which are not measurable in \(\bar M_\ords\) and are not limit points of the set of critical points of \(k\).
The term \(\accpt_f(\xi,\gamma)\) gives accumulation points. Note that \(\accpt_f(\xi,\gamma)\) is not measurable in \(K(\reals)^\chang\), but \(k_{\nu}(f)(\xi)\) would have been a member of \(K(\reals)^\chang\) if \(\lambda\) had cofinality \(\omega\).][5]
”Proof” As in [1] it is sufficient to verify that every generator is so definable. Each \(\kappa_\nu\in I\) of uncountable cofinality other than \(\kappa_0\) is accounted for, as is \(\bar\kappa_{\nu}=\lastgen(\kappa_{\nu+1})\) for all ordinals \(\nu\). For \(\cof(\nu)=\omega\), then, \(\kappa_\nu=\bar\kappa_\nu\) is also accounted for.
The ordinal \(\kappa_0=\kappa\) is a special case, since it is not included in \(I\). It is equal to \(\nextgen_{U,\kappa}(0,0)\) where \(U\) is the order \(0\) measure on \(\kappa\).[10]
To conclude, we will deal with ordinals \(\alpha\) in the interval \((\bar\kappa_\nu,\bar\kappa_{\nu+1})\). By taking \(\bar\kappa_{-1}=0\) we can deal with the interval \([0, \bar\kappa_0)\) at the same time. The argument is by induction on \(\alpha\): Suppose \(\alpha\) is a generator at \(\lambda=\crit(k_{\nu,\nu+1})\), i.e., \(\alpha\in\supp(F_\nu)\). Let \(F_\nu=k_\nu(f)(\xi)\) and \(\alpha=k_\nu(g)(\xi)\), with \(f,g\in M\) and \(\xi\lt\lambda\).
If \(\alpha>\lambda\) then \(\alpha=\nextgen_{f,g}(\xi,\lambda)\), so we can assume \(\alpha=\lambda\). If there are only boundedly many \(\nu'\lt\nu\) such that \(F_{\nu'}=k_{\nu'}(f)(\xi)\) then \(\lambda=\nextgen_{f,g}(\xi,\lambda')\) for sufficiently large \(\lambda'\lt\lamda\), where \(g(\xi)=\crit(f(\xi))\).
If \(\alpha\) is not as in the last paragraph, then we must have \(\cof(\nu)>\omega\): It cannot be a successor, since otherwise \(\crit(k_{\nu-1,\nu})\) would certainly be such a bound, and if \(\cof(\nu)=\omega\) then \(F_\nu=k_\nu(f)(\xi)\) would be in \(\changcore\) by Gitik’s argument.
Now \(F_\nu\) is the order \(0\) measure, which is not help, so let \(F\) be the least extender on \(\lambda\) in \(\bar M_\nu\) such that there are only boundedly many \(\nu'\lt\nu\) such that \(F=k_{\nu',\nu}(F_\nu')\). Let \(F=k_\nu(f)(\xi)\), or if \(F\) does not exist then let \(k_\nu(f)(\xi)=\lambda\). Then there is \(\gamma\lt\lambda\) such that \(\lambda=\accpt_f(\xi,\gamma)\). Otherwise for any \(\gamma\lt\lambda\), if we define a sequence by \(\gamma_0=\gamma\) and \(\gamma_{n+1}=\accpt_f(\xi,\gamma_n)\) then \(\gamma_\omega\lt\lambda\) and, if \(F\) exists, then setting \(\gamma_\omega=\crit(k_{\nu',\nu'+1})\), we have \(k_{\nu'}(f)(\xi)\in \changcore\) by Gitik’s argument. This contradicts the choice of \(F\). If \(F\) does not exist, then every extender on \(\gamma_\omega\) is in \(\changcore\), so that \(\crit(k_{\nu',\ords})\gt\gamma_\omega\). \(\qed\)
Theorem. If \(\phi\) is any restricted formula, and \(B\) and \(B'\) are equivalent suitable sequences, then \(\chang\models\phi(B)\iff \phi(B')\).
Possible proof. Do the forcing over \(\bar M_\ords\) instead of \(M_\ords\). This means it has to be changed so the forcing adds indiscernibles for all measurable cardinals in \(\bar M_B\), not just for \(\ords\), but the iteration gives all these indiscernibles so that should work. Now why does this eliminate the problem with gaps I had in[1]? First, \(\kappa_0\) is now definable, so the case I worked out explicitly goes away. Now the problem in general was that I was trying to stuff indiscernibles from a longer gap into the interval \((\kappa_\nu,\kappa_{\nu+1})\). Since the iteration didn’t give indiscernibles in this gap, I had to use up a non-direct extension to provide indiscernibles to fill in. However, I can’t do that for infinitely many gaps because I don’t have any Prikry generic sequences to use. In this case the relevant gap is \((\bar\kappa_\nu,\bar\kappa_{\nu+1})\). Since \(\bar\kappa_\nu\) is the result of iterating to give indiscernibles for all its measures, I have the required Prikry generic sequences. \(\qed\)
Mitchell, W. The sharp of the Chang model is small, http://dracontium.org/wp-content/uploads/publications/changmodel-paper.pdf (February 9, 2014).
Although, as I mention below, there is some reason tho think that fine structural consideration could usefully be brought into play for the first of these. ↩
I don’t think I’m using here the assumption that \(M\) is minimal, which means that \(M\) is an active mouse, with \(E\) the final extender. However I think that using this assumption may further improve the set of terms. ↩
I will also write \(\nextgen\) with subscripts which are members of \(M\) and are not functions; it should be understood that in this case the constant function or functions are meant. ↩
But the converse is not true: ↩
Of course the quantification is not allowed over the subscripts of \(\nextgen\): different subscripts yield different terms. ↩
This doesn’t necessarily mean that we are dependent on the assumption that the sharp is the minimal mouse, as we could explicitly disallow \(E\). ↩
Or so I hope. ↩
Note that this is not actually a proof, as I assume that an extender is in \(\changcore\) only if it is in there by Gitik’s argument. I have no doubt that this is true, but I don’t have a proof, and I actually expect the proof to come as a consequence of the theory of \(\changsharp\), and hence not to be available at this point. ↩
Here I am using \(U\) and \(\kappa\) to mean the respective constant functions. ↩