\[$\newcommand\seq[1]{\langle#1\rangle} \newcommand{\chang}{\mathbb C} \newcommand\changsharp{\chang^\sharp} \newcommand\changcore{K(\reals)^\chang} \newcommand\reals{\mathcal R} \newcommand\ords{\Omega} \DeclareMathOperator\ult{Ult} \DeclareMathOperator\supp{supp} \DeclareMathOperator\cof{cf} \newcommand\cut{\vert} \newcommand\ecut{\vert} \DeclareMathOperator\crit{crit} \newcommand\set[1]{\{#1\}} \newcommand\belongsto{\sim} \DeclareMathOperator{\genterm}{\mathcal G} \DeclareMathOperator{\lastgen}{\mathcal{Base}} \DeclareMathOperator{\nextgen}{\mathcal{N\!extB}} \DeclareMathOperator{\accpt}{\mathcal{Accpt}} \DeclareMathOperator{\Mfunc}{\mathcal{Mterm}} \]

An Attempt at Better Terms for \(\changsharp\)

Introduction

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]changpaper. 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]changpaper 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]changpaper. 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 \(\kappa_{\nu+1}\).

Definition of the terms

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]changpaper, 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]

• We set \(I=\set{\kappa_\nu\mid\nu\in\ords\setminus\{\emptyset\}}\).
• A suitable sequence is a countable increasing subsequence \(\seq{\kappa_{\nu_\xi}\mid\xi\in\alpha}\) of \(I\) such that \(\cof(\xi)\not=\omega\) for each \(\xi\in\alpha\).
• Two suitable sequences \(\seq{\kappa_{\nu_\xi}\mid\xi\in\alpha}\) and \(\seq{\kappa_{\nu'_\xi}\mid\xi\in\alpha'}\) are equivalent if \(\alpha=\alpha'\) and for each \(\xi\in\alpha\), \(\nu_\xi\) is a limit ordinal if and only if \(\nu_{\xi'}\) is.

Remarks

• Notice that \(\accpt_f(\xi,\gamma)\) is actually definable from \(\nextgen_{f,\kappa_\nu'}(\cdot,\cdot)\).
• If I am understanding things properly, then this schema is sufficiently homogeneous that equivalent suitable sequences will satisfy the same formulas, where the terms are allowed to be used freely in the formulas.[6]
• This seems to eliminate the significance which gaps had in [1]changpaper. However the significance of limit points of \(I\) is strengthened: a member \(\lambda\) if \(I\) is a limit point of \(I\) if and only if \(\lastgen(\lambda)\) is not defined. In particular the allowance for finitely many exceptions is eliminated. That’s too bad: I liked it.
• This contrasts with what Woodin has announced, that for his sharp, any two suitable sequences would satisfy the same formulas. For the model \(\chang^{+}\), which he is primarily interested in, I can see that this might make sense: The inclusion of the non stationary ideals on \(P_\omega(\lambda)\) for an ordinal \(\lambda\in I\) of uncountable cofinality may cause \(\lambda\) to be a large cardinal in \(K(\reals)^{\chang^+}\). I don’t understand how this could apply to the Chang model itself.
• The class \(I\) was chosen partly so that it would be closed and unbounded and so that it would not depend on the universe \(V\) (so long as the set of reals is unchanged). A possible alternative would be to take the set of ordinals \(\bar\kappa_\nu\); however apart from the considerations just mentioned, I don’t see any definition of terms which would safely yield either of \(\kappa_\nu\) or \(\bar\kappa_\nu\) as a function of the other.
• \(\kappa_0\) is omitted from \(I\) because it is definable: \(\kappa_0=\nextgen{U,\kappa}(0,0)\).
• The terms from clause 2 in the definition 3.7 of the terms in [1]changpaper are a special case of \(\nextgen_{f,g}\), obtained by letting \(f\) and \(g\) each be constant functions, with the constant value of \(f\) an initial segment of \(E\). It is important to understand that the case that \(f\) is constantly equal to \(E\) is not allowed under this definition, since the assumption of the minimality of \(M\) implies that \(M\) is active and thus \(E\) is a predicate on \(M\), rather than a member of \(M\).[7] This means that no countable set of terms is sufficient to define the class \(I\)[8].

“Theorems”

Proposition: The set of terms is sufficient: every member of \(\chang\) is denoted by some term, using countable sequences from \(I\) as parameters.[9]

”Proof” As in [1]changpaper 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]changpaper? 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. \(\qed\)