
## 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]. 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 $$\kappa_{\nu+1}$$.

### 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], 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]. 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] 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] 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. $$\qed$$

1. Mitchell, W. The sharp of the Chang model is small, http://dracontium.org/wp-content/uploads/publications/changmodel-paper.pdf (February 9, 2014).

2. 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.  ↩

3. 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.  ↩

4. 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.  ↩

5. But the converse is not true:  ↩

6. Of course the quantification is not allowed over the subscripts of $$\nextgen$$: different subscripts yield different terms.  ↩

7. This doesn’t necessarily mean that we are dependent on the assumption that the sharp is the minimal mouse, as we could explicitly disallow $$E$$.  ↩

8. Or so I hope.  ↩

9. 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.  ↩

10. Here I am using $$U$$ and $$\kappa$$ to mean the respective constant functions.  ↩