- E. Carmody, V. Gitman, and M. Habič, “Mitchell order for Ramsey and Ramsey-like cardinals.” (Manuscript under review)
`@ARTICLE{CarmodyGitmanHabic:MitchellOrder, AUTHOR= {Carmody, Erin and Gitman, Victoria and Habi\v{c}, Miha}, TITLE= {Mitchell order for {R}amsey and {R}amsey-like cardinals}, Note ={Manuscript under review}, PDF={http://boolesrings.org/victoriagitman/files/2016/09/MitchellOrderRamseyCardsArxiv.pdf}, }`

Mitchell introduced the *Mitchell order* on normal measures on a measurable cardinal

$\kappa$ in [1], where he defined that $U\vartriangleleft W$ for two normal measures $U$ and $W$ on $\kappa$ whenever $U\in{\rm Ult}(V,W)$, the ultrapower of the universe $V$ by $W$. Since $\vartriangleleft$ is easily seen to be well-founded, we can define the ordinal rank $o(U)$ of a normal measure and define $o(\kappa)$, the Mitchell rank of $\kappa$, to be the supremum of $o(U)$ over all normal measures $U$ on $\kappa$. The Mitchell rank of $\kappa$ tells us to what extent measurability is reflected below $\kappa$. Mitchell used the Mitchell order to study coherent sequences of normal measures, which allowed him to generalize Kunen’s $L[U]$ construction to canonical inner models with many measures [1]. The Mitchell rank of a measurable cardinal has also proved instrumental in calibrating consistency strength of set theoretic assertions. Gitik showed, for instance, that the consistency strength of a measurable cardinal at which the ${\rm GCH}$ fails is a measurable cardinal $\kappa$ with $o(\kappa)=\kappa^{++}$ ([2]). The notion of Mitchell order generalizes to extenders, where it has played a role in constructions of core models.

In this article, we introduce a Mitchell-like order for Ramsey and Ramsey-like cardinals.

Although we tend to associate smaller large cardinals $\kappa$ with combinatorial definitions, many of them have characterizations in terms of existence of elementary embeddings. The domains of these embeddings are *weak $\kappa$-models*, transitive models of ${\rm ZFC}^-$ of size $\kappa$ and height above $\kappa$, or some stronger version of these such as *$\kappa$-models*, which are additionally closed under <$\kappa$-sequences. Usually, the embeddings are ultrapower or extender embeddings by mini-measures or mini-extenders that apply only to the $\kappa$-sized domain of the embedding. If $M$ is a transitive model of ${\rm ZFC}^-$ and $\kappa$ is a cardinal in $M$, then we call $U\subseteq \mathcal P(\kappa)^M$ an $M$-ultrafilter if it is an ultrafilter on $\mathcal P(\kappa)^M$ that is normal for sequences in $M$. In most interesting cases, an $M$-ultrafilter is external to $M$, but we can still form the ultrapower by using functions on $\kappa$ that are elements of $M$. A prototypical characterization of a smaller large cardinal $\kappa$ states that every $A \subseteq\kappa$ is an element of a weak $\kappa$-model $M$ (with additional requirements) for which there is an $M$-ultrafilter on $\kappa$ (with additional requirements). The additional requirements on $M$ and the $M$-ultrafilter are dictated by the large cardinal property. The simplest such characterization belongs to weakly compact cardinals, where there is the minimal requirement on the $M$-ultrafilter, namely that the ultrapower of $M$ is well-founded.

Given a large-cardinal property $\mathscr P$ with an embedding characterization as discussed above (such as weak compactness, Ramseyness, etc.), let us say that an $M$-ultrafilter is a *$\mathscr P$ measure* if it, together with $M$, witnesses $\mathscr P$ and that a $\mathscr P$-measure is *$A$-good* for some $A\subseteq\kappa$ if $A\in M$. (For technical reasons we also require that $V_\kappa\in M$. Note that if $M$ is a $\kappa$-model, then $V_\kappa\in M$ follows.) To avoid having to specify which model $M$ we associate to a given $\mathscr P$-measure $U$, we will always associate it with the unique minimal model $M_U$, namely the $H_{\kappa^+}$ of any such model $M$. Let us say that a collection $\mathcal U$ of $\mathscr P$-measures is a *witness* for $\mathscr P$ if for every $A\subseteq\kappa$, it contains some $A$-good $\mathscr P$-measure. So while a normal measure on $\kappa$ witnesses the measurability of $\kappa$, a witness collection of $\mathscr P$-measures is precisely what witnesses $\mathscr P$ for one of these smaller large cardinals. This suggests that a reasonable Mitchell-like order should not be comparing the tiny $\mathscr P$-measures, but rather witness collections of $\mathscr P$-measures in a way that ensures that the corresponding rank $o_{\mathscr P}(\kappa)$ of $\kappa$ measures the extent to which $\mathscr P$ is reflected below $\kappa$. We will call this order the M-order in honor of Mitchell.

**Definition**: (M-order) Suppose that $\kappa$ has a large-cardinal property $\mathscr P$ with an embedding characterization. Given two witness collections $\mathcal U$ and $\mathcal W$ of $\mathscr P$-measures, we define that $\mathcal U\vartriangleleft \mathcal W$ if

- For every $W\in \mathcal W$ and $A\subseteq\kappa$ in the ultrapower $N_W$ of $M_W$ by $W$, there is an $A$-good $U\in\mathcal U\cap N_W$ such that $N_W$ agrees that $U$ is an $A$-good $\mathscr P$-measure on $\kappa$.
- $\mathcal U\subseteq \bigcup_{W\in\mathcal W}N_W$.

The key part of the definition is clause (1). It states that the elements of $\mathcal U$ witness that $\kappa$ retains the property $\mathscr P$ in the ultrapowers by the elements of $\mathcal W$. It is tempting to say that $\mathcal U$ itself should witness $\mathscr P$ in those ultrapowers, but note that $\mathcal U$ is too large to be an element of a weak $\kappa$-model. Clause (2) ensures that the collections under consideration do not contain superfluous $\mathscr P$-measures.

Mitchell proved that Ramsey cardinals have an embedding characterization and I used generalizations of it to define the Ramsey-like cardinals: *$\alpha$-iterable*, *strongly Ramsey*, and *super Ramsey cardinals* ([3] and [4]). Thus, a *Ramsey measure* $U$ is a weakly amenable $\omega_1$-intersecting $M_U$-ultrafilter, a *strongly Ramsey measure* $U$ is a weakly amenable $M_U$-ultrafilter, where $M_U$ is a $\kappa$-model, and a *super Ramsey measure* is a weakly amenable $M_U$-ultrafilter where $M_U$ is a $\kappa$-model elementary in $H_{\kappa^+}$.

We will show that the M-order and the corresponding notion of M-rank share all the desirable features of the Mitchell order on normal measures on a measurable cardinal. For example, the order is transitive and well-founded. Note that since an ultrapower of a weak $\kappa$-model has size at most $\kappa$, the M-rank of a large cardinal $\kappa$ can be at most $\kappa^+$, in contrast with the upper bound of $(2^\kappa)^+$ in the case of the usual Mitchell rank for a measurable cardinal.

**Theorem**: Suppose $\mathcal U$ is a witness collection of $\mathscr P$-measures, where $\mathscr P$ is Ramsey or Ramsey-like, such that $o_{\mathscr P}(\mathcal U)\geq\alpha$. Then:

- For every $U\in \mathcal U$, the ultrapower $N_U$ of $M_U$ by $U$ satisfies $o_{\mathscr P}(\kappa)\geq\alpha$.
- There is a good collection $\mathcal W$ with $o_{\mathscr P}(\mathcal W)=\alpha$ such that

$N_W\models o_{\mathscr P}(\kappa)=\alpha$ for all $W\in\mathcal W$.

We should not expect an analogue of (1) above with equality because we are now dealing with collections of measures instead of a single measure and so (2) above is the best possible result.

**Theorem**: Any strongly Ramsey cardinal $\kappa$ has the maximum Ramsey M-rank $o_{\text{Ram}}(\kappa)=\kappa^+$, any super Ramsey cardinal $\kappa$ has the maximum strongly Ramsey M-rank $o_{\text{stRam}}(\kappa)=\kappa^+$, and any measurable cardinal $\kappa$ has the maximum super Ramsey M-rank $o_{\text{supRam}}(\kappa)=\kappa^+$.

We will show that the new Mitchell order behaves robustly with respect to forcing constructions. We show that extensions with cover and approximation properties cannot create new Ramsey or Ramsey-like cardinals or increase their M-rank. Hamkins showed, in [5], that most large cardinals cannot be created in extensions with cover and approximation properties and we provide several modifications of his techniques to the embeddings characterizing Ramsey and Ramsey-like cardinals. This result is of independent interest since it was not previously known whether Ramsey

cardinals can be created in extensions with cover and approximation properties.

**Theorem**: If $V\subseteq V’$ has the $\delta$-cover and $\delta$-approximation properties for some regular cardinal $\delta<\kappa$ of $V'$, then $o_{\mathscr P}^V(\kappa)\geq o_{\mathscr P}^{V'}(\kappa)$, where $\mathscr P$ is strongly or super Ramsey, and if we additionally assume that $V^\omega\subseteq V$ in $V'$, then $o_{\text{Ram}}^V(\kappa)\geq o_{\text{Ram}}^{V'}(\kappa)$.

Using the results about extensions with approximation and cover properties together with new techniques recently developed in Carmody’s dissertation [6] about softly killing degrees of large cardinals with forcing, we show how to softly kill the M-rank of a Ramsey or Ramsey-like cardinal by forcing.

**Theorem**: If $\kappa$ has $o_{\mathscr P}(\kappa)=\alpha$, where $\mathscr P$ is Ramsey or Ramsey-like and $\beta<\alpha$, then there is a cofinality-preserving forcing extension in
which $o_{\mathscr P}(\kappa)=\beta$.

Although the general framework of the M-order we have sketched here applies to many smallish large cardinals, we focus in this paper on its application to Ramsey, strongly Ramsey and super Ramsey cardinals. Other instances of it definitely warrant further research.

[Bibtex]

```
@article {mitchell:mitchellRank,
AUTHOR = {Mitchell, William J.},
TITLE = {Sets constructible from sequences of ultrafilters},
JOURNAL = {J. Symbolic Logic},
FJOURNAL = {The Journal of Symbolic Logic},
VOLUME = {39},
YEAR = {1974},
PAGES = {57--66},
ISSN = {0022-4812},
MRCLASS = {02K30},
MRNUMBER = {0344123 (49 \#8863)},
MRREVIEWER = {Keith Devlin},
}
```

[Bibtex]

```
@article {gitik:measurablenotCH,
AUTHOR = {Gitik, Moti},
TITLE = {On measurable cardinals violating the continuum hypothesis},
JOURNAL = {Ann. Pure Appl. Logic},
FJOURNAL = {Annals of Pure and Applied Logic},
VOLUME = {63},
YEAR = {1993},
NUMBER = {3},
PAGES = {227--240},
ISSN = {0168-0072},
CODEN = {APALD7},
MRCLASS = {03E55 (03E50)},
MRNUMBER = {1237232 (94m:03086)},
MRREVIEWER = {James Baumgartner},
DOI = {10.1016/0168-0072(93)90149-8},
URL = {http://dx.doi.org/10.1016/0168-0072(93)90149-8},
}
```

[Bibtex]

```
@article {mitchell:ramsey,
AUTHOR = {Mitchell, William},
TITLE = {Ramsey cardinals and constructibility},
JOURNAL = {J. Symbolic Logic},
FJOURNAL = {The Journal of Symbolic Logic},
VOLUME = {44},
YEAR = {1979},
NUMBER = {2},
PAGES = {260--266},
ISSN = {0022-4812},
CODEN = {JSYLA6},
MRCLASS = {03E55},
MRNUMBER = {534574 (80f:03063)},
MRREVIEWER = {Thomas J. Jech},
DOI = {10.2307/2273732},
URL = {http://dx.doi.org.ezproxy.gc.cuny.edu/10.2307/2273732},
}
```

[Bibtex]

```
@ARTICLE {gitman:ramsey,
AUTHOR = {Victoria Gitman},
TITLE = {{R}amsey-like cardinals},
JOURNAL = {The Journal of Symbolic Logic},
VOLUME = {76},
YEAR = {2011},
NUMBER = {2},
PAGES = {519-540},
EPRINT={0801.4723},
PDF={http://boolesrings.org/victoriagitman/files/2011/08/ramseylikecardinals.pdf},
ISSN = {0022-4812},
CODEN = {JSYLA6},
MRCLASS = {03E55},
MRNUMBER = {2830415 (2012e:03110)},
MRREVIEWER = {Bernhard A. K{\"o}nig},
DOI = {10.2178/jsl/1305810762},
URL = {http://dx.doi.org/10.2178/jsl/1305810762},
}
```

[Bibtex]

```
@article {hamkins:coverandapproximations,
AUTHOR = {Hamkins, Joel David},
TITLE = {Extensions with the approximation and cover properties have no
new large cardinals},
JOURNAL = {Fund. Math.},
FJOURNAL = {Fundamenta Mathematicae},
VOLUME = {180},
YEAR = {2003},
NUMBER = {3},
PAGES = {257--277},
ISSN = {0016-2736},
MRCLASS = {03E55 (03E40)},
MRNUMBER = {2063629 (2005m:03100)},
DOI = {10.4064/fm180-3-4},
URL = {http://dx.doi.org/10.4064/fm180-3-4},
}
```

[Bibtex]

```
@PHDTHESIS{carmody:thesis,
AUTHOR={Erin Carmody},
TITLE={Forcing to change large cardinal strength},
SCHOOL={CUNY Graduate Center},
YEAR ={2015}}
```