Ramsey-like cardinals

This is a talk at the Algebra and Logic Seminar, University of Denver, January 14, 2020.
Slides

Ramsey-like cardinals originated with the study of elementary embedding properties of smaller large cardinals. Measurable cardinals and stronger large cardinals $\kappa$ are defined by the existence of elementary embeddings $j:V\to M$ with critical point $\kappa$ from the universe into an inner model. Elementary embeddings provide a unifying framework in the study of these large cardinals and set theorists have developed a toolbox of techniques for working with elementary embeddings, particularly for showing indestructibility under forcing.

Smaller large cardinals such as weakly compact cardinals and Ramsey cardinals were originally defined in terms of combinatorial Ramsey-type properties, but it turns out that they also have elementary embedding characterizations involving elementary embeddings of set-sized models.

Elementary embeddings $j:V\to M$ are connected with the existence of certain ultrafilters. The ultrapower of $V$ by a non-principal countably complete ultrafilter is well-founded, and therefore produces a non-trivial elementary embedding $j:V\to M$, and such an embedding, if its critical point is $\kappa$, can be used to obtain a non-principal $\kappa$-complete ultrafilter $$U=\{A\subseteq\kappa\mid \kappa\in j(A)\}.$$ We will say that $U$ is generated by $\kappa$ via $j$. Thus, in particular, there is a non-trivial elementary embedding $j:V\to M$ if and only if there is a non-principal countably complete ultrafilter on some cardinal $\kappa$.

The ultrapower construction with a countably complete ultrafilter can be iterated through all the ordinals to produce ${\rm Ord}$-many well-founded iterated ultrapowers. Given such an ultrafilter $U$, the first step is the original ultrapower $j:V\to M$, the second step is the ultrapower by $j(U)$, and we proceed in this manner, using the image of the original ultrafilter at successor stages and direct limits at limit stages. The successor stages are well-founded because all ultrafilters are countably complete (by elementarity) from the perspective of the model in which the ultrapower is constructed, and the direct limits are well-founded by a theorem of Gaifman [1]. Thus, we say that countably complete ultrafilters are iterable.

Elementary embeddings of set-sized models of set theory are also connected with the existence of certain ultrafilters. So let's talk about these models and ultrafilters. We say that a weak $\kappa$-model is a transitive set model $M\models{\rm ZFC}^-$ of size $\kappa$ and height above $\kappa$. A weak $\kappa$-model $M$ is a $\kappa$-model if it is additionally closed under sequences of length less than $\kappa$, the maximum possible closure for a model of size $\kappa$. Natural weak $\kappa$-models arise as elementary substructures of $H_{\kappa^+}$. Smaller large cardinals $\kappa$ are characterized by the existence of elementary embeddings of weak $\kappa$-models or $\kappa$-models with critical point $\kappa$. Given a weak $\kappa$-model $M$, we say that $U\subseteq P^M(\kappa)$ is an $M$-ultrafilter if the structure $\langle M,\in,U\rangle$, that is $M$ together with a predicate for $U$, satisfies that $U$ is a normal ultrafilter on $\kappa$. What this means is that $U$ is an ultrafilter measuring $P^M(\kappa)$ that is closed under diagonal intersections of sequences from $M$. The set $U$ is allowed to be completely external to $M$, and while normality for sequences from $M$ implies $\kappa$-completeness for sequences from $M$, an $M$-ultrafilter $U$ may not even be countably complete because $M$, having no closure, can be missing many countable sequences. Note that, while the weak $\kappa$-model $M$ is required to satisfy ${\rm ZFC}^-$, replacement almost certainly fails in the structure $\langle M,\in,U\rangle$ once we add the $M$-ultrafilter. How do we know this? Read on.

In the context of $M$-ultrafilters, we will weaken the notion of countable completeness to say only that the intersection of countably many sets from the $M$-ultrafilter is non-empty. The two notions are equivalent for ultrafilters on $\kappa$, but for $M$-ultrafilters it is too strong to require that the intersection of countably many sets in the ultrafilter is itself in the ultrafilter because the intersection may not even be an element of $M$ and therefore not measured by the $M$-ultrafilter.

We can build the ultrapower of a weak $\kappa$-model $M$ by an $M$-ultrafilter using functions $f$ on $\kappa$ from $M$. While we have a complete characterization of when an ultrafilter on $\kappa$ has a well-founded ultrapower, namely countable completeness, the situation with $M$-ultrafilters is much messier. Countable completeness of the $M$-ultrafilter suffices for well-foundedness, but this condition, as we will see, is too strong. Let's call an $M$-ultrafilter good if it has a well-founded ultrapower. There is no known elegant characterization of good $M$-ultrafilters. The existence of good $M$-ultrafilters is equivalent to the existence of elementary embeddings $j:M\to N$ because the set $U$ generated by $\kappa$ via $j$ as above is a good $M$-ultrafilter.

Next, let's talk about iterability of $M$-ultrafilters. After a moment's thought one realizes that given a good $M$-ultrafilter $U$ we cannot even perform the second step of the iterated ultrapower construction because, since $U$ is not an element of $M$, we cannot take the image of it under the ultrapower embedding. To have any hope of being iterable, an $M$-ultrafilter first needs to be internal enough to $M$ to make it possible to define successor stage ultrafilters. An $M$-ultrafilter is said to be weakly amenable if for every set $S\in M$ with $|S|^M=\kappa$, $S\cap U\in M$. The condition of weak amenability suffices to define successor stage ultrafilters. It also has another nice characterization. We will say that an elementary embedding $j:M\to N$ with critical point $\kappa$ is $\kappa$-powerset preserving if $M$ and $N$ have the same subsets of $\kappa$. It is always the case that $P^M(\kappa)\subseteq P^N(\kappa)$, but the other inclusion need not hold. Indeed, if a good $M$-ultrafilter $U$ is weakly amenable, then the ultrapower embedding $j:M\to N$ is $\kappa$-powerset preserving and conversely an $M$-ultrafilter generated by $\kappa$ via a $\kappa$-powerset preserving embedding $j:M\to N$ is weakly amenable. We will see below that while weak amenability makes it possible to define the stages of the iterated ultrapower construction, it does not guarantee the well-foundeness even of the second step ultrapower.

We are finally ready to see some elementary embedding characterizations of smaller large cardinals. The simplest such characterization belongs to weakly compact cardinals. An inaccessible cardinal $\kappa$ is weakly compact if and only if every $A\subseteq\kappa$ is an element of a weak $\kappa$-model $M$ which has a good $M$-ultrafilter $U$. It turns out that this characterization can be strengthened in a number of ways. We can replace weak $\kappa$-model by $\kappa$-model, even by $\kappa$-model that is elementary in $H_{\kappa^+}$. Indeed, if $\kappa$ is weakly compact, then every weak $\kappa$-model has a good $M$-ultrafilter! Can we further strengthen the characterization by asking that all or at least some weak $\kappa$-models have weakly amenable good $M$-ultrafilters? The answer turns out to be no.

Let's make another definition. We will say that a weakly amenable $M$-ultrafilter is $\alpha$-good if it has $\alpha$-many well-founded iterated ultrapowers. Note that $1$-good equals weakly amenable plus good. Some observations are in order. By a theorem of Gaifman [1], only countable stages of the iteration matter. If an $M$-ultrafilter is $\omega_1$-good, then it is $\alpha$-good for every $\alpha$. By a theorem of Kunen [2], a countably complete $M$-ultrafilter is $\omega_1$-good. We will say that a cardinal $\kappa$ is $\alpha$-iterable if every $A\subseteq\kappa$ is an element of a weak $\kappa$-model $M$ which has an $\alpha$-good $M$-ultrafilter. Thus, we can rephrase the question about weakly compacts above as asking whether a weakly compact cardinal is 1-iterable. In fact, 1-iterable cardinals are limits of weakly compact cardinals, even limits of ineffable cardinals [3]. The $\alpha$-iterable cardinals for $\alpha\leq\omega_1$ form a hierarchy of strength where a $\beta$-iterable cardinal is a limit of $\alpha$-iterable cardinals for every $\alpha\lt\beta$. The $\alpha$-iterable cardinals for $\alpha$ countable are downward absolute to $L$, but $\omega_1$-iterable cardinals already imply $0^{\#}$ [4].

Next, up are Ramsey cardinals, which also have an unexpectedly nice elementary embedding characterization. A cardinal $\kappa$ is Ramsey if and only if every $A\subseteq\kappa$ is an element of a weak $\kappa$-model $M$ which has a weakly amenable countably complete $M$-ultrafilter [4]. Can we strengthen the Ramsey elementary embedding characterization by replacing weak $\kappa$-model with $\kappa$-model or even $\kappa$-model elementary in $H_{\kappa^+}$, or even ask that it holds for all weak $\kappa$-models? The answer is no.

We say that a cardinal $\kappa$ is strongly Ramsey if every $A\subseteq\kappa$ is an element of a $\kappa$-model $M$ which has a weakly amenable $M$-ultrafilter and a cardinal $\kappa$ is super Ramsey if we replace $\kappa$-model with $\kappa$-model elementary in $H_{\kappa^+}$. Strongly Ramsey cardinals are limits of Ramsey cardinals and super Ramsey cardinals are limits of strongly Ramsey cardinals. The assertion that every weak $\kappa$-model has a weakly amenable $M$-ultrafilter is inconsistent! [3] Broadly speaking, the reason for this contrast in relationships with the introduction of weak amenability is that having the same subsets of $\kappa$ creates a great deal of reflection between $M$ and its ultrapower $N$.

Are there other natural large cardinal notions between super Ramsey and measurable cardinals? Super Ramsey cardinals require embeddings on $\kappa$-model elementary in $H_{\kappa^+}$ because such models look like an initial segment of the universe. A natural extension of the idea of having embeddings on models reflecting an initial segment of the universe is to require that the weak $\kappa$-model is elementary in some larger $H_\theta$. But this is impossible because large $H_\theta$ cannot have transitive elementary submodels of size $\kappa$. So let's define that an imperfect weak $\kappa$-model is a weak $\kappa$-model $M$ where we replace transitivity with the weaker requirement that $\kappa+1\subseteq M$, and similarly define imperfect $\kappa$-models. Now we can consider the large cardinal notion $\kappa$-Ramsey, introduced by Holy and Schlicht, where for every set $A$ and arbitrarily large cardinals $\theta$, $A$ is an element of an imperfect $\kappa$-model $M\prec H_\theta$ which has a weakly amenable $M$-ultrafilter. $\kappa$-Ramsey cardinals have a natural game characterization. The game is played by the challenger and the judge, with the challenger playing an increasing sequence of imperfect $\kappa$-models $M\prec H_\theta$ and the judge responding with nested $M$-ultrafilters. For the judge to win a game of length $\alpha$ she has to be able to play for $\alpha$-many steps, with the challenger winning otherwise. The reason the large cardinal is called $\kappa$-Ramsey is that it is characterized by the challenger failing to have a winning strategy in this game of length $\kappa$. $\kappa$-Ramsey cardinals are limits of super Ramsey cardinals. [5]

Are there natural large cardinal notions between $\kappa$-Ramsey cardinals and measurable cardinals? Yes! Remember I remarked earlier that the structure $\langle M,\in,U\rangle$, where $M$ is a weak $\kappa$-model and $U$ is an $M$-ultrafilter almost certainly fails to satisfy replacement? Well, let's define that a cardinal $\kappa$ is weakly baby measurable if every $A\subseteq\kappa$ is an element of a weak $\kappa$-model $M$ which has a good $M$-ultrafilter such that $\langle M,\in, U\rangle\models{\rm ZFC}^-$. Similar large cardinal notions were introduced and investigated recently by McKenzie and Bovykin [6]. We recently realized with Philipp Schlicht that if $\kappa$ is weakly baby measurable, then $V_\kappa$ is a model of proper class many cardinals $\alpha$ that are $\alpha$-Ramsey. Baby measurable cardinals (McKenzie and Bovykin call them simultaneously baby measurable) would be cardinals where we replace weak $\kappa$-model with $\kappa$-model. These are even stronger, but still some ways below a measurable!

References

1. H. Gaifman, “Elementary embeddings of models of set-theory and certain subtheories,” in Axiomatic set theory (Proc. Sympos. Pure Math., Vol. XIII, Part II, Univ. California, Los Angeles, Calif., 1967), Providence R.I.: Amer. Math. Soc., 1974, pp. 33–101.
2. K. Kunen, “Some applications of iterated ultrapowers in set theory,” Ann. Math. Logic, vol. 1, pp. 179–227, 1970.
3. V. Gitman, “Ramsey-like cardinals,” The Journal of Symbolic Logic, vol. 76, no. 2, pp. 519–540, 2011.
4. V. Gitman and P. D. Welch, “Ramsey-like cardinals II,” The Journal of Symbolic Logic, vol. 76, no. 2, pp. 541–560, 2011.
5. P. Holy and P. Schlicht, “A hierarchy of Ramsey-like cardinals,” Fund. Math., vol. 242, no. 1, pp. 49–74, 2018.
6. A. Bovykin and Z. McKenzie, “Ramsey-like cardinals that characterize the exact consistency strength of NFUM,” Manuscript.