Completely ineffable cardinals

I have to admit that I always thought completely ineffable cardinals to be a rather boring technical notion. But I recently came across several equivalent characterizations of complete ineffability that are quite surprising and natural. I also think I stumbled upon an interesting consistency strength implication from completely ineffable cardinals.

Completely ineffable cardinals sit atop the ineffability hierarchy. A cardinal $\kappa$ is ineffable if every coloring $f:[\kappa]^2\to 2$, coloring pairs of ordinals in 2 colors, has a stationary homogeneous subset. A cardinal $\kappa$ is $n$-ineffable if every coloring $f:[\kappa]^n\to 2$ has a stationary homogeneous subset. A cardinal $\kappa$ is totally ineffable if it is $n$-ineffable for every $n$. It is not easy to remember the definition of completely ineffable cardinals. A cardinal $\kappa$ is completely ineffable if there is a collection $\mathcal R$ of stationary subsets of $\kappa$ closed under supersets such that every coloring $f:[A]^2\to 2$ for $A\in\mathcal R$ has a homogeneous subset in $\mathcal R$. So a completely ineffable $\kappa$ has a set of stationary subsets of $\kappa$ that is closed under the operation of obtaining homogeneous sets for colorings. Completely ineffable cardinals are limits of totally ineffable cardinals. They are also totally indescribable. Completely ineffable cardinals lie relatively low in the large cardinal hierarchy. They are much weaker than Ramsey or even $\omega$-Erdos cardinals, in particular, they can exist in $L$.

I call a transitive model $M\models{\rm ZFC}^-$ (${\rm ZFC}$ with the powerset axiom removed) a weak $\kappa$-model if it has size $\kappa$ and $\kappa\in M$. Many of the smaller large cardinals, those below a measurable cardinal, have characterizations in terms of existence of elementary embeddings of weak $\kappa$-models. For example, $\kappa$ is weakly compact if (it is inaccessible and) every $A\subseteq\kappa$ is an element of a weak $\kappa$-model $M$ for which there is an elementary embedding $j:M\to N$ with critical point $\kappa$ and $N$ transitive. Equivalently, every $A\subseteq\kappa$ is an element of a weak $\kappa$-model $M$ for which there is an $M$-ultrafilter $U$ on $\kappa$ with a well-founded ultrapower. An $M$-ultrafilter on $\kappa$ is an ultrafilter on $P(\kappa)^M$, not necessarily from $M$, that is normal for sequences from $M$. A well-founded ultrapower by an $M$-ultrafilter yields an elementary embedding $j:M\to N$ with critical point $\kappa$ and conversely given an elementary embedding $j:M\to N$ with critical point $\kappa$ and $N$ transitive, we have that $$U=\{A\in M\mid A\subseteq\kappa\text{ and }\kappa\in j(A)\}$$ is an $M$-ultrafilter with a well-founded ultrapower (the ultrapower embeds into $N$).

Although we can take the ultrapower by an $M$-ultrafilter, we cannot necessarily iterate the ultrapower construction. If $U$ is not in $M$, how do we define $j(U)$? It turns out that you can do this if the $M$-ultrafilter has the additional property of being weakly amenable, that is being partially internal to $M$. An $M$-ultrafilter $U$ on $\kappa$ is weakly amenable if for every $A\in M$ which $M$ thinks has size $\kappa$, $A\cap U\in M$. A well-founded ultrapower $j:M\to N$ by a weakly amenable $M$-ultrafilter $U$ on $\kappa$ has the property that $P(\kappa)^M=P(\kappa)^N$ and if $j:M\to N$ is any embedding with critical point $\kappa$ such that $P(\kappa)^M=P(\kappa)^N$, then the derived $M$-ultrafilter $U$ (defined as above) is weakly amenable. In my dissertation, I introduced the $\alpha$-iterability hierarchy by exploring iterability properties of weakly amenable $M$-ultrafilters. A cardinal $\kappa$ is 1-iterable if every $A\subseteq\kappa$ is an element of a weak $\kappa$-model $M$ for which there is a weakly amenable $M$-ultrafilter with a well-founded ultrapower. More generally, a cardinal $\kappa$ is $\alpha$-iterable for $1\leq\alpha\leq\omega_1$ if every $A\subseteq\kappa$ is an element of a weak $\kappa$-model $M$ for which there is a weakly amenable $M$-ultrafilter with $\alpha$-many well-founded iterated ultrapowers (once you have $\omega_1$-many well-founded iterated ultrapowers, then all the rest are well-founded as well).

The point here is that a $1$-iterable cardinal $\kappa$ is a already a limit of completely ineffable cardinals. To see this, let $M$ be a weak $\kappa$-model with $V_\kappa\in M$ for which there is a weakly amenable $M$-ultrafilter $U$ with a well-founded ultrapower. First, let's argue that for every $A\in U$ and coloring $f:[A]^2\to 2$ in $M$, there is a homogeneous set for $f$ in $U$. Let $h:M\to K$ be the (possibly ill-founded) ultrapower by $U\times U$, which exists by weak amenability. It is still the case that $A\subseteq \kappa\times\kappa$ is in $U\times U$ if and only if $[\text{id}]_{U\times U}\in h(A)$. Say $h(f)([\text{id}]_{U\times U})=1$. Then $$A=\{(\xi_1,\xi_2)\mid f(\xi_1,\xi_2)=1\}\in U\times U.$$ An easy argument shows that whenever $X\in U\times U$, then there is a set $\bar X\in U$ such that for all $\xi_1<\xi_2$ in $\bar X$, $(\xi_1,\xi_2)\in X$. So $U$ has a set $\bar A$ such that whenever $\xi_1<\xi_2$ are in $\bar A$, then $(\xi_1,\xi_2)\in A$, and so $f(\xi_1,\xi_2)=1$, meaning that $\bar A$ is homogeneous for $f$. Now let $j:M\to N$ be the ultrapower by $U$. We will argue that $\kappa$ is completely ineffable in $N$, and hence by elementarity $M$ thinks that $\kappa$ is a limit of completely ineffable cardinals, but it must be correct about it because it has the true $V_\kappa$. We work now inside $N$. Let $\mathcal R_0$ be the collection of all stationary subsets of $\kappa$. Given $R_\xi$, let $R_{\xi+1}$ be the collection of all sets $A$ in $R_\xi$ such that for every coloring $f:[A]^2\to 2$, $R_\xi$ has a homogeneous set for $A$. At limits take intersections. There must be some $\theta$ such that $R_\theta=R_{\theta+1}$ and if $R_\theta$ is not empty, then by construction it will witness that $\kappa$ is completely ineffable. But this is the case because $U\subseteq R_\xi$ for every $\xi$. Note though that a 1-iterable cardinals may not be completely ineffable itself because it is $\Pi^1_2$-describable.

Next, let's talk about one of the game Ramsey cardinals introduced by Holy and Schlicht in [1]. First, let's modify the definition of a weak $\kappa$-model to allow non-transitive $\in$-models. We need this because we want to consider models of size $\kappa$ elementary in large $H_\theta$ (collection of all sets whose transitive closure has size less than $\theta$). Let's call a weak $\kappa$-model $M$ a $\kappa$-model if it is closed under sequences of length less than $\kappa$. Now fix an inaccessible cardinal $\kappa$ and a regular cardinal $\theta>\kappa$. Consider a two player game of perfect information of length $\omega$ between the challenger and the judge, where at each step $n$ of the game, the challenger plays a $\kappa$-model $M_n\prec H_\theta$ and the judge responds with an $M_n$-ultrafilter $U_n$. Additionally, at each step the challenger needs to make sure that $M_n\subseteq M_{n+1}$ and $M_n,U_n\cap M_n\in M_{n+1}$. The judge wins the game if she is able to play for $\omega$-many steps and otherwise the challenger wins. A cardinal $\kappa$ has the $\omega$-filter property if the challenger doesn't have a winning strategy for any regular $\theta$.

Finally, let's talk about Paul Corazza's Wholeness Axiom [2]. Kunen's Inconsistency, which says that there cannot be an elementary embedding $j:V\to V$ can be formalized in several ways. The proof shows that in a model of the second-order Kelley-Morse set theory $(V,\in,\mathcal S)$ there cannot be a class $j\in \mathcal S$ which is an elementary embedding of the first-order part $(V,\in)$. We can also consider expanding the language $\{\in\}$ of set theory by a binary predicate $j$ and considering the theory ${\rm ZFC}$ in this expanded language (with separation and replacement applying to formulas with $j$) together with a scheme of assertions that $j$ is a non-trivial elementary embedding. Kunen's result can also be viewed as asserting that such a theory is inconsistent. What fragment of this theory in the language with $j$ can we salvage? Suppose we just assert that $j$ is non-trivial and elementary. This theory is equiconsistent with ${\rm ZFC}$ because if there is a model of ${\rm ZFC}$, then there is a model of ${\rm ZFC}$ that is computably saturated and hence has plenty of not just elementary embeddings, but automorphisms (see this post). How about the theory saying that $j$ is elementary and has a critical point (meaning that some ordinal $\kappa$ is moved and every ordinal $\alpha<\kappa$ is fixed)? Corazza calls this theory ${\rm BTEE}$ (basic theory of elementary embeddings). So there is a model $M\models{\rm ZFC}$ and an elementary embedding $j:M\to M$ with critical point $\kappa\in M$. It is easy to see that $\kappa$ must be at least inaccessible in $M$. The argument we gave for 1-iterable cardinals shows that $\kappa$ is $n$-ineffable in $M$ for every natural number $n$ of the metatheory because we can use $j$ to derive an $M$-ultrafilter $U$, which must be weakly amenable, and use product ultrafilters $U^n$ to obtain homogeneous sets. If we additionally assume that $\Sigma_0$-separation holds for formulas with $j$, then it follows that $j$ is amenable to $M$ (for every $a\in M$, $j\upharpoonright a\in M$), which implies that $\kappa$ is at least super $n$-huge in $M$ for every $n\in\omega$. That's quite a jump in consistency strength! It is not known whether assuming full separation for $j$ gives a consistency-wise stronger theory although it is known that weaker and stronger separation have different consequences. Stronger separation implies for example that iterate embeddings $j^n$ are definable in $M$ for every natural number $n\in M$, but this is known to fail with just $\Sigma_0$-separation (the issues here arise for nonstandard $n$).

The following theorem gives several very different characterizations of completely ineffable cardinals.

Theorem: The following are equivalent to $\kappa$ being completely ineffable.

  1. (Nielsen and Welch [3]) $\kappa$ has the $\omega$-filter property.
  2. ([4]) In a forcing extension $V[G]$, there is a weakly amenable $V$-ultrafilter and hence a generic elementary embedding $j:V\to N$ with critical point $\kappa$ such that $V_{\kappa+1}^V=V_{\kappa+1}^N$ (it follows that $\kappa^+$ is in the well-founded part of $j$).
  3. For every regular $\theta>\kappa$, every $A\subseteq\kappa$ is an element of a weak $\kappa$-model $M\prec H_\theta$ for which there is a weakly amenable $M$-ultrafilter $U$ (but the ultrapower may fail to be well-founded).

Proof: For equivalence with (1) see [3]. For equivalence with (2), see the the argument in [4] for forcing over countable models and instead force over $V$. (3) follows from (2) by playing the game for $\omega$-many steps to build $U$. Finally suppose (3) and let's argue that $\kappa$ is completely ineffable. The same argument which shows that $\kappa$ is completely ineffable in $N$ for a 1-iterable embedding $j:M\to N$ will show that $\kappa$ is completely ineffable in $M$ and this suffices because $M\prec H_\theta$ is correct about complete ineffability.

Completely ineffable cardinals now look very similar to 1-iterable cardinals, but the true analogue is actually the $\omega$-Ramsey cardinals, another game Ramsey cardinal notion of Holy and Schlicht from [1]. A cardinal $\kappa$ is $\omega$-Ramsey if the challenger has no winning strategy in the games described as above only with the additional condition that in order for the judge to win, after the $\omega$-many steps, $U=\bigcup_{n<\omega}U_n$ has to have a well-founded ultrapower. Equivalently, for every regular $\theta>\kappa$, every $A\subseteq\kappa$ is an element of a weak $\kappa$-model $M\prec H_\theta$ for which there is a weakly amenable $M$-ultrafilter $U$ with a well-founded ultrapower. $\omega$-Ramsey cardinals are between 1-iterable and 2-iterable cardinals in consistency strength.

Theorem: Consistency of a completely ineffable cardinal implies consistency of the theory ${\rm BTEE}$.

Proof: Suppose that $\kappa$ is completely ineffable in a model $V\models{\rm ZFC}$ and move to a forcing extension $V[G]$, in which there is a generic embedding $j:V\to N$ by a weakly amenable $V$-ultrafilter $U$. Since $U$ is weakly amenable, we can iterate the ultrapower construction $\omega$-many times, obtaining iterate embeddings $j_{mn}:N_m\to N_n$ with critical point $\kappa_n$. Let $\mathcal N$ be the direct limit of this system. Standard arguments show that the critical sequence $\{\kappa_n\mid n<\omega\}$ are indiscernibles for $\mathcal N$ even for formulas with parameters $\alpha<\kappa$. (Note that $V_{\kappa+1}^{\mathcal N}=V_{\kappa+1}^V$.) Now let $M$ be generated in $\mathcal N$ by $\alpha<\kappa$ and the critical sequence $\kappa_n$ for $n<\omega$. Define $h:M\to M$ to be the elementary embedding generated by the shift of indiscernibles map taking $\kappa_n$ to $\kappa_{n+1}$. In other words, a term $t(\alpha_0,\ldots,\alpha_n,\kappa_0,\ldots,\kappa_m)$, with $\alpha_i<\kappa$, gets mapped to the term $t(\alpha_0,\ldots\alpha_n,\kappa_1,\ldots,\kappa_{n+1})$. Because the $\kappa_n$ are indiscernibles for formulas with parameters $\alpha<\kappa$, $h$ is a well-defined elementary embedding with critical point $\kappa=\kappa_0$.

This squizes the consistency strength of ${\rm BTEE}$ between $n$-ineffable cardinals for every $n$ and completely ineffable cardinals.

References

  1. P. Holy and P. Schlicht, “A hierarchy of Ramsey-like cardinals.”
  2. P. Corazza, “The spectrum of elementary embeddings $j\:V\to V$,” Ann. Pure Appl. Logic, vol. 139, no. 1-3, pp. 327–399, 2006. Available at: https://doi.org/10.1016/j.apal.2005.06.014
  3. D. S. Nielsen and P. D. Welch, “Games and Ramsey-like cardinals,” Submitted.
  4. F. G. Abramson, L. A. Harrington, E. M. Kleinberg, and W. S. Zwicker, “Flipping properties: a unifying thread in the theory of large cardinals,” Ann. Math. Logic, vol. 12, no. 1, pp. 25–58, 1977. Available at: http://dx.doi.org/10.1016/0003-4843(77)90005-5