# $n$-ineffable and $n$-Ramsey hierarchies

The paper Games and Ramsey-like cardinals by Nielsen and Welch 2018 defines $$n$$-Ramsey cardinals as follows (this is not quite the same definition but it's equivalent): $$\kappa$$ is $$n-1$$-Ramsey if player II has a winning strategy in a game where player I plays $$\kappa$$-models $$M_i$$ (a $$\kappa$$-model is a transitive model of $$ZFC^-$$ of cardinality $$\kappa$$ which contains all of its subsets of cardinality less than $$\kappa$$ as elements) and player II plays $$M_i$$-ultrafilters $$U_i$$ for $$0 \le i \lt n$$, where the models must contain previous models and filters as elements and the filters $$U_i$$ must measure all subsets of $$\kappa$$ in $$M_i$$ and extend the previous filters. If the filters are required to be genuine, that is all diagonal intersections of sets in the filter, even those indexed by sets not in the model, are non-empty, $$\kappa$$ is said to be genuine $$n-1$$-Ramsey. If the filters are required to be normal, that is all diagonal intersections, and thus all elements of the filters, are stationary, $$\kappa$$ is said to be normal $$n-1$$-Ramsey. (Update: According to comments by Miha Habic, a paywalled paper has proven that genuine filters are normal, so genuine $$n-1$$-Ramsey cardinals are normal $$n-1$$-Ramsey, and the theorem characterizing weakly ineffable cardinals by genuine filters is wrong.) The paper mentions the following open question:

Are genuine $$n$$-Ramsey cardinals limits of $$n$$-Ramsey cardinals? We conjecture this to be true, in analogy with the weakly ineffables being limits of weakly compacts. Since “weakly ineffable = $$\Pi^1_1$$-indescribability + subtlety”, this might involve some notion of “n-iterated subtlety”. The difference here is that $$n$$-Ramseys cannot be equivalent to $$\Pi^1_{2n+1}$$-indescribables for consistency reasons, so there is some work to be done.

For $$n$$-subtle, $$n$$-weakly ineffable and $$n$$-ineffable I will use the following definitions from the paper Subtle cardinals and linear orderings by Friedman 1998: $$\kappa$$ is $$n$$-subtle if for every function $$f: \mathcal{P}_n(\kappa) \to \mathcal{P}(\kappa)$$ (where the $$\mathcal{P}_n\kappa$$ is the set of size $$n$$ subsets of $$\kappa$$) such that $$f(\vec{\alpha}) \subseteq \min(\vec{\alpha})$$ for every $$\vec{\alpha}$$, there is an $$f$$-homogeneous set of size $$n+1$$ (an $$f$$-homogeneous set is a set $$A$$ such that $$f(B_1) = f(B_2) \cap \min(B_1)$$ for all $$B_1, B_2 \in \mathcal{P}_n(A)$$ such that $$\min(B_1) \le \min(B_2)$$). $$\kappa$$ is $$n$$-weakly ineffable if for every such function there is an $$f$$-homogeneous set of size $$\kappa$$. $$\kappa$$ is $$n$$-ineffable if for every such function there is a stationary $$f$$-homogeneous set.

What is the proof that $$\Pi^1_1$$-indescribable (weakly compact) subtle cardinals are weakly ineffable (or am I misunderstanding the paper)? Does it also show that $$\Pi^1_2$$-indescribable subtle cardinals are ineffable, that $$\Pi^1_1$$-indescribable $$n$$-subtle cardinals are $$n$$-weakly ineffable and that $$\Pi^1_2$$-indescribable $$n$$-subtle cardinals are $$n$$-ineffable?

I would also like to know how the two hierarchies relate to each other. I know the following:

A genuine $$n$$-Ramsey cardinal is $$n+1$$-weakly ineffable and, if $$0 \lt n$$, a limit of $$n+1$$-weakly ineffable cardinals. A normal $$n$$-Ramsey cardinal is $$n+1$$-ineffable and, if $$0 \lt n$$, a limit of $$n+1$$-ineffable cardinals. Proof (inspired by lemma 5.8 of Sato 2007): Suppose that $$\kappa$$ is $$n$$-Ramsey and $$f$$ is as in the definition of $$n$$-subtle, $$n$$-weakly ineffable and $$n$$-ineffable cardinals. Since $$\kappa$$ is 0-Ramsey there is a $$\kappa$$-model $$M_0$$ such that $$f \in M_0$$ and an $$M_0$$-ultrafilter $$U_0$$. For every $$\vec{\alpha} \in \mathcal{P}_n(\kappa)$$ there is a set $$\{\beta \gt \max(\vec{\alpha}) | f(\vec{\alpha} \cup \{\beta\}) = j(f)(\vec{\alpha} \cup \{\kappa\})\} \in U_0$$ (where j is the ultrapower embedding of $$M_0$$ by $$U_0$$) By induction on $$i \lt n$$ suppose that there are $$\kappa$$-models $$M_k$$ and $$M_k$$-ultrafilters $$U_k$$ for $$k \lt i$$ such that $$f \in M_0$$, $$M_k \in M_{k+1}$$, $$U_k \in M_{k+1}$$, $$U_k \subset U_{k+1}$$ and for every $$\vec{\alpha} \in \mathcal{P}_{n-k}\kappa$$ there is a set $$T_{\vec{\alpha}} \in U_{k}$$ such that $$\max(\vec{\alpha}) \lt \min(T_{\vec{\alpha}})$$ and $$f(\vec{\alpha} \cup \vec{\beta})$$ is the same for every $$\vec{\beta} \in \mathcal{P}_{k+1}T_{\vec{\alpha}}$$. Call that value $$f'(\vec{\alpha})$$. There is a $$\kappa$$-model $$M_i$$ such that $$\{(\vec{\alpha}, T_{\vec{\alpha}}) | \vec{\alpha} \in \mathcal{P}_{n-i+1}\kappa\} \in M_i$$ and since $$\kappa$$ is $$i$$-Ramsey there is an $$M_i$$-ultrafilter $$U_i$$. For every $$\vec{\alpha} \in \mathcal{P}_{n-i}\kappa$$, define $$S_{\vec{\alpha}}$$ using the ultrapower embedding of $$M_i$$ by $$U_i$$ as $$\{\beta \gt \max(\vec{\alpha}) | f'(\vec{\alpha} \cup \{\beta\}) = j(f')(\vec{\alpha} \cup \{\kappa\}) \}$$ which is an element of $$U_i$$, and define $$T_{\vec{\alpha}}$$ as the diagonal intersection $$\Delta_{\beta \in S_{\vec{\alpha}}} (T_{\vec{\alpha} \cup \{\beta\}})$$ (for $$\beta \notin S_{\vec{\alpha}}$$ the diagonal intersection uses $$S_{\vec{\alpha}}$$ in place of $$T_{\vec{\alpha} \cup \{\beta\}}$$), which is an element of $$U_i$$. At stage $$n$$, we find the sets $$T_\alpha$$ for single $$\alpha$$ and $$T_{\emptyset}$$. Since $$T_{\emptyset} \in U_n$$, If $$\kappa$$ in genuine $$n$$-Ramsey, the diagonal intersection $$T_{\emptyset} = \Delta_{\beta \in S_{\vec{\alpha}}} (T_{\vec{\alpha} \cup \{\beta\}})$$ has size $$\kappa$$, and if $$U_n$$ is normal, $$T_{\emptyset}$$ is stationary. If $$T_{\emptyset}$$ has size $$\kappa$$, it is $$f$$-homogeneous Thus genuine $$n$$-Ramsey cardinals are $$n+1$$-weakly ineffable and normal $$n$$-Ramsey cardinals are $$n+1$$-ineffable. Nielsen and Welch proved that $$n$$-Ramsey cardinals are $$\Pi^1_{2n+1}$$-indescribable and normal $$n$$-Ramsey cardinals are $$\Pi^1_{2n+2}$$-indescribable, so since $$n+1$$-weak ineffability is $$\Pi^1_2$$-describable and $$n+1$$-ineffability is $$\Pi^1_3$$-describable, $$n$$-Ramsey cardinals are limits of $$n+1$$-weakly ineffable cardinals and normal $$n$$-Ramsey cardinals are limits of $$n+1$$-ineffable cardinals. (edit: this does not work for $$n$$-Ramsey cardinals without the assumption that they are genuine $$n$$-Ramsey).

What I don't know can be summed up in the following two extremes that I can't rule out: Are $$n+2$$-subtle cardinals limits of normal $$n$$-Ramsey $$n+1$$-Ramsey cardinals and are $$n+1$$-ineffable cardinals limits of genuine $$n$$-Ramsey cardinals? Or are 1-Ramsey cardinals totally ineffable limits of totally ineffable cardinals? I would also mention a possibility inspired by my first question: Is genuine $$n$$-Ramseyness equivalent to $$n+1$$-subtlety combined with $$\Pi^1_{2n+1}$$-indescribability and is normal $$n$$-Ramseyness equivalent to $$n+1$$-subtlety combined with $$\Pi^1_{2n+2}$$-indescribability?

• Holy and Lücke showed that genuine $M$-ultrafilters, where $M$ is a weak $\kappa$-model, are normal, so the genuine and normal Ramsey hierarchies are the same (see Proposition 17.2 of their paper). Commented Nov 10, 2021 at 16:29
• @MihaHabič That paper is paywalled. The Arxiv version has no proposition 17.2. In fact it has no theorems in chapter 17, only open questions. Does the paywalled version add answers to some of the open questions to the chapter? Commented Nov 10, 2021 at 21:06
• @MihaHabič Also, that claim is inconsistent with theorem 4.3 of Nielsen and Welch 2018, which says that genuine 0-Ramsey is equivalent to weakly ineffable and normal 0-Ramsey is eqivalent to ineffable. Does that mean that there can't be weakly ineffable cardinals? Commented Nov 10, 2021 at 21:17
• As far as I can tell, the only difference in section 17 between the arxiv and the published version is the added proposition about genuine and normal $M$-ultrafilters. They claim that Theorem 3.2 of Nielsen--Welch is false and that the proof relies on a faulty reading of Theorem 1.2.1 from Abramson et al. Commented Nov 10, 2021 at 21:49
• @MihaHabič Yes, I meant theorem 3.2, not 4.3, and it was indeed attributed to Flipping properties: A unifying thread in the theory of large cardinals by Abramson et al. Commented Nov 10, 2021 at 22:07

What is the proof that $$\Pi^1_1$$-indescribable (weakly compact) subtle cardinals are weakly ineffable (or am I misunderstanding the paper)? Does it also show that $$\Pi^1_2$$-indescribable subtle cardinals are ineffable
Theorem 2.4 of the paper A refinement of the Ramsey hierarchy via indesceribility by Brent Cody 2019, which originally comes from a paper by Baumgartner 1975, answers that part of my question: If a cardinal $$\kappa$$ is ineffable, its ineffable ideal is the ideal generated by the union of the subtle and $$\Pi^1_2$$-indescribable ideals. Similarly, the weakly ineffable ideal is the ideal generated by the union of the subtle and $$\Pi^1_1$$-indescribable ideals. That can be proven in the same way as theorem 6.1 of Cody's paper (to prove lemma 2.20 for the ineffable ideal, note that, since $$\kappa$$ is inaccessible, there is a club of cardinals $$\alpha$$ such that $$\langle V_\alpha, \in, A \cap \alpha, Y \cap \alpha \rangle \prec \langle V_\kappa, \in, A, Y \rangle$$ for every $$A, Y \subseteq \kappa$$, and to prove lemma 2.20 for the weakly ineffable ideal, additionally note that the subtle ideal is normal, so if $$S$$ is a subtle subset of $$\kappa$$, so is $$S \cap C$$ for any club $$C \subseteq \kappa$$).
However, a weakly ineffable cardinal is a limit of subtle and $$\Pi^1_2$$-indescribable cardinals since the set of $$\Pi^1_2$$-indescribable cardinals less than $$\kappa$$ is in the subtle filter and the set of subtle cardinals less than $$\kappa$$ is in the $$\Pi^1_1$$-indescribable filter, and thus their intersection is in the filter generated by the union of the two filters.
• In the statement "$\langle V_\alpha, \in, A \cup \alpha, Y \cup \alpha \rangle \prec \langle V_\kappa, \in, A, Y \rangle$" in the first paragraph after the block quote, should the unions be replaced with intersections?