Small models, large cardinals, and induced ideals

Abstract

We show that many large cardinal notions up to measurability can be characterized through the existence of certain filters for small models of set theory. This correspondence will allow us to obtain a canonical way in which to assign ideals to many large cardinal notions. This assignment coincides with classical large cardinal ideals whenever such ideals had been defined before. Moreover, in many important cases, relations between these ideals reflect the ordering of the corresponding large cardinal properties both under direct implication and consistency strength.

Introduction

The work presented in this paper is motivated by the aim to develop general frameworks for large cardinal properties and their ordering under both direct implication and consistency strength. We develop such a framework for large cardinal notions up to measurability, that is based on the existence of set-sized models and ultrafilters for these models satisfying certain degrees of amenability and normality. This will cover several classical large cardinal concepts like inaccessibility, weak compactness, ineffability, Ramseyness and measurability, and also many of the Ramsey-like cardinals that are an increasingly popular subject of recent set-theoretic research (see, for example, [4], [5], [6], [9], [10], [16], [27], and [29]), but in addition, it canonically yields a number of new large cardinal notions. We then use these large cardinal characterizations to canonically assign ideals to large cardinal notions, in a way that generalizes all such assignments previously considered in the set-theoretic literature, like the classical definition of the weakly compact ideal, the ineffable ideal, the completely ineffable ideal and the Ramsey ideal. In a great number of cases, we can show that the ordering of these ideals under inclusion directly corresponds to the ordering of the underlying large cardinal notions under direct implication. Similarly, the ordering of these large cardinal notions under consistency strength can usually be read off these ideals in a simple and canonical way. In combination, these results show that the framework developed in this paper provides a natural setting to study the lower reaches of the large cardinal hierarchy.

Starting from measurability upwards, many important large cardinal notions are defined through the existence of certain ultrafilters that can be used in ultrapower constructions in order to produce elementary embeddings $j:\mathrm{\text{V}}⟶M$ of the set-theoretic universe V into some transitive class M with the large cardinal in question as their critical point. A great variety of results shows that many prominent large cardinal properties below measurability can be characterized through the existence of filters that only measure sets contained in set-sized models M of set theory.¹ For example, the equivalence of weak compactness to the filter property (see [1, Theorem 1.1.3]) implies that an uncountable cardinal κ is weakly compact if and only if for every model M of ${\mathrm{ZFC}}^{-}$ of cardinality at most κ that contains κ, there exists an uniform² M-ultrafilter U on κ that is <κ-complete in V.³ Isolating what was implicit in folklore results (see, for example, [26]), Gitman, Sharpe and Welch showed that Ramseyness can be characterized through the existence of countably complete ultrafilters for transitive ${\mathrm{ZFC}}^{-}$-models of cardinality κ (see [9, Theorem 1.3] or [29, Theorem 5.1]). More examples of such characterizations are provided by results of Kunen [24], Kleinberg [23] and Abramson–Harrington–Kleinberg–Zwicker [1]. Their characterizations can be formulated through the following scheme, which is hinted at in the paragraph before [1, Lemma 3.5.1]: An uncountable cardinal κ has the large cardinal property $\mathit{\Phi }(\mathit{\kappa })$ if and only if for some (equivalently, for all) sufficiently large regular cardinal(s) θ and for some (equivalently, for all) countable $M\prec \mathit{\text{H}}(\mathit{\theta })$ with $\mathit{\kappa }\in M$, there exists a uniform M-ultrafilter U on κ with the property that the statement $\mathit{\Psi }(M,U)$ holds. Their results show that this scheme holds true in the following cases:

• [1] $\mathrm{\Phi }(\kappa )\equiv \text{“}\mathit{\text{κ}}\mathit{\text{is inaccessible}}\text{”}$ and $\mathrm{\Psi }(M,U)\equiv \text{“}\mathit{\text{U}}\mathit{\text{is}}\mathrm{<}\mathit{\text{κ}}\mathit{\text{-amenable and}}\mathrm{<}\mathit{\text{κ}}\mathit{\text{-complete for}}\mathit{\text{M}}\text{”}$.⁴

• [24] $\mathrm{\Phi }(\kappa )\equiv \text{“}\mathit{\text{κ}}\mathit{\text{is weakly compact}}\text{”}$ and $\mathrm{\Psi }(M,U)\equiv \text{“}\mathit{\text{U}}\mathit{\text{is}}\mathit{\text{κ}}\mathit{\text{-amenable and}}\mathrm{<}\mathit{\text{κ}}\mathit{\text{-complete for}}\mathit{\text{M}}\text{”}$.

• [23] $\mathrm{\Phi }(\kappa )\equiv \text{“}\mathit{\text{κ}}\mathit{\text{is completely ineffable}}\text{”}$ and $\mathrm{\Psi }(M,U)\equiv \text{“}\mathit{\text{U}}\mathit{\text{is}}\mathit{\text{κ}}\mathit{\text{-amenable for}}\mathit{\text{M}}\mathit{\text{and}}\mathit{\text{M}}\mathit{\text{-normal}}\text{”}$.

Generalizing the above scheme, our large cardinal characterizations will be based on three schemes that are introduced below. In order to phrase these schemes in a compact way, we introduce some terminology. As usual, we say that some statement $\phi (\beta )$ holds for sufficiently large ordinals β if there is an $\alpha \in \mathrm{Ord}$ such that $\phi (\beta )$ holds for all $\alpha \le \beta \in \mathrm{Ord}$. Given infinite cardinals $\lambda \le \kappa$, a ${\mathrm{ZFC}}^{-}$-model M is a $(\mathit{\lambda },\mathit{\kappa })$-model if it has cardinality λ and $(\lambda +1)\cup \{\kappa \}\subseteq M$. A $(\kappa ,\kappa )$-model is called a weak κ-model. A κ-model is a weak κ-model that is closed under <κ-sequences.⁵ Moreover, given an infinite cardinal θ and a class $\mathcal{S}$ of elementary submodels of $\mathrm{\text{H}}(\theta )$, we say that some statement $\psi (M)$ holds for many models in $\mathcal{S}$ if for every $x\in \mathrm{\text{H}}(\theta )$, there exists an M in $\mathcal{S}$ with $x\in M$ for which $\psi (M)$ holds. Finally, we say that a statement $\psi (M)$ holds for many transitive weak κ-models M if for every $x\subseteq \kappa$, there exists a transitive weak κ-model M with $x\in M$ for which $\psi (M)$ holds.

Scheme A

An uncountable cardinal κ has the large cardinal property $\mathrm{\Phi }(\kappa )$ if and only if for all sufficiently large regular cardinals θ and all infinite cardinals $\lambda <\kappa$, there are many $(\lambda ,\kappa )$-models $M\prec \mathrm{\text{H}}(\theta )$ for which there exists a uniform M-ultrafilter U on κ with $\mathrm{\Psi }(M,U)$.

Scheme B

An uncountable cardinal κ has the large cardinal property $\mathrm{\Phi }(\kappa )$ if and only if for many transitive weak κ-models M there exists a uniform M-ultrafilter U on κ with $\mathrm{\Psi }(M,U)$.

Scheme C

An uncountable cardinal κ has the large cardinal property $\mathrm{\Phi }(\kappa )$ if and only if for all sufficiently large regular cardinals θ, there are many weak κ-models $M\prec \mathrm{\text{H}}(\theta )$ for which there exists a uniform M-ultrafilter U on κ with $\mathrm{\Psi }(M,U)$.

Trivial examples of valid instances of the Scheme A, Scheme C can be obtained by considering the properties $\mathrm{\Phi }(\kappa )\equiv {\mathrm{\Phi }}_{ms}(\kappa )\equiv \text{“}\mathit{\text{κ}}\mathit{\text{is measurable}}\text{”}$ and

$$\mathrm{\Psi }(M,U)\equiv {\mathrm{\Psi }}_{ms}(M,U)\equiv \text{“}U\mathit{\text{ is }}M\mathit{\text{-normal and }}U=F\cap M\mathit{\text{ for some }}F\in M\text{”}.$$

In contrast, Scheme B cannot provably hold true for $\mathrm{\Phi }(\kappa )\equiv {\mathrm{\Phi }}_{ms}(\kappa )$ and a property $\mathrm{\Psi }(M,U)$ of models M and M-ultrafilters U whose restriction to κ-models and filters on κ is definable by a ${\mathrm{\Pi }}_1^2$-formula over ${\mathrm{\text{V}}}_{\kappa }$, because the statement that for many transitive weak κ-models M there exists a uniform M-ultrafilter U on κ with $\mathrm{\Psi }(M,U)$ could then again be formulated by a ${\mathrm{\Pi }}_1^2$-sentence over ${\mathrm{\text{V}}}_{\kappa }$, and measurable cardinals are ${\mathrm{\Pi }}_1^2$-indescribable (see [22, Proposition 6.5]). Since the measurability of κ can be expressed by a ${\mathrm{\Sigma }}_1^2$-formula over ${\mathrm{\text{V}}}_{\kappa }$, this shows that there is no reasonable⁶ characterization of measurability through Scheme B. In order to have a trivial example for a valid instance of Scheme B available, we make the following definition:

Definition 1.1

An uncountable cardinal κ is locally measurable if and only if for many transitive weak κ-models M there exists a uniform M-normal M-ultrafilter U on κ with $U\in M$.

By the transitivity of the models M involved, Scheme B then holds true for the properties

$$\mathrm{\Phi }(\kappa )\equiv {\mathrm{\Phi }}_{lms}(\kappa )\equiv \text{“}\mathit{\kappa }\mathit{\text{is locally measurable}}\text{”}$$

and $\mathrm{\Psi }(M,U)\equiv {\mathrm{\Psi }}_{ms}(M,U)$. Standard arguments show that measurable cardinals are locally measurable limits of locally measurable cardinals. In addition, we will show that consistency-wise, locally measurable cardinals are strictly above all other large cardinal notions discussed in this paper. We will also show that locally measurable cardinals are Ramsey. In contrast, they are not necessarily ineffable, for ineffable cardinals are known to be ${\mathrm{\Pi }}_2^1$-indescribable, while local measurability is a ${\mathrm{\Pi }}_2^1$-property of κ.

In combination with existing results, the work presented in this paper will yield a complete list of large cardinal properties that can be characterized through the above schemes by considering filters possessing various degrees of amenability and normality. In order to present these results in a compact way (and also for later usage), we introduce abbreviations for the relevant properties of cardinals, models and filters. All relevant definitions will be provided in the later sections of our paper.

• ${\mathrm{\Phi }}_{ia}(\kappa )\equiv \text{“}\mathit{\text{κ}}\mathit{\text{is inaccessible}}\text{”}$, ${\mathrm{\Psi }}_{ia}(M,U)\equiv \text{“}\mathit{\text{U}}\mathit{\text{is}}\mathrm{<}\mathit{\text{κ}}\mathit{\text{-amenable and}}\mathrm{<}\mathit{\text{κ}}\mathit{\text{-complete for}}\mathit{\text{M}}\text{”}$.

• ${\mathrm{\Phi }}_{wc}(\kappa )\equiv \text{“}\mathit{\text{κ}}\mathit{\text{is weakly compact}}\text{”}$, ${\mathrm{\Psi }}_{wc}(M,U)\equiv \text{“}\mathit{\text{U}}\mathit{\text{is}}\mathit{\text{κ}}\mathit{\text{-amenable and}}\mathrm{<}\mathit{\text{κ}}\mathit{\text{-complete for}}\mathit{\text{M}}\text{”}$.

• ${\mathrm{\Psi }}_{\delta }(M,U)\equiv \text{“}\mathit{\text{U}}\mathit{\text{is}}\mathit{\text{M}}\mathit{\text{-normal}}\text{”}$.

• ${\mathrm{\Psi }}_{WC}(M,U)\equiv \text{“}\mathit{\text{U}}\mathit{\text{is}}\mathrm{<}\mathit{\text{κ}}\mathit{\text{-amenable for}}\mathit{\text{M}}\mathit{\text{and}}\mathit{\text{M}}\mathit{\text{-normal}}\text{”}$.

• ${\mathrm{\Phi }}_{ie}(\kappa )\equiv \text{“}\mathit{\text{κ}}\mathit{\text{is ineffable}}\text{”}$, ${\mathrm{\Psi }}_{ie}(M,U)\equiv \text{“}\mathit{\text{U}}\mathit{\text{is normal}}\text{”}$.

• ${\mathrm{\Phi }}_{cie}(\kappa )\equiv \text{“}\mathit{\text{κ}}\mathit{\text{is completely ineffable}}\text{”}$, ${\mathrm{\Psi }}_{cie}(M,U)\equiv \text{“}\mathit{\text{U}}\mathit{\text{is}}\mathit{\text{κ}}\mathit{\text{-amenable for}}\mathit{\text{M}}\mathit{\text{and}}\mathit{\text{M}}\mathit{\text{-normal}}\text{”}$.

• ${\mathrm{\Phi }}_{\mathrm{\text{w}}R}(\kappa )\equiv \text{“}\mathit{\text{κ}}\mathit{\text{is weakly Ramsey}}\text{”}$,

  ${\mathrm{\Psi }}_{\mathrm{\text{w}}R}(M,U)\equiv$“ U is κ-amenable for M, M-normal, and $\mathrm{Ult}(M,U)$ is well-founded ”.

• ${\mathrm{\Phi }}_{\omega R}(\kappa )\equiv \text{“}\mathit{\text{κ}}\mathit{\text{is}}\mathit{\text{ω}}\mathit{\text{-Ramsey}}\text{”}$.

• ${\mathrm{\Phi }}_R(\kappa )\equiv \text{“}\mathit{\text{κ}}\mathit{\text{is Ramsey}}\text{”}$,

  ${\mathrm{\Psi }}_R(M,U)\equiv \text{“}\mathit{\text{U}}\mathit{\text{is}}\mathit{\text{κ}}\mathit{\text{-amenable for}}\mathit{\text{M}}\mathit{\text{,}}\mathit{\text{M}}\mathit{\text{-normal and countably complete}}\text{”}$.

• ${\mathrm{\Phi }}_{iR}(\kappa )\equiv \text{“}\mathit{\text{κ}}\mathit{\text{is ineffably Ramsey}}\text{”}$,

  ${\mathrm{\Psi }}_{iR}(M,U)\equiv$“ U is κ-amenable for M, M-normal and stationary-complete ”.

• ${\mathrm{\Phi }}_{nR}(\kappa )\equiv \text{“}\mathit{\text{κ}}\mathit{\text{ is }}{\mathit{\Delta }}_{\mathit{\omega }}^{\forall }\mathit{\text{-Ramsey}}\text{”}$, ${\mathrm{\Psi }}_{nR}(M,U)\equiv \text{“}\mathit{\text{U}}\mathit{\text{is}}\mathit{\text{κ}}\mathit{\text{-amenable for}}\mathit{\text{M}}\mathit{\text{and normal}}\text{”}$.

• ${\mathrm{\Phi }}_{stR}(\kappa )\equiv \text{“}\mathit{\text{κ}}\mathit{\text{is strongly Ramsey}}\text{”}$,

  ${\mathrm{\Psi }}_{stR}(M,U)\equiv \text{“}\mathit{\text{M}}\mathit{\text{is a}}\mathit{\text{κ}}\mathit{\text{-model,}}\mathit{\text{U}}\mathit{\text{is}}\mathit{\text{κ}}\mathit{\text{-amenable for}}\mathit{\text{M}}\mathit{\text{and}}\mathit{\text{M}}\mathit{\text{-normal}}\text{”}$.

• ${\mathrm{\Phi }}_{suR}(\kappa )\equiv \text{“}\mathit{\text{κ}}\mathit{\text{is super Ramsey}}\text{”}$,

  ${\mathrm{\Psi }}_{suR}(M,U)\equiv \text{“}M\prec \mathit{\text{H}}({\mathit{\kappa }}^{+})\mathit{\text{is a}}\mathit{\text{κ}}\mathit{\text{-model,}}\mathit{\text{U}}\mathit{\text{is}}\mathit{\text{κ}}\mathit{\text{-amenable for}}\mathit{\text{M}}\mathit{\text{and}}\mathit{\text{M}}\mathit{\text{-normal}}\text{”}$.

First, note that some of the large cardinal properties appearing in the above list are already defined through one of the above schemes, yielding the following trivial correspondences:

• Scheme B holds true in the following cases:

  • $\mathrm{\Phi }(\kappa )\equiv {\mathrm{\Phi }}_{stR}(\kappa )$ and $\mathrm{\Psi }(M,U)\equiv {\mathrm{\Psi }}_{stR}(M,U)$.

  • $\mathrm{\Phi }(\kappa )\equiv {\mathrm{\Phi }}_{suR}(\kappa )$ and $\mathrm{\Psi }(M,U)\equiv {\mathrm{\Psi }}_{suR}(M,U)$.

• Scheme C holds true in the following cases:

  • $\mathrm{\Phi }(\kappa )\equiv {\mathrm{\Phi }}_{\omega R}(\kappa )$ and $\mathrm{\Psi }(M,U)\equiv {\mathrm{\Psi }}_{\mathrm{\text{w}}R}(M,U)$.

  • $\mathrm{\Phi }(\kappa )\equiv {\mathrm{\Phi }}_{nR}(\kappa )$ and $\mathrm{\Psi }(M,U)\equiv {\mathrm{\Psi }}_{nR}(M,U)$.

The following theorem summarizes our results, together with a number of known results. Items (1), (2a) and (4(b)i) extend the classical results of Kunen, Kleinberg, and Abramson–Harrington–Kleinberg–Zwicker from [1] mentioned above.⁷ Item (3a) is the result from Gitman, Sharpe and Welch mentioned above (in [9], M-normality is not mentioned, however this is easily seen to be irrelevant – see also our Corollary 3.9). Both Item (3a) and Item (3b) are special cases of a result of Sharpe and Welch ([29, Theorem 3.3]). Item (5b) is due to Abramson, Harrington, Kleinberg and Zwicker ([1, Corollary 1.3.1]).

Theorem 1.2

• Scheme A, Scheme B, Scheme C hold true in case $\mathit{\Phi }(\kappa )\equiv {\mathit{\Phi }}_{wc}(\kappa )$ and $\mathit{\Psi }(M,U)\equiv {\mathit{\Psi }}_{wc}(M,U)$.

• Scheme A, Scheme C both hold true in the following cases:

  • $\mathit{\Phi }(\kappa )\equiv {\mathit{\Phi }}_{cie}(\kappa )$ and $\mathit{\Psi }(M,U)\equiv {\mathit{\Psi }}_{cie}(M,U)$.

  • $\mathit{\Phi }(\kappa )\equiv {\mathit{\Phi }}_{\omega R}(\kappa )$ and $\mathit{\Psi }(M,U)\equiv {\mathit{\Psi }}_{\mathrm{\text{w}}R}(M,U)$.

  • $\mathit{\Phi }(\kappa )\equiv {\mathit{\Phi }}_{nR}(\kappa )$ and either

    • $\mathit{\Psi }(M,U)\equiv {\mathit{\Psi }}_{iR}(M,U)$, or

    • $\mathit{\Psi }(M,U)\equiv {\mathit{\Psi }}_{nR}(M,U)$.

• Scheme B holds true in the following cases:

  • $\mathit{\Phi }(\kappa )\equiv {\mathit{\Phi }}_R(\kappa )$ and $\mathit{\Psi }(M,U)\equiv {\mathit{\Psi }}_R(M,U)$.

  • $\mathit{\Phi }(\kappa )\equiv {\mathit{\Phi }}_{iR}(\kappa )$ and $\mathit{\Psi }(M,U)\equiv {\mathit{\Psi }}_{iR}(M,U)$.

• Scheme A holds true in the following cases:

  • $\mathit{\Phi }(\kappa )\equiv \mathit{\text{“}}\mathit{\text{κ}}\mathit{\text{is regular}}\mathit{\text{”}}$ and either

    • $\mathit{\Psi }(M,U)\equiv \mathit{\text{“}}\mathit{\text{U}}\mathit{\text{is}}\mathrm{<}\mathit{\text{κ}}\mathit{\text{-complete for}}\mathit{\text{M}}\mathit{\text{”}}$, or

    • $\mathit{\Psi }(M,U)\equiv {\mathit{\Psi }}_{ie}(M,U)$.

  • $\mathit{\Phi }(\kappa )\equiv {\mathit{\Phi }}_{ia}(\kappa )$ and either

    • $\mathit{\Psi }(M,U)\equiv {\mathit{\Psi }}_{ia}(M,U)$, or

    • $\mathit{\Psi }(M,U)\equiv \mathit{\text{“}}\mathit{\text{U}}\mathit{\text{is}}\mathrm{<}\mathit{\text{κ}}\mathit{\text{-amenable for}}\mathit{\text{M}}\mathit{\text{and normal}}\mathit{\text{”}}$.

• Scheme B, Scheme C hold true in the following cases:

  • $\mathit{\Phi }(\kappa )\equiv {\mathit{\Phi }}_{wc}(\kappa )$ and either

    • $\mathit{\Psi }(M,U)\equiv {\mathit{\Psi }}_{ia}(M,U)$, or

    • $\mathit{\Psi }(M,U)\equiv \mathit{\text{“}}\mathit{\text{U}}\mathit{\text{is stationary-complete,}}\mathit{\text{M}}\mathit{\text{-normal and}}\mathrm{<}\mathit{\text{κ}}\mathit{\text{-amenable for}}\mathit{\text{M}}\mathit{\text{”}}$.

  • $\mathit{\Phi }(\kappa )\equiv {\mathit{\Phi }}_{ie}(\kappa )$ and either

    • $\mathit{\Psi }(M,U)\equiv {\mathit{\Psi }}_{ie}(M,U)$, or

    • $\mathit{\Psi }(M,U)\equiv \mathit{\text{“}}{\mathit{\Psi }}_{ie}(M,U)\mathit{\text{and}}U\mathit{\text{is}}<\kappa \mathit{\text{-amenable for}}M\mathit{\text{”}}$.

The above results are summarized in abbreviated form in Table 1, Table 2 on the next page.⁸ The meaning of the tables should be clear to the reader when compared with the results presented in Theorem 1.2.

All entries in Table 1, Table 2 that are not mentioned within the statement of Theorem 1.2 will be immediate consequences of the definitions of the large cardinal notions that will be introduced later in our paper. Furthermore, let us remark that our results (some of which are mentioned already within Theorem 1.2) will show that the size of the models considered is irrelevant once we consider elementary submodels of (sufficiently large) $\mathrm{\text{H}}(\theta )$'s together with κ-amenability as in Table 2.⁹

We next want to study the large cardinal ideals that are canonically induced by our characterizations.

Definition 1.3

Let $\mathrm{\Psi }(M,U)$ be a property of models M and filters U, and let κ be an uncountable cardinal.

• We define $\mathrm{\text{I}}{}_{\mathrm{\Psi }}^{<\kappa }$ to be the collection of all $A\subseteq \kappa$ with the property that for all sufficiently large regular cardinals θ, there exists a set $x\in \mathrm{\text{H}}(\theta )$ such that for all infinite cardinals $\lambda <\kappa$, if $M\prec \mathrm{\text{H}}(\theta )$ is a $(\lambda ,\kappa )$-model with $x\in M$ and U is a uniform M-ultrafilter on κ with $\mathrm{\Psi }(M,U)$, then $A\notin U$.

• We define $\mathrm{\text{I}}{}_{\mathrm{\Psi }}^{\kappa }$ to be the collection of all $A\subseteq \kappa$ with the property that there exists $x\subseteq \kappa$ such that if M is a transitive weak κ-model with $x\in M$ and U is a uniform M-ultrafilter on κ with $\mathrm{\Psi }(M,U)$, then $A\notin U$.

• We define $\mathrm{\text{I}}{}_{\prec \mathrm{\Psi }}^{\kappa }$ to be the collection of all $A\subseteq \kappa$ with the property that for all sufficiently large regular cardinals θ, there exists a set $x\in \mathrm{\text{H}}(\theta )$ such that if $M\prec \mathrm{\text{H}}(\theta )$ is a weak κ-model with $x\in M$ and U is a uniform M-ultrafilter on κ with $\mathrm{\Psi }(M,U)$, then $A\notin U$.

It is easy to see that the collections $\mathrm{\text{I}}{}_{\mathrm{\Psi }}^{<\kappa }$, $\mathrm{\text{I}}{}_{\mathrm{\Psi }}^{\kappa }$ and $\mathrm{\text{I}}{}_{\prec \mathrm{\Psi }}^{\kappa }$ always form ideals on κ. Moreover, if Scheme A, Scheme B or C holds for some large cardinal property $\mathrm{\Phi }(\kappa )$ and some property $\mathrm{\Psi }(M,U)$ of models M and filters U, then the statement that $\mathrm{\Phi }(\kappa )$ holds for some uncountable cardinal κ implies the properness of the ideal $\mathrm{\text{I}}{}_{\mathrm{\Psi }}^{<\kappa }$, $\mathrm{\text{I}}{}_{\mathrm{\Psi }}^{\kappa }$, or $\mathrm{\text{I}}{}_{\prec \mathrm{\Psi }}^{\kappa }$ respectively. In addition, in all cases covered by Theorem 1.2 (and also in most other natural situations), the converse of this implication also holds true. This is trivial for instances of Scheme B. For instances of Scheme A, Scheme C, this is an easy consequence of the observation that all properties Ψ listed in the theorem are restrictable, i.e. given uncountable cardinals $\overline{\theta }<\theta$, if $M\prec \mathrm{\text{H}}(\theta )$ with $\overline{\theta }\in M$, $\kappa \in M\cap \overline{\theta }$ is a cardinal and U is a uniform M-ultrafilter on κ with $\mathrm{\Psi }(M,U)$, then $\mathrm{\Psi }(\overline{M},U)$ holds, where $\overline{M}=M\cap \mathrm{\text{H}}(\overline{\theta })\prec \mathrm{\text{H}}(\overline{\theta })$. Moreover, in Lemma 2.2 below, we will see that in most cases these ideals are in fact normal ideals.

The above definitions provide uniform ways to assign ideals to large cardinal properties. The next theorem shows that, in the cases where such canonical ideals were already defined, these assignments turn out to coincide with the known notions. In the following, given an abbreviation ${\mathrm{\Psi }}_{\dots }$ for a property of models and filters from the above list, we will write $\mathrm{\text{I}}{}_{\dots }^{<\kappa }$ instead of $\mathrm{\text{I}}{}_{{\mathrm{\Psi }}_{\dots }}^{<\kappa }$, $\mathrm{\text{I}}{}_{\prec \dots }^{\kappa }$ instead of $\mathrm{\text{I}}{}_{\prec {\mathrm{\Psi }}_{\dots }}^{\kappa }$, and $\mathrm{\text{I}}{}_{\dots }^{\kappa }$ instead of $\mathrm{\text{I}}{}_{{\mathrm{\Psi }}_{\dots }}^{\kappa }$. Item (3) below is essentially due to Baumgartner (see [3, Section 2]). The ineffable ideal was introduced by Baumgartner in [2]. The completely ineffable ideal was introduced by Johnson in [20]. Item (5) below is a generalization of a result for countable models by Kleinberg mentioned in [23] after the proof of its Theorem 2. The Ramsey ideal and the ineffably Ramsey ideal were introduced by Baumgartner in [3]. Item (6) is essentially due to Mitchell (see for example [5, Theorem 2.10]), and both Item (6) and Item (7) are special cases of [29, Theorem 3.3].

Theorem 1.4

• If κ is inaccessible, then $\mathit{\text{I}}{}_{ia}^{<\kappa }$ is the bounded ideal on κ.

• If κ is a regular and uncountable cardinal, then $\mathit{\text{I}}{}_{\delta }^{<\kappa }$ is the non-stationary ideal on κ.

• If κ is a weakly compact cardinal, then $\mathit{\text{I}}{}_{WC}^{\kappa }$ is the weakly compact ideal on κ.

• If κ is an ineffable cardinal, then $\mathit{\text{I}}{}_{ie}^{\kappa }$ is the ineffable ideal on κ.

• If κ is a completely ineffable cardinal, then $\mathit{\text{I}}{}_{\prec cie}^{\kappa }$ is the completely ineffable ideal on κ.

• If κ is a Ramsey cardinal, then $\mathit{\text{I}}{}_R^{\kappa }$ is the Ramsey ideal on κ.

• If κ is an ineffably Ramsey cardinal, then $\mathit{\text{I}}{}_{iR}^{\kappa }$ is the ineffably Ramsey ideal on κ.

• If κ is a measurable cardinal, then the complement of $\mathit{\text{I}}{}_{\prec ms}^{\kappa }$ is the union of all normal ultrafilters on κ.

We show that many aspects of the relationship between different large cardinal notions are reflected in the relationship of their corresponding ideals.

First, our results will show that, for many important examples, the ordering of large cardinal properties under direct implication turns out to be identical to the ordering of their corresponding ideals under inclusion.

Next, our approach to show that the ordering of large cardinal properties by their consistency strength can also be read off from the corresponding ideals is motivated by the fact that the well-foundedness of the Mitchell order (see [17, Lemma 19.32]) implies that for every measurable cardinal κ, there is a normal ultrafilter U on κ with the property that κ is not measurable in $\mathrm{Ult}(\mathrm{\text{V}},U)$. Translated into the context of our paper (using Theorem 1.4.(8)) this shows that the set of all non-measurables below κ is not contained in the ideal $\mathrm{\text{I}}{}_{\prec ms}^{\kappa }$.¹⁰ To generalize this to other large cardinal properties Φ, if κ is a cardinal, we let

$$\mathrm{\text{N}}{}_{\mathrm{\Phi }}^{\kappa }=\{\alpha <\kappa |\neg \mathrm{\Phi }(\alpha )\}.$$

If ${\mathrm{\Phi }}_{\dots }$ is an abbreviation for a large cardinal property, then we write $\mathrm{\text{N}}{}_{\dots }^{\kappa }$ instead of $\mathrm{\text{N}}{}_{{\mathrm{\Phi }}_{\dots }}^{\kappa }$. We show that the above result for measurable cardinals can be generalized to many other important large cardinal notions.¹¹ More precisely, for various instances of our characterization schemes, we will show that the above set of ordinals without the given large cardinal property is not contained in the corresponding ideal. These results can be seen as indicators that the derived characterization and the associated ideal canonically describe the given large cardinal property, as one would expect these cardinals to lose some of their large cardinal properties in their ultrapowers. Moreover, our results also show that, in many important cases, the fact that some large cardinal property ${\mathrm{\Phi }}^{⁎}$ has a strictly higher consistency strength than some other large cardinal property Φ is equivalent to the fact that ${\mathrm{\Phi }}^{⁎}(\kappa )$ implies that the set $\mathrm{\text{N}}{}_{\mathrm{\Phi }}^{\kappa }$ is an element of the ideal on κ corresponding to ${\mathrm{\Phi }}^{⁎}$. This allows us to reconstruct the consistency strength ordering of these properties from structural properties of their corresponding ideals. Together with the correspondence described in the last paragraph, it also shows that, in many cases, the fact that some large cardinal property ${\mathrm{\Phi }}^{⁎}$ provably implies a large cardinal property Φ of strictly lower consistency strength yields that $\mathrm{\Phi }(\kappa )$ implies the ideal on κ corresponding to Φ to be a proper subset of the ideal on κ corresponding to ${\mathrm{\Phi }}^{⁎}$.

Finally, we consider the question whether there are fundamental differences between the ideal $\mathrm{\text{I}}{}_{\prec ms}^{\kappa }$ induced by measurability and the ideals induced by weaker large cardinal notions. By classical results of Kunen (see [22, Theorem 20.10]), it is possible that there is a unique normal measure on a measurable cardinal κ. In this case, the ideal $\mathrm{\text{I}}{}_{\prec ms}^{\kappa }$ is equal to the complement of this measure and hence the induced partial order $\mathcal{P}(\kappa )/\mathrm{\text{I}}{}_{\prec ms}^{\kappa }$ is trivial, hence in particular atomic. We study the question whether the partial orders induced by other large cardinal ideals can also be atomic, conjecturing that the possible atomicity of the quotient partial order is a property that separates measurability from all weaker large cardinal properties (this is motivated by Lemma 16.3 below). This conjecture turns out to be closely related to the previous topics, and we will verify it for many prominent large cardinal properties.

The following theorem provides selected instances of our results, namely those concerning large cardinal notions that had already been introduced in the set theoretic literature. Item (1) and the statement that $\mathrm{\text{I}}{}_{ia}{}^{<\kappa }\subseteq \mathrm{\text{I}}{}_{WC}^{\kappa }$ in Item (2) below are of course trivial consequences of Theorem 1.4. The statement that $\mathrm{\text{N}}{}_{ia}^{\kappa }$ belongs to the weakly compact ideal in Item (2) has been shown by Baumgartner in [3, Theorem 2.8]. That $\mathrm{\text{N}}{}_{ie}{}^{\kappa }\in \mathrm{\text{I}}{}_{\prec cie}^{\kappa }$ in Item (4) was shown by Johnson in [19, Corollary 4], however we will also provide an easy self-contained argument of this result later on for the benefit of our readers. Gitman has shown that weakly Ramsey cardinals (which are also known under the name of 1-iterable cardinals) are weakly compact limits of completely ineffable cardinals (see [9, Theorem 3.3 and Theorem 3.7]). Her arguments in the proof of [9, Theorem 3.7] actually show that if κ is a weakly Ramsey cardinal, then $\mathrm{\text{N}}{}_{cie}{}^{\kappa }\in \mathrm{\text{I}}{}_{\mathrm{\text{w}}R}^{\kappa }$, as in Item (5). That $\mathrm{\text{I}}{}_{\mathrm{\text{w}}R}{}^{\kappa }\subseteq \mathrm{\text{I}}{}_R^{\kappa }$ in Item (6) is already immediate from our above definitions. The proof of [10, Theorem 4.1] shows that $\mathrm{\text{N}}{}_{\mathrm{\text{w}}R}{}^{\kappa }\in \mathrm{\text{I}}{}_R^{\kappa }$, as in Item (6). That $\mathrm{\text{N}}{}_R{}^{\kappa }\notin I_R^{\kappa }$ in Item (6), and $\mathrm{\text{I}}{}_R{}^{\kappa }\cup \{\mathrm{\text{N}}{}_R^{\kappa }\}\subseteq \mathrm{\text{I}}{}_{iR}^{\kappa }$ and $\mathrm{\text{N}}{}_{iR}{}^{\kappa }\notin I_{iR}^{\kappa }$ in Item (7) are due to Feng (see [8, Corollary 4.4 and Theorem 4.5]). Moreover, Theorem 1.4 directly shows that $\mathrm{\text{I}}{}_{ie}{}^{\kappa }\subseteq \mathrm{\text{I}}{}_{iR}^{\kappa }$ holds for ineffably Ramsey cardinals κ. That $\mathrm{\text{N}}{}_{stR}{}^{\kappa }\notin I_{stR}^{\kappa }$ in Item (8) and that $\mathrm{\text{N}}{}_{suR}{}^{\kappa }\notin I_{suR}^{\kappa }$ in Item (9) follows easily from the results of [4], and these statements will also be immediate consequences of fairly general results from our paper. That $\mathrm{\text{I}}{}_{\prec cie}{}^{\kappa }⊈\mathrm{\text{I}}{}_{suR}^{\kappa }$ in Item (9) was brought to our attention by Gitman, after we had posed this as an open question in an early version of this paper. The final statement of Item (11) is an immediate consequence of the above-mentioned result of Kunen.

Theorem 1.5

• If κ is an inaccessible cardinal, then $\mathit{\text{N}}{}_{ia}{}^{\kappa }\notin \mathit{\text{I}}{}_{ia}^{<\kappa }$, and $\mathcal{P}(\kappa )/\mathit{\text{I}}{}_{ia}^{<\kappa }$ is not atomic.

• If κ is a weakly compact cardinal, then $\mathit{\text{I}}{}_{ia}{}^{<\kappa }\cup \{\mathit{\text{N}}{}_{ia}^{\kappa }\}\subseteq \mathit{\text{I}}{}_{WC}^{\kappa }$, $\mathit{\text{N}}{}_{wc}{}^{\kappa }\notin \mathit{\text{I}}{}_{WC}^{\kappa }$, and $\mathcal{P}(\kappa )/\mathit{\text{I}}{}_{WC}^{\kappa }$ is not atomic.

• If κ is an ineffable cardinal, then $\mathit{\text{I}}{}_{WC}{}^{\kappa }\cup \{\mathit{\text{N}}{}_{wc}^{\kappa }\}\subseteq \mathit{\text{I}}{}_{ie}^{\kappa }$, $\mathit{\text{N}}{}_{ie}{}^{\kappa }\notin \mathit{\text{I}}{}_{ie}^{\kappa }$, and $\mathcal{P}(\kappa )/\mathit{\text{I}}{}_{ie}^{\kappa }$ is not atomic.

• If κ is a completely ineffable cardinal, then $\mathit{\text{I}}{}_{ie}{}^{\kappa }\cup \{\mathit{\text{N}}{}_{ie}^{\kappa }\}\subseteq \mathit{\text{I}}{}_{\prec cie}^{\kappa }$, $\mathit{\text{N}}{}_{cie}{}^{\kappa }\notin \mathit{\text{I}}{}_{\prec cie}^{\kappa }$, and $\mathcal{P}(\kappa )/\mathit{\text{I}}{}_{\prec cie}^{\kappa }$ is not atomic.

• If κ is a weakly Ramsey cardinal, then $\mathit{\text{I}}{}_{WC}{}^{\kappa }\cup \{\mathit{\text{N}}{}_{cie}^{\kappa }\}\subseteq \mathit{\text{I}}{}_{\mathrm{\text{w}}R}^{\kappa }$, $\mathit{\text{N}}{}_{\mathrm{\text{w}}R}{}^{\kappa }\notin \mathit{\text{I}}{}_{\mathrm{\text{w}}R}^{\kappa }$, $\mathit{\text{I}}{}_{ie}{}^{\kappa }⊈\mathit{\text{I}}{}_{\mathrm{\text{w}}R}^{\kappa }$, and $\mathcal{P}(\kappa )/\mathit{\text{I}}{}_{\mathrm{\text{w}}R}^{\kappa }$ is not atomic.

• If κ is a Ramsey cardinal, then $\mathit{\text{I}}{}_{\mathrm{\text{w}}R}{}^{\kappa }\cup \{\mathit{\text{N}}{}_{\mathrm{\text{w}}R}^{\kappa }\}\subseteq \mathit{\text{I}}{}_R^{\kappa }$, $\mathit{\text{N}}{}_R{}^{\kappa }\notin \mathit{\text{I}}{}_R^{\kappa }$, $\mathit{\text{I}}{}_{ie}{}^{\kappa }⊈\mathit{\text{I}}{}_R^{\kappa }$, and $\mathcal{P}(\kappa )/\mathit{\text{I}}{}_R^{\kappa }$ is not atomic.

• If κ is an ineffably Ramsey cardinal, then $\mathit{\text{I}}{}_{ie}{}^{\kappa }\cup \mathit{\text{I}}{}_R{}^{\kappa }\cup \{\mathit{\text{N}}{}_R^{\kappa }\}\subseteq \mathit{\text{I}}{}_{iR}^{\kappa }$, $\mathit{\text{N}}{}_{iR}{}^{\kappa }\notin \mathit{\text{I}}{}_{iR}^{\kappa }$, $\mathit{\text{I}}{}_{\prec cie}{}^{\kappa }⊈\mathit{\text{I}}{}_{iR}^{\kappa }$, and $\mathcal{P}(\kappa )/\mathit{\text{I}}{}_{iR}^{\kappa }$ is not atomic.

• If κ is strongly Ramsey, then $\mathit{\text{I}}{}_R{}^{\kappa }\cup \{\mathit{\text{N}}{}_{iR}^{\kappa }\}\subseteq \mathit{\text{I}}{}_{stR}^{\kappa }$, $\mathit{\text{N}}{}_{stR}{}^{\kappa }\notin \mathit{\text{I}}{}_{stR}^{\kappa }$, $\mathit{\text{I}}{}_{ie}{}^{\kappa }⊈\mathit{\text{I}}{}_{stR}^{\kappa }$, and $\mathcal{P}(\kappa )/\mathit{\text{I}}{}_{stR}^{\kappa }$ is not atomic.

• If κ is super Ramsey, then $\mathit{\text{I}}{}_{iR}{}^{\kappa }\cup \mathit{\text{I}}{}_{stR}{}^{\kappa }\cup \{\mathit{\text{N}}{}_{stR}^{\kappa }\}\subseteq \mathit{\text{I}}{}_{suR}^{\kappa }$, $\mathit{\text{N}}{}_{suR}{}^{\kappa }\notin \mathit{\text{I}}{}_{suR}^{\kappa }$, $\mathit{\text{I}}{}_{\prec cie}{}^{\kappa }⊈\mathit{\text{I}}{}_{suR}^{\kappa }$, and $\mathcal{P}(\kappa )/\mathit{\text{I}}{}_{suR}^{\kappa }$ is not atomic.

• If κ is locally measurable, then $\mathit{\text{I}}{}_{stR}{}^{\kappa }\cup \{\mathit{\text{N}}{}_{suR}^{\kappa }\}\subseteq \mathit{\text{I}}{}_{ms}^{\kappa }$, $\mathit{\text{N}}{}_{lms}{}^{\kappa }\notin \mathit{\text{I}}{}_{ms}^{\kappa }$, $\mathit{\text{I}}{}_{ie}{}^{\kappa }⊈\mathit{\text{I}}{}_{ms}^{\kappa }$, and $\mathcal{P}(\kappa )/\mathit{\text{I}}{}_{ms}^{\kappa }$ is not atomic.

• If κ is measurable, then $\mathit{\text{I}}{}_{\prec cie}{}^{\kappa }\cup \mathit{\text{I}}{}_{suR}{}^{\kappa }\cup \mathit{\text{I}}{}_{ms}{}^{\kappa }\cup \{\mathit{\text{N}}{}_{lms}^{\kappa }\}\subseteq \mathit{\text{I}}{}_{\prec ms}^{\kappa }$, $\mathit{\text{N}}{}_{ms}{}^{\kappa }\notin \mathit{\text{I}}{}_{\prec ms}^{\kappa }$, and $\mathcal{P}(\kappa )/\mathit{\text{I}}{}_{\prec ms}^{\kappa }$ may be atomic.

Note that the above statements show that the linear ordering of the mentioned large cardinal properties by their consistency strength can be read off from the containedness of sets of the form $\mathrm{\text{N}}{}_{\mathrm{\Phi }}^{\kappa }$ in the induced ideals. Moreover, all provable implications and consistent non-implications can be read of from the ordering of the corresponding ideals under inclusion. For example, the fact that ineffability and Ramseyness do not provably imply each other corresponds to the fact that $\mathrm{\text{I}}{}_{ie}{}^{\kappa }⊈\mathrm{\text{I}}{}_R{}^{\kappa }⊈\mathrm{\text{I}}{}_{ie}^{\kappa }$ holds whenever κ is both ineffable and Ramsey, where the second non-inclusion is a consequence of $\mathrm{\text{N}}{}_{ie}{}^{\kappa }\subseteq \mathrm{\text{N}}{}_{cie}{}^{\kappa }\in \mathrm{\text{I}}{}_{\mathrm{\text{w}}R}{}^{\kappa }\subseteq \mathrm{\text{I}}{}_R^{\kappa }$ and $\mathrm{\text{N}}{}_{ie}{}^{\kappa }\notin \mathrm{\text{I}}{}_{ie}^{\kappa }$.

Fig. 1 summarizes the structural statements listed in Theorem 1.5. In this diagram, a provable inclusion $I_1\subseteq I_0$ of large cardinal ideals is represented by a solid arrow

. Moreover, if $I_1$ is an ideal induced by a large cardinal property Φ, then a dashed arrow

represents the statement that $\mathrm{\text{N}}{}_{\mathrm{\Phi }}{}^{\kappa }\in I_0$ provably holds.