Rank into rank axioms
A nontrivial elementary embedding $j:V_\lambda\to V_\lambda$ for some infinite ordinal $\lambda$ is known as a rank into rank embedding and the axiom asserting that such an embedding exists is usually denoted by $\text{I3}$, $\text{I2}$, $\text{I1}$, $\mathcal{E}(V_\lambda)\neq \emptyset$ or some variant thereof. The term applies to a hierarchy of such embeddings increasing in large cardinal strength reaching toward the Kunen inconsistency. The axioms in this section are in some sense a technical restriction falling out of Kunen's proof that there can be no nontrivial elementary embedding $j:V\to V$ in $\text{ZFC}$). An analysis of the proof shows that there can be no nontrivial $j:V_{\lambda+2}\to V_{\lambda+2}$ and that if there is some ordinal $\delta$ and nontrivial rank to rank embedding $j:V_\delta\to V_\delta$ then necessarily $\delta$ must be a strong limit cardinal of cofinality $\omega$ or the successor of one. By standing convention, it is assumed that rank into rank embeddings are not the identity on their domains.
There are really two cardinals relevant to such embeddings: The large cardinal is the critical point of $j$, often denoted $\mathrm{crit}(j)$ or sometimes $\kappa_0$, and the other (not quite so large) cardinal is $\lambda$. In order to emphasize the two cardinals, the axiom is sometimes written as $E(\kappa,\lambda)$ (or $\text{I3}(\kappa,\lambda)$, etc.) as in [1]. The cardinal $\lambda$ is determined by defining the critical sequence of $j$. Set $\kappa_0 = \mathrm{crit}(j)$ and $\kappa_{n+1}=j(\kappa_n)$. Then $\lambda = \sup \langle \kappa_n : n <\omega\rangle$ and is the first fixed point of $j$ that occurs above $\kappa_0$. Note that, unlike many of the other large cardinals appearing in the literature, the ordinal $\lambda$ is not the target of the critical point; it is the $\omega^{th}$ $j$-iterate of the critical point.
As a result of the strong closure properties of rank into rank embeddings, their critical points are huge and in fact $n$-huge for every $n$. This aspect of the large cardinal property is often called $\omega$-hugeness and the term $\omega$-huge cardinal is sometimes used to refer to the critical point of some rank into rank embedding.
Contents
The $\text{I3}$ Axiom and Natural Strengthenings
The $\text{I3}$ axiom asserts, generally, that there is some embedding $j:V_\lambda\to V_\lambda$. $\text{I3}$ is also denoted as $\mathcal{E}(V_\lambda)\neq\emptyset$ where $\mathcal{E}(V_\lambda)$ is the set of all elementary embeddings from $V_\lambda$ to $V_\lambda$, or sometimes even $\text{I3}(\kappa,\lambda)$ when mention of the relevant cardinals is necessary. In its general form, the axiom asserts that the embedding preserves all first-order structure but fails to specify how much second-order structure is preserved by the embedding. The case that no second-order structure is preserved is also sometimes denoted by $\text{I3}$. In this specific case $\text{I3}$ denotes the weakest kind of rank into rank embedding and so the $\text{I3}$ notation for the axiom is somewhat ambiguous. To eliminate this ambiguity we say $j$ is $E_0(\lambda)$ when $j$ preserves only first-order structure.
The axiom can be strengthened and refined in a natural way by asserting that various degrees of second-order correctness are preserved by the embeddings. A rank into rank embedding $j$ is said to be $\Sigma^1_n$ or $\Sigma^1_n$ correct if, for every $\Sigma^1_n$ formula $\Phi$ and $A\subseteq V_\lambda$ the elementary schema holds for $j,\Phi$, and $A$: $$V_\lambda\models\Phi(A) \Leftrightarrow V_\lambda\models\Phi(j(A)).$$ The more specific axiom $E_n(\lambda)$ asserts that some $j\in\mathcal{E}(V_\lambda)$ is $\Sigma^1_{2n}$.
The ``$2n$" subscript in the axiom $E_n(\lambda)$ is incorporated so that the axioms $E_m(\lambda)$ and $E_n(\lambda)$ where $m<n$ are strictly increasing in strength. This is somewhat subtle. For $n$ odd, $j$ is $\Sigma^1_n$ if and only if $j$ is $\Sigma^1_{n+1}$. However, for $n$ even, $j$ being $\Sigma^1_{n+1}$ is significantly stronger than a $j$ being $\Sigma^1_n$[2].
The $\text{I2}$ Axiom
Any $j:V_\lambda\to V_\lambda$ can be extended to a $j^+:V_{\lambda+1}\to V_{\lambda+1}$ but in only one way: Define for each $A\subseteq V_\lambda$ $$j^+(A)=\bigcup_{\alpha < \lambda}(j(V_\alpha\cap A)).$$ $j^+$ is not necessarily elementary. The $\text{I2}$ axiom asserts the existence of some elementary embedding $j:V\to M$ with $V_\lambda\subseteq M$ where $\lambda$ is defined as the $\omega^{th}$ $j$-iterate of the critical point. Although this axiom asserts the existence of a class embedding with a very strong closure property, it is in fact equivalent to an embedding $j:V_\lambda\to V_\lambda$ with $j^+$ preserving well-founded relations on $V_\lambda$. So this axioms preserves some second-order structure of $V_\lambda$ and is in fact equivalent to $E_1(\lambda)$ in the hierarchy defined above. A specific property of $\text{I2}$ embeddings is that they are iterable (i.e. the direct limit of directed system of embeddings is well-founded). In the literature, $IE(\lambda)$ asserts that $j:V_\lambda\to V_\lambda$ is iterable and $IE(\lambda)$ falls strictly between $E_0(\lambda)$ and $E_1(\lambda)$.
As a result of the strong closure property of $\text{I2}$, the equivalence mentioned above cannot be through an analysis of some ultrapower embedding. Instead, the equivalence is established by constructing a directed system of embeddings of various ultrapowers and using reflection properties of the critical points of the embeddings. The direct limit is well-founded since well-founded relations are preserved by $j^+$. The use of both direct and indirect limits, in conjunction with reflection arguments, is typical for establishing the properties of rank into rank embeddings.
The $\text{I1}$ Axiom
$\text{I1}$ asserts the existence of a nontrivial elementary embedding $j:V_{\lambda+1}\to V_{\lambda+1}$. This axiom is sometimes denoted $\mathcal{E}(V_{\lambda+1})\neq\emptyset$. Any such embedding preserves all second-order properties of $V_\lambda$ and so is $\Sigma^1_n$ for all $n$. To emphasize the preservation of second-order properties, the axiom is also sometimes written as $E_\omega(\lambda)$. In this case, restricting the embedding to $V_\lambda$ and forming $j^+$ as above yields the original embedding.
Strengthening this axiom in a natural way leads the $\text{I0}$ axiom, i.e. asserting that embeddings of the form $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$ exist. There are also other natural strengthenings of $\text{I1}$, though it is not entirely clear how they relate to the $\text{I0}$ axiom. For example, one can assume the existence of elementary embeddings satisfying $\text{I1}$ which extend to embeddings $j:M\to M$ where $M$ is a transitive class inner model and add various requirements to $M$. These requirements must not entail that $M$ satisfies the axiom of choice by the Kunen inconsistency. Requirements that have been considered include assuming $M$ contains $V_{\lambda+1}$, $M$ satisfies $DC_\lambda$, $M$ satisfies replacement for formulas containing $j$ as a parameter, $j(\mathrm{crit}(j))$ is arbitrarily large in $M$, etc.
Large Cardinal Properties of Critical Points
The critical points of rank into rank embeddings have many strong reflection properties. They are measurable, $n$-huge for all $n$ (hence the terminology $\omega$-huge mentioned in the introduction) and partially supercompact.
Using $\kappa_0$ as a seed, one can form the ultrafilter $$U=\{X\subseteq\mathcal{P}(\kappa_0): j``\kappa_0\in j(X)\}.$$ Thus, $\kappa_0$ is a measurable cardinal.
In fact, for any $n$, $\kappa_0$ is also $n$-huge as witnessed by the ultrafilter $$U=\{X\subseteq\mathcal{P}(\kappa_n): j``\kappa_n\in j(X)\}.$$ This motivates the term $\omega$-huge cardinal mentioned in the introduction.
Letting $j^n$ denote the $n^{th}$ iteration of $j$, then $$V_\lambda\models ``\lambda\text{ is supercompact"}.$$ To see this, suppose $\kappa_0\leq \theta <\kappa_n$, then $$U=\{X\subseteq\mathcal{P}_{\kappa_0}(\theta): j^n``\theta\in j^n(X)\}$$ winesses the $\theta$-compactness of $\kappa_0$ (in $V_\lambda$). For this last claim, it is enough that $\kappa_0(j)$ is $<\lambda$-supercompact, i.e. not *fully* supercompact in $V$. In this case, however, $\kappa_0$ *could* be fully supercompact.
Critical points of rank-into-rank embeddings also exhibit some *upward* reflection properties. For example, if $\kappa$ is a critical point of some embedding witnessing $\text{I3}(\kappa,\lambda)$, then there must exist another embedding witnessing $\text{I3}(\kappa',\lambda)$ with critical point above $\kappa$! This upward type of reflection is not exhibited by large cardinals below extendible cardinals in the large cardinal hierarchy.
Algebras of elementary embeddings
If $j,k\in\mathcal{E}_{\lambda}$, then $j^+(k)\in\mathcal{E}_{\lambda}$ as well. We therefore define a binary operation $*$ on $\mathcal{E}_{\lambda}$ called application defined by $j*k=j^{+}(k)$. The binary operation $*$ together with composition $\circ$ satisfies the following identities:
1. $(j\circ k)\circ l=j\circ(k\circ l),\,j\circ k=(j*k)\circ j,\,j*(k*l)=(j\circ k)*l,\,j*(k\circ l)=(j*k)\circ(j*l)$
2. $j*(k*l)=(j*k)*(j*l)$ (self-distributivity).
Identity 2 is an algebraic consequence of the identities in 1.
If $j\in\mathcal{E}_{\lambda}$ is a nontrivial elementary embedding, then $j$ freely generates a subalgebra of $(\mathcal{E}_{\lambda},*,\circ)$ with respect to the identities in 1, and $j$ freely generates a subalgebra of $(\mathcal{E}_{\lambda},*)$ with respect to the identity 2.
If $j_{n}\in\mathcal{E}_{\lambda}$ for all $n\in\omega$, then $\sup\{\textrm{crit}(j_{0}*\dots*j_{n})\mid n\in\omega\}=\lambda$ where the implied parentheses a grouped on the left (for example, $j*k*l=(j*k)*l$).
Suppose now that $\gamma$ is a limit ordinal with $\gamma<\lambda$. Then define an equivalence relation $\equiv^{\gamma}$ on $\mathcal{E}_{\lambda}$ where $j\equiv^{\gamma}k$ if and only if $j(x)\cap V_{\gamma}=k(x)\cap V_{\gamma}$ for each $x\in V_{\gamma}$. Then the equivalence relation $\equiv^{\gamma}$ is a congruence on the algebra $(\mathcal{E}_{\lambda},*,\circ)$. In other words, if $j_{1},j_{2},k\in \mathcal{E}_{\lambda}$ and $j_{1}\equiv^{\gamma}j_{2}$ then $j_{1}\circ k\equiv^{\gamma} j_{2}\circ k$ and $j_{1}*k\equiv^{\gamma}j_{2}*k$, and if $j,k_{1},k_{2}\in\mathcal{E}_{\lambda}$ and $k_{1}\equiv^{\gamma}k_{2}$ then $j\circ k_{1}\equiv^{\gamma}j\circ k_{2}$ and $j*k_{1}\equiv^{j(\gamma)}j*k_{2}$.
If $\gamma<\lambda$, then every finitely generated subalgebra of $(\mathcal{E}_{\lambda}/\equiv^{\gamma},*,\circ)$ is finite.
References
- Kanamori, Akihiro. The higher infinite. Second, Springer-Verlag, Berlin, 2009. (Large cardinals in set theory from their beginnings, Paperback reprint of the 2003 edition) www bibtex
- Laver, Richard. Implications between strong large cardinal axioms. Ann Math Logic 90(1--3):79--90, 1997. MR bibtex