Rank-into-rank cardinal

The rank-into-rank cardinals are uncountable cardinal numbers \(\rho\) that satisfy one of these axioms:


 * I3. There exists a nontrivial elementary embedding \(j : V_\rho \to V_\rho\).
 * I2. There exists a nontrivial elementary embedding \(j : V \to M\), where \(V_\rho \subseteq M\) and \(\rho\) is the first fixed point above the critical point of \(j\).
 * I1. There exists a nontrivial elementary embedding \(j : V_{\rho + 1} \to V_{\rho + 1}\).
 * I0. There exists a nontrivial elementary embedding \(j : L(V_{\rho + 1}) \to L(V_{\rho + 1})\) with critical point below \(\rho\).

The four axioms are numbered in increasing strength: I0 implies I1, I1 implies I2, and I2 implies I3. A cardinal satisfying axiom I0 is called an I0 cardinal, and so forth.

The axioms asserting the existence of the rank-into-ranks are extremely strong, so strong that there are a few specialists who doubt the consistency of the system. They are certainly not provable in ZFC (if it's consistent). If ZFC + "there exists a rank-into-rank cardinal" is consistent, then I0 rank-into-ranks are the largest kind of cardinals known that are compatible with ZFC.

The existence of rank-into-ranks implies the divergence of the slow-growing \(p\) function from Laver tables (and the totality of its fast-growing pseudo-inverse \(q\)).

Von Neumann universe
The von Neumann universe is a proper class (that is, a collection of sets which is "too large" to be set itself) that incrementally defines all possible sets in ZFC. It could be thought of as "the set of all sets," although it is not actually itself a set. It is defined by so-called cumulative hierarchy, that is, transfinite sequence of sets defined as follows:


 * \(V_0 = \{\}\)
 * \(V_{\alpha} = \bigcup_{\beta < \alpha} \mathcal{P}(V_\beta)\) (where \(\mathcal{P}(X)\) denotes power set of \(X\), i.e. the set of all subsets of \(X\))

If we combine \(V_\alpha\) for all ordinals \(\alpha\), we get

\[V = \bigcup_{\alpha \in \text{On}} V_\alpha\]

which is the von Neumann universe.

Constructible universe
To define the constructible universe, we need to define following relation:


 * \(\text{Def}(X)\) is set containing all subsets of \(X\) which can be defined by a first-order formula with parameters from \(X\).

A first-order formula with parameters from \(X\) is a first order formula (that is, containing symbols ∃, ⊼, ∈) of the form \(\Phi(y,z_1,...,z_n)\) where y is free variable (is not quantified within the formula) and \(z_1,...,z_n\in X\)

A set is defined by a formula iff its elements are exactly sets \(y\) satisfying that formula.

Now, given a class \(A\), we can make its constructible universe \(L(A)\) like so:


 * \(L_0(A) = \) the smallest transitive class containing \(A\). (A transitive class is a class containing only its own subclasses.)
 * \(L_\alpha(A) = \bigcup_{\beta < \alpha} \text{Def}(L_{\beta}(A))\)
 * \(L(A) = \bigcup_{\alpha \in \text{On}} L_{\alpha}(A)\)

Elementary embedding
A structure of a formal language is a set (or proper class) together with constants, functions, and relations corresponding to constant term symbols, function symbols, and relation symbols in the lamguage. Special cases of structures include groupoids, groups, rings, and fields etc. The relations collectively form the signature of that structure. If \(\sigma\) is a signature, a first-order \(\sigma\)-sentence is a first-order formula using the relations in \(\sigma\); specifically we are allowed to use variables, quantifiers, logical operators, and relations.

We are dealing with set theory, so \(\sigma\) consists of a single relation \(\in\) and our first-order language has the following symbols:


 * variables,
 * ∃, the existential quantifier, which can only quantify over elements (not relations) over the domain,
 * ⊼, the logical NAND, which is the minimum needed to specify all logical operations,
 * ∈, our relation (in other first-order languages, there may be other relations)

(Googologists will notice that this is first-order set theory, the language used in Rayo's number.) From these tools we can create predicates over the domain.

An elementary embedding from structures N to M is a map h: N → M such that, if formula \(\phi(a_1,...,a_n)\) is true in N, then \(\phi(h(a_1),...,h(a_n))\) is true in M. Note that the identity map \(h(x) = x\) over N → N is elementary; we say an elementary embedding is nontrivial iff it is not the identity map.

If an elementary embedding is nontrivial, it must have a critical point. For example, each elementary embedding must map empty set to empty set, so 0 is fixed point. Similarly for all finite ordinals, and even incredibly large portion of transfinite ones (actually, it is consistent that all ordinals must be fixed points, that is, there are no nontrivial elementary embeddings). But, as we assumed, our elementary embedding is nontrivial, so there is an ordinal which maps to some other ordinal - smallest such ordinal is exactly the critical point of the embedding.