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 $$\rho$$.

Here, $$V$$ denotes von Neumann universe, and $$L$$ denotes the relativised constructible universe. 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.

If an elementary embedding $$j$$ of transitive models of a sufficiently strong set theory is nontrivial, then it must have a critical point. i.e. an ordinal $$\lambda$$ such that $$j(\lambda) \neq \lambda$$. For example, each elementary embedding must map empty set to empty set, so 0 is a fixed point of $$j$$. Similarly for all finite ordinals, and even incredibly large portion of transfinite ones. Although it is not obvious from the assumption of the non-triviality, there is an ordinal which maps to some other ordinal - smallest such ordinal is exactly the critical point of the embedding.

## Application to Googology

I0 is used in the definition of $$\textrm{I}0$$ function, and the $$\Sigma_1$$-soundness of I0 implies its totality. I0 function is considered as one of the fastest-growing functions in computable googology. Indeed, it outgrows all computable functions whose totality is provable under $$\textrm{ZFC}+\textrm{I}0$$.

I3 implies the divergence of the slow-growing $$p(n)$$ function from Laver tables, and hence the totality of its fast-growing pseudo-inverse $$q(n)$$. $$q(n)$$ is expected to grow very fast.