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 \(\rho\).
- IE. there is \(e: V_\delta\ \prec V_\delta\) who’s \(\alpha\)-th iteration is well founded for all \(\alpha \in \mathrm{Ord}\)[1]
- IE\(\omega\). There is an nontrivial embedding \(e: V_\delta\ \prec V_\delta\) with \(\text{Crit}(e)\) = \(\kappa\) such that the direct limit of \(f\) \(\langle e^n: V_\delta\ \prec V_\delta \in \omega \rangle\) is well founded[1]
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[2] - smallest such ordinal is exactly the critical point of the embedding.
En axioms[]
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 \(A \subseteq V\)λ the elementary schema holds for \(j\), \(phi\), And \(A\): V_Lambdamodelsphi(A)LeftrightarrowV_lambdamodelsphi(j(A)
The more specific axiom En(λ) asserts that some \(j \in \mathcal{E}(V_\Lambda\)\) is \(\Sigma^1_{2n}\)
The “\(2\)\(n\)” subscript in the axiom En(λ) is incorporated so that the axioms Em(λ) and En(λ) 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 \(\Sigma^1_{n+1}\) (shown by Donald Martin). However, for \(n\) even, \(j\) being \(\Sigma^1_n+1\) is significantly stronger than a \(j\) being \(\Sigma^1_n\) .
En+1 strongly implies En . It also implies the consistency En of strengthened by adding “with an arbitrarily large first target”[1]
Virtually rank into rank cardinal[]
A cardinal \(\kappa\) is virtually rank-into-rank iff in a set-forcing extension it is the critical point of an elementary embedding \(j : V_\Lambda \to V_\Lambda\). for some λ > \(\kappa\)
This notion does not require stratification, because Kunen’s Inconsistency does not hold for virtual embeddings
Results:
- Every virtually rank-into-rank cardinal is a virtually \(n\)-huge* limit of virtually \(n\)-huge* cardinals for every \(n\) < \(\omega\)
- The least \(\omega\)-Erdős cardinal \(\eta\)\(\omega\) is a limit of virtually rank-into-rank cardinals
- Every virtually rank-into-rank cardinal is an \(\omega\)-iterable limit of \(\omega\)-iterable cardinals
- Every element of a club \(C\) witnessing that \(\kappa\) is a Silver cardinal is virtually rank-into-rank.
- If gVP(Σn+1) holds, then either there is a proper class of \(n\)-remarkable cardinals or there is a proper class of virtually rank-into-rank cardinals[1]
B-En, P-En, and W-En cardinals[]
- \(\kappa\) is B-En if and only if En(\(\kappa\)); IE there is some j: Vλ ≺ Vλ such that j+ preserves \(\Sigma^1_{2n}\)-properties
- \(\kappa\) is W-En if and only if for every ƒ: \(\kappa\) -> \(\kappa\) there is some j:Vλ ≺ Vλ there is some \(\alpha\) < \(\kappa\) such that ƒ’’\(\alpha\) ⊆ \(\alpha\) and En(\(\alpha\))[1]
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.
See also[]
Basics: cardinal numbers · ordinal numbers · limit ordinals · fundamental sequence · normal form · transfinite induction · ordinal notation · Absolute infinity
Theories: Robinson arithmetic · Presburger arithmetic · Peano arithmetic · KP · second-order arithmetic · ZFC
Model Theoretic Concepts: structure · elementary embedding
Countable ordinals: \(\omega\) · \(\varepsilon_0\) · \(\zeta_0\) · \(\eta_0\) · \(\Gamma_0\) (Feferman–Schütte ordinal) · \(\varphi(1,0,0,0)\) (Ackermann ordinal) · \(\psi_0(\Omega^{\Omega^\omega})\) (small Veblen ordinal) · \(\psi_0(\Omega^{\Omega^\Omega})\) (large Veblen ordinal) · \(\psi_0(\varepsilon_{\Omega + 1}) = \psi_0(\Omega_2)\) (Bachmann-Howard ordinal) · \(\psi_0(\Omega_\omega)\) with respect to Buchholz's ψ · \(\psi_0(\varepsilon_{\Omega_\omega + 1})\) (Takeuti-Feferman-Buchholz ordinal) · \(\psi_0(\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}})\) (countable limit of Extended Buchholz's function) · \(\omega_1^\mathfrak{Ch}\) (Omega one of chess) · \(\omega_1^{\text{CK}}\) (Church-Kleene ordinal) · \(\omega_\alpha^\text{CK}\) (admissible ordinal) · recursively inaccessible ordinal · recursively Mahlo ordinal · reflecting ordinal · stable ordinal · \(\lambda,\gamma,\zeta,\Sigma\) (Infinite time Turing machine ordinals) · gap ordinal · List of countable ordinals
Ordinal hierarchies: Fast-growing hierarchy · Hardy hierarchy · Slow-growing hierarchy · Middle-growing hierarchy · N-growing hierarchy
Ordinal functions: enumeration · normal function · derivative · Veblen function · ordinal collapsing function · Weak Buchholz's function · Bachmann's function · Madore's function · Feferman's \(\theta\) function · Buchholz's function · Extended Weak Buchholz's function · Extended Buchholz's function · Jäger-Buchholz function · Jäger's function · Rathjen's \(\psi\) function · Rathjen's \(\Psi\) function · Stegert's \(\Psi\) function · Arai's \(\psi\) function
Uncountable cardinals: \(\omega_1\) · \(\omega\) fixed point · inaccessible cardinal \(I\) · Mahlo cardinal \(M\) · weakly compact cardinal \(K\) · indescribable cardinal · rank-into-rank cardinal)
Classes: \(\textrm{Card}\) · \(\textrm{On}\) · \(V\) · \(L\) · \(\textrm{Lim}\) · \(\textrm{AP}\) · Class (set theory)