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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[1] - 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.


See also

Basics: cardinal numbers · ordinal numbers · limit ordinals · fundamental sequence · normal form · transfinite induction · ordinal notation
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)‎ · \(\omega_1^\mathfrak{Ch}\) · \(\omega_1^{\text{CK}}\) (Church-Kleene ordinal) · \(\omega_\alpha^\text{CK}\) (admissible ordinal) · recursively inaccessible ordinal · recursively Mahlo ordinal · reflecting ordinal · stable ordinal · \(\lambda,\zeta,\Sigma,\gamma\) (ordinals on infinite time Turing machine) · gap ordinal · List of countable ordinals
Ordinal hierarchies: Fast-growing hierarchy · Slow-growing hierarchy · Hardy hierarchy · Middle-growing hierarchy · N-growing hierarchy
Ordinal functions: enumeration · normal function · derivative · Veblen function · ordinal collapsing function · Bachmann's function · Madore's function · Feferman's theta function · 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
Uncountable cardinals: \(\omega_1\) · omega fixed point · inaccessible cardinal \(I\) · Mahlo cardinal \(M\) · weakly compact cardinal \(K\) · indescribable cardinal · rank-into-rank cardinal
Classes: \(V\) · \(L\) · \(\textrm{On}\) · \(\textrm{Lim}\) · \(\textrm{AP}\) · Class (set theory)


Sources

  1. Dehornoy, Patrick. "Elementary embeddings and algebra." Handbook of Set Theory. Springer, Dordrecht, 2010. 737-774.
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