Let \(\Pi_n\) and \(\Sigma_n\) denote the Levy hierarchy of formulae. A cardinal \(\kappa\) is \(\Pi^n_m\)-indescribable (\(\Sigma^n_m\)-indescribable resp.) if for every \(\Pi_m\) (\(\Sigma_m\) resp.) proposition \(\varphi\), and set \(\text{A} \subseteq V_{\kappa}\) with \((V_{\kappa+n}, \in , \text{A}) \models \varphi\) there exists an \(\alpha<\kappa\) with \((\text{V}_{\alpha+n}, \in , \text{A} \cap \text{V}) \models \varphi\)[1].

\(\Pi^1_1\)-indescribable cardinals are the same as weakly compact cardinals.

Levy hierarchy

Main article: Levy hierarchy

The hierarchies \(\Sigma_n\) and \(\Pi_n\), which are called Levy hierarchy, are defined inductively on \(n\) in the following way:

  1. If \(\phi\) is equivalent to a first order formula in set theory without unbounded quantifiers, then \(\phi\) is \(\Pi_0\) and \(\Sigma_0\)
  2. If \(\phi\) is equivalent to \(\exists n_0 \exists n_1 \exists n_2...\exists n_k \psi\) for some natural numbers \(n_0, n_1, n_2...n_k\) where \(\psi\) is \(\Pi_n\), then \(\phi\) is \(\Sigma_{n+1}\)
  3. If \(\phi\) is equivalent to \(\forall n_0 \forall n_1 \forall n_2...\forall n_k \psi\) for some natural numbers \(n_0, n_1, n_2...n_k\) where \(\psi\) is \(\Sigma_n\), then \(\phi\) is \(\Pi_{n+1}\)

Readers should be careful not to confound Levy hierarchy with arithmetic hierarchy, which is also denoted by \(\Sigma_n\) and \(\Pi_n\).

Although the short definition above looks like an ill-defined predicate requiring the "quantification of formulae", which is not allowed in the set theory itself because formulae are objects in the metatheory rather than the set theory, it is actually formalisable as a predicate in set theory itself using a non-trivial coding. In particular, the Levy hierarchy is a well-defined hierarchy of codes which is formalisable in the set theory itself.

There is a common misconception that a \(\Pi^n_m\)-indescribable cardinal is one that can't be uniquely described using a \(\Pi^n_m\)-formula, however this is false. For example, power admissibilty[2] is known to be at least a \(\Pi_3\) condition[3], however the least \(\Pi_2\)-indescribable cardinal is much larger. Furthermore, the condition of being an inaccessible cardinal is known to be \(\Pi_1\)[4] (also note that a \(\Pi_4\)-indescribable cardinal isn't necessarily a huge cardinal, because \(\kappa\) need not satisfy the formula \(\phi\); and for \(n>2\) \(\Sigma_n\)-correct cardinals are much larger than \(\Sigma_n\)-indescribable cardinals, because correct cardinals are defined using biconditional elementary substructures instead of implication)

If a cardinal \(\kappa\) is \(\Pi^n_m\)-indescribable, then it is also \(\Pi^n_m\)-indescribable in \(L\). This means that if \(\text{ZFC + there is a } \Pi^n_m\text{-indescribable cardinal}\) is consistent, then it is also consistent with the axiom \(V = L\). This is not the case for every kind of large cardinal.

Size

The \(\Pi^0_m\)-indescribable cardinals are the same as the inaccessible cardinals for \(m \geq 2\).

If a cardinal is \(\Pi^n_m\)-indescribable, then it is also \(\Pi^i_j\)-indescribable for \(i < n\) and \(j < m\).

In general, for \(n \geq 1\), the least \(\Pi^n_{m+1}\)-indescribable cardinal contains many \(\Pi^n_m\)-indescribable cardinals below it.

Totally Indescribable Cardinals

A cardinal \(\kappa\) is totally indescribable if it is \(\Pi^n_m\)-indescribable for every \(n\) and \(m\).

Sources

  1. F. Drake, Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics ; V. 76 (1974)
  2. L. Moss, Power set recursion (1993) (p.260)
  3. W. Richter and P. Aczel, Inductive Definitions and Reflecting Properties of Admissible Ordinals (1973) (p.20)
  4. [1], non-wiki citation needed

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'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)

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