Indescribable cardinal

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 \((\text{V}_{\alpha+n}, \in , \text{A} \cap \text{V}) \models \varphi\)F. Drake, Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics ; V. 76 (1974).
\(\Pi^1_1\)-indescribable cardinals are the same as weakly compact cardinals. T. Jech, Set Theory (Third Millennium Edition) (2002)
==Levy hierarchy==
The hierarchies \(\Sigma_n\) and \(\Pi_n\), which are called Levy hierarchy, are defined inductively on \(n\) in the following way:
# If \(\phi\) is equivalent to a first order formula in set theory without unbounded quantifiers, then \(\phi\) is \(\Pi_0\) and \(\Sigma_0\)
# 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 \(\Sigma_{n+1}\)
# 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.
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==
== See also ==
ja:記述不可能基数
Category:Transfinite ordinals
Category:Set theory