, a totally indescribable cardinal
is a certain kind of large cardinal
Formally, a cardinal number κ is called totally indescribable iff for every natural number n, proposition φ, and set A ⊆ Vκ with (Vκ+n, ∈, A) ⊧ φ there exists an α < κ with (Vα+n, ∈, A ∩ Vα) ⊧ φ.
The idea is that κ cannot be distinguished (looking from below) from smaller cardinals by any formula of n+1-th order logic even with the advantage of an extra unary predicate symbol (for A). This implies that it is large because it means that there must be many smaller cardinals with similar properties.
More generally, a cardinal number κ is called Πnm-indescribable if for every Πm proposition φ, and set A ⊆ Vκ with (Vκ+n, ∈, A) ⊧ φ there exists an α < κ with (Vα+n, ∈, A ∩ Vα) ⊧ φ.
Here one looks at formulas with m-1 alternations of quantifiers with the outermost quantifier being universal.
- Drake, F. R. (1974). Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics ; V. 76). Elsevier Science Ltd. ISBN 0-444-10535-2.
- Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings. 2nd ed, Springer. ISBN 3-540-00384-3.