# Difference between revisions of "User:Muhammad Bukhari Noor/Karauwan cardinal"

Unlike most other large cardinal axioms, Karauwan cardinals are defined in an inpredicative way using the notion of restricted undefinability itself. More precisely, it's defined as uncountable cardinals $$\kappa$$ that are not definable in 1st order set theory using the infinitary language $$L_{\kappa,\omega}$$ with $$\alpha$$ many symbols for all $$\alpha \in \kappa$$, with the usual alphabet in first order set theory, namely the symbols $$\forall$$, $$\exists$$, $$\in$$, $$\neg$$, $$\cup$$, $$\cap$$, parethesis, in addition to ordinal, cardinal, and set variables. The definition of infinitary logic being used is given below:
The infinitary logic $$L_{\alpha,\beta}$$ for regular $$\alpha$$ and $$\beta = 0$$ or $$\omega \le \beta \le \alpha$$ use the same set of symbols as a finitary logic and may use the same rule of formulation as formulae of a finitary logic, in addition to:
• Given a set of formulae $$A = \{A_{\gamma}\mid\gamma<\delta<\alpha\}$$, then $$(A_0 \lor A_1 \lor \cdots)$$ and $$(A_0 \land A_1 \land \cdots)$$ are formulae
• Given a set of variables $$V = \{V_{\gamma}\mid\gamma<\delta<\beta\}$$ and a formula $$A_0$$, then $$\forall V_0:\forall V_1:\cdots (A_0)$$ and $$\exists V_0:\exists V_1:\cdots(A_0)$$ are formulae.
In other words, the infinitary logic $$L_{\kappa,\omega}$$ allows for logical concatenation of up to but not including $$\kappa$$ many formulae and Levy formula of arbitrarily (finitely) many alternations of logical quantifiers. In some sense, Karauwan cardinals can be informaly thought of as the catching points of the inversed Rayo function. i.e, the values where $$\text{Rayo}^{-1}(\alpha)$$ returns $$\alpha$$, where the inversed Rayo function $$\text{Rayo}^{-1}(\alpha)$$ is defined as the minimum amount of symbols required to define $$\alpha$$ in first-order set theory.