User:Muhammad Bukhari Noor/Karauwan cardinal

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(All information from [1])

Karauwan cardinal is a large cardinal axiom first proposed by the Indonesian math hobbyist whose name is Muhammad Bukhari Noor in April 2021 during a brainstorm while he was consternating due to boredom while sitting in his bedroom. At first it's intended as a joke to create the largest large cardinal axiom ever invented, but he later realized that it's not as strong as he thought. However, it still remains an important concept due to its unusual property, namely being the weakest known impredicative large cardinal axiom. Since then, it has gotten multiple revisions and modifications in its definitions. The original definition of Karauwan cardinal is written with paper, as opposed to having an online publication, due to the fact that the inventor prefers to write stuff on paper over digital media. The origin of the name of the large cardinal axiom is believed to be one of the inventor's best friends in church

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.