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

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(All information from [https://github.com/MuhammadBukhariNoor/karauwancardinals/blob/main/Ordinal%20Collapsing%20Function%20Kr.pdf])
 
(All information from [https://github.com/MuhammadBukhariNoor/karauwancardinals/blob/main/Ordinal%20Collapsing%20Function%20Kr.pdf])
  
'''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 milestone for set theorists to study due to the fact that it's 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
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'''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 milestone for set theorists to study due to the fact that it's 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 <math>\kappa</math> that is not definable in 1st order set theory using the infinitary language <math>L_{\kappa,\omega}</math> with <math>\alpha</math> many symbols for all <math>\alpha \in \kappa</math>, with the usual alphabet in first order set theory, namely the symbols <math>\forall</math>, <math>\exists</math>, <math>\in</math>, <math>\neg</math>, <math>\cup</math>, <math>\cap</math>, parethesis, in addition to ordinal, cardinal, and set variables. The definition of infinitary logic being used given below:
 
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 <math>\kappa</math> that is not definable in 1st order set theory using the infinitary language <math>L_{\kappa,\omega}</math> with <math>\alpha</math> many symbols for all <math>\alpha \in \kappa</math>, with the usual alphabet in first order set theory, namely the symbols <math>\forall</math>, <math>\exists</math>, <math>\in</math>, <math>\neg</math>, <math>\cup</math>, <math>\cap</math>, parethesis, in addition to ordinal, cardinal, and set variables. The definition of infinitary logic being used given below:
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In other words, the infinitary logic <math>L_{\kappa,\omega}</math> allows for logical concatenation of up to but not including <math>\kappa</math> 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 <math>\text{Rayo}^{-1}(\alpha)</math> returns <math>\alpha</math>, where the inversed Rayo function <math>\text{Rayo}^{-1}(\alpha)</math> is defined as the minimum amount of symbols required to define <math>\alpha</math> in first-order set theory.
 
In other words, the infinitary logic <math>L_{\kappa,\omega}</math> allows for logical concatenation of up to but not including <math>\kappa</math> 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 <math>\text{Rayo}^{-1}(\alpha)</math> returns <math>\alpha</math>, where the inversed Rayo function <math>\text{Rayo}^{-1}(\alpha)</math> is defined as the minimum amount of symbols required to define <math>\alpha</math> in first-order set theory.
  
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==Size and consistency strength==
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The precise size of the least Karauwan cardinal is not known, but it's conjectured to be strictly between the least [[Aleph | <math>\aleph</math>-fixed point]] and the smallest worldly cardinal. More precisely, it's believed in the community that the least Karauwan cardinal is exactly the least cardinal that is <math>\Pi_2</math>-reflecting onto the class of regular cardinals, although the proof of this previous statement is yet to be given. Assuming [[continuum hypothesis | generalized continuum hypothesis]] and the previous conjecture, this implies that the existence of a Karauwan cardinal with ZFC is roughly equipotent with unboundedly many iterations of powerset axiom starting from the set of natural numbers, A Karauwan cardinal doesn't necessarily have to be regular, and in fact its definition is intended to have nothing to do with regularity but instead based on restricted undefinability, which is a completely different concept. Because of this, if by chance Karauwan cardinals are ever proven to be regular, it would be due to its emergent property rather than as a definition. However, it's widely believed that the least Karauwan cardinal (if it exist) is singular.
 
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Revision as of 20:08, 26 July 2021

(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 milestone for set theorists to study due to the fact that it's 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 is 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 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 \(\forall A_0:\forall A_1:\cdots (A_0)\) 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.

Size and consistency strength

The precise size of the least Karauwan cardinal is not known, but it's conjectured to be strictly between the least \(\aleph\)-fixed point and the smallest worldly cardinal. More precisely, it's believed in the community that the least Karauwan cardinal is exactly the least cardinal that is \(\Pi_2\)-reflecting onto the class of regular cardinals, although the proof of this previous statement is yet to be given. Assuming generalized continuum hypothesis and the previous conjecture, this implies that the existence of a Karauwan cardinal with ZFC is roughly equipotent with unboundedly many iterations of powerset axiom starting from the set of natural numbers, A Karauwan cardinal doesn't necessarily have to be regular, and in fact its definition is intended to have nothing to do with regularity but instead based on restricted undefinability, which is a completely different concept. Because of this, if by chance Karauwan cardinals are ever proven to be regular, it would be due to its emergent property rather than as a definition. However, it's widely believed that the least Karauwan cardinal (if it exist) is singular.

References

    Main library