# Difference between revisions of "Middle attic"

From Cantor's Attic

(25 intermediate revisions by 2 users not shown) | |||

Line 4: | Line 4: | ||

Welcome to the middle attic, where the uncountable cardinals, that solid stock of mathematics, begin their endless upward structural progession. Here, we survey the infinite cardinals whose existence can be proved in, or is at least equiconsistent with, the ZFC axioms of set theory. | Welcome to the middle attic, where the uncountable cardinals, that solid stock of mathematics, begin their endless upward structural progession. Here, we survey the infinite cardinals whose existence can be proved in, or is at least equiconsistent with, the ZFC axioms of set theory. | ||

− | * | + | * into the [[upper attic]] |

− | * the [[reflecting#Feferman theory | Feferman theory]] | + | * [[correct]] cardinals, [[reflecting | $V_\delta\prec V$]] and the [[reflecting#Feferman theory | Feferman theory]] |

− | * [[reflecting | $\ | + | * [[reflecting#Sigma_2 correct cardinals | $\Sigma_2$ correct]] and [[reflecting | $\Sigma_n$-correct]] cardinals |

− | * [[ | + | * [[extendible#-extendible cardinals | 0-extendible]] cardinal |

+ | * [[extendible#Sigma_n-extendible cardinals|$\Sigma_n$-extendible]] cardinal | ||

* [[beth#beth_fixed_point | $\beth$-fixed point]] | * [[beth#beth_fixed_point | $\beth$-fixed point]] | ||

− | * [[beth omega]] and the [[strong limit]] cardinals | + | * the [[beth]] numbers and the [[beth | $\beth_\alpha$ hierarchy]] |

− | * | + | * [[beth omega | $\beth_\omega$]] and the [[strong limit]] cardinals |

+ | * [[Theta | $\Theta$]] | ||

* the [[continuum]] | * the [[continuum]] | ||

+ | * [[cardinal characteristics]] of the continuum | ||

+ | ** the [[bounding number | bounding number $\frak{b}$]], the [[dominating number | dominating number $\frak{d}$]], the [[covering number | covering numbers]], [[additivity number | additivity numbers]] and many more | ||

+ | * the [[descriptive set theory | descriptive set-theoretic]] cardinals | ||

* [[aleph#aleph fixed point | $\aleph$-fixed point]] | * [[aleph#aleph fixed point | $\aleph$-fixed point]] | ||

− | * the [[aleph | $\aleph_\alpha$ hierarchy]] | + | * the [[aleph]] numbers and the [[aleph | $\aleph_\alpha$ hierarchy]] |

+ | * [[Buchholz's ψ functions]] | ||

* [[aleph#aleph omega | $\aleph_\omega$]] and [[singular]] cardinals | * [[aleph#aleph omega | $\aleph_\omega$]] and [[singular]] cardinals | ||

* [[aleph#aleph two | $\aleph_2$]], the second uncountable cardinal | * [[aleph#aleph two | $\aleph_2$]], the second uncountable cardinal | ||

− | * [[ | + | * [[uncountable]], [[regular]] and [[successor]] cardinals |

− | + | * [[aleph#aleph one | $\aleph_1$]], the first [[uncountable]] cardinal | |

− | * [[aleph#aleph one | $\aleph_1$]], the first uncountable cardinal | + | * [[cardinal | cardinals]], [[infinite]] cardinals |

− | * [[ | + | * [[Aleph zero | $\aleph_0$]] and the rest of the [[lower attic]] |

## Latest revision as of 07:24, 19 May 2018

Welcome to the middle attic, where the uncountable cardinals, that solid stock of mathematics, begin their endless upward structural progession. Here, we survey the infinite cardinals whose existence can be proved in, or is at least equiconsistent with, the ZFC axioms of set theory.

- into the upper attic
- correct cardinals, $V_\delta\prec V$ and the Feferman theory
- $\Sigma_2$ correct and $\Sigma_n$-correct cardinals
- 0-extendible cardinal
- $\Sigma_n$-extendible cardinal
- $\beth$-fixed point
- the beth numbers and the $\beth_\alpha$ hierarchy
- $\beth_\omega$ and the strong limit cardinals
- $\Theta$
- the continuum
- cardinal characteristics of the continuum
- the bounding number $\frak{b}$, the dominating number $\frak{d}$, the covering numbers, additivity numbers and many more

- the descriptive set-theoretic cardinals
- $\aleph$-fixed point
- the aleph numbers and the $\aleph_\alpha$ hierarchy
- Buchholz's ψ functions
- $\aleph_\omega$ and singular cardinals
- $\aleph_2$, the second uncountable cardinal
- uncountable, regular and successor cardinals
- $\aleph_1$, the first uncountable cardinal
- cardinals, infinite cardinals
- $\aleph_0$ and the rest of the lower attic