# Difference between revisions of "Middle attic"

From Cantor's Attic

m (Fixed reference typo) |
|||

(2 intermediate revisions by the same user not shown) | |||

Line 17: | Line 17: | ||

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

+ | * [[Karauwan cardinal| Karauwan]] cardinals, and Rayo length catching points | ||

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

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

## Revision as of 04:49, 26 July 2021

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
- Karauwan cardinals, and Rayo length catching points
- $\aleph$-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