# Difference between revisions of "Middle attic"

From Cantor's Attic

Line 5: | Line 5: | ||

* Up to the [[upper attic]] | * Up to the [[upper attic]] | ||

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

− | * [[reflecting | $\Sigma_n$- | + | * [[reflecting | $\Sigma_n$-reflecting]] cardinals |

* [[reflecting#Sigma2 reflecting | $\Sigma_2$ reflecting cardinal]] | * [[reflecting#Sigma2 reflecting | $\Sigma_2$ reflecting cardinal]] | ||

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

## Revision as of 17:32, 29 December 2011

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.

- Up to the upper attic
- fully reflecting cardinals $V_\delta\prec V$ and the Feferman theory
- $\Sigma_n$-reflecting cardinals
- $\Sigma_2$ reflecting cardinal
- $\beth$-fixed point
- the $\beth_\alpha$ hierarchy
- $\beth_\omega$ and the strong limit cardinals
- the continuum
- $\aleph$-fixed point
- the $\aleph_\alpha$ hierarchy
- $\aleph_\omega$ and singular cardinals
- $\aleph_2$, the second uncountable cardinal
- successor cardinals
- regular cardinals
- $\aleph_1$, the first uncountable cardinal
- $\aleph_0$ and the rest of the lower attic