# The middle attic

From Cantor's Attic

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
- $\Sigma_2$ reflecting and $\Sigma_n$-reflecting cardinals
- $\beth$-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