The middle attic

St. Augustine Lighthouse photo by Madrigar

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