# The upper attic

From Cantor's Attic

Welcome to the upper attic, the transfinite realm of large cardinals, the higher infinite, carrying us upward from the merely inaccessible and indescribable to the subtle and endlessly extendible concepts beyond, towards the calamity of inconsistency.

- The Kunen inconsistency
- Reinhardt cardinal
- $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$
- rank+1 into rank+1 cardinal $j:V_{\lambda+1}\to V_{\lambda+1}$
- rank into rank cardinal $j:V_\lambda\to V_\lambda$
- The wholeness axiom
- super $n$-huge cardinal
- superhuge cardinal
- huge cardinal
- almost huge cardinal
- Vopěnka cardinal, Vopěnka's principle
- extendible cardinal
- grand reflection cardinal
- supercompact cardinal
- strongly compact cardinal
- nearly supercompact and nearly strongly compact cardinals
- indestructible weakly compact cardinal
- subcompact cardinal
- superstrong cardinal
- Shelah cardinal
- Woodin cardinal
- strong cardinal and the $\theta$-strong and hypermeasurability hierarchy
- tall cardinal
- $0^\dagger$
- Nontrivial Mitchell rank, $o(\kappa)=1$, $o(\kappa)=\kappa^{++}$
- measurable cardinal
- weakly measurable cardinal
- strongly Ramsey cardinal
- weakly Ramsey cardinal
- iterably Ramsey cardinal
- Ramsey cardinal
- Rowbottom cardinal
- Jónsson cardinal
- $\omega_1$-Erdős cardinal and $\gamma$-Erdős cardinals for uncountable $\gamma$
- $0^\sharp$
- Erdős cardinal, and the $\alpha$-Erdős hierarchy for countable $\alpha$
- remarkable cardinal
- ineffible cardinal
- subtle cardinal
- ethereal cardinal
- unfoldable cardinal, strongly unfoldable cardinal
- totally indescribable cardinal
- indescribable cardinal
- weakly compact cardinal
- $1$-Mahlo, the $\alpha$-Mahlo hierarchy and hyper-Mahlo cardinals
- Mahlo cardinal
- ORD is Mahlo
- inaccessible $\Sigma_2$-reflecting, inaccessible $\Sigma_n$-reflecting and inaccessible reflecting cardinals
- $1$-inaccessible, the $\alpha$-inaccessible hierarchy and hyper-inaccessible cardinals
- Grothendieck universe axiom, equivalent to the existence of a proper class of inaccessible cardinals
- inaccessible cardinal, also known as strongly inaccessible
- weakly inaccessible cardinal
- worldly cardinal and the $\alpha$-wordly hierarchy, hyper-worldly cardinal
- the transitive model universe axiom
- Transitive ZFC model
- the minimal transitive model
- Con(ZFC) and $\text{Con}^\alpha(\text{ZFC})$, the iterated consistency hierarchy

- down to the middle attic