# 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
- Berkeley cardinal
- 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, Woodin for supercompactness cardinal
- Vopěnka's principle
- extendible cardinal
- grand reflection cardinal
- supercompact cardinal
- PFA 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
- Ramsey cardinal
- virtually 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 $\alpha$
- $1$-iterable cardinal, and the $\alpha$-iterable cardinals hierarchy for $1\leq \alpha\leq \omega_1$
- remarkable cardinal
- completely ineffable cardinal
- ineffable cardinal, and the $n$-ineffable cardinals hierarchy
- weakly ineffable cardinal
- subtle cardinal
- ethereal cardinal
- superstrongly unfoldable cardinal, strongly uplifting cardinal
- weakly superstrong cardinal
- strongly unfoldable cardinal
- unfoldable cardinal
- Totally indescribable cardinal
- indescribable cardinal
- weakly compact cardinal
- hyper-Mahlo cardinals
- the $\alpha$-Mahlo hierarchy
- $1$-Mahlo
- Mahlo cardinal
- uplifting cardinal
- psuedo uplifting cardinal
- ORD is Mahlo
- $\Sigma_2$-reflecting, $\Sigma_n$-reflecting and 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
- Kelly-Morse set theory
- 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