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, super Reinhardt cardinal, Berkeley cardinal
- Rank into rank cardinals $j:V_\lambda\to V_\lambda$, rank+1 into rank+1 cardinal $j:V_{\lambda+1}\to V_{\lambda+1}$, I0 cardinal $j:L(V_{\lambda+1})\to L(V_{\lambda+1})$
- The wholeness axioms
- n-fold supercompact, n-fold strong, n-fold extendible, n-fold Woodin
- n-huge cardinal, almost n-huge cardinal, n-superhuge cardinal, (n+1)-superstrong cardinal
- high-jump cardinal, almost high-jump cardinal, super high-jump cardinal, high-jump with unbounded excess closure cardinal
- Vopěnka's principle, Vopěnka cardinal, Woodin for supercompactness cardinal
- extendible cardinal, $\alpha$-extendible cardinal
- hypercompact cardinal
- supercompact cardinal, $\lambda$-supercompact cardinal, PFA cardinal
- strongly compact cardinal
- nearly supercompact and nearly strongly compact cardinals
- indestructible weakly compact cardinal
- subcompact cardinal
- superstrong cardinal
- Shelah cardinal
- The axiom of determinacy
- Woodin cardinal
- strong cardinal and the $\theta$-strong and hypermeasurability hierarchy
- tall cardinal
- $0^\dagger$, $j:L[U]\to L[U]$ cardinal
- Nontrivial Mitchell rank, $o(\kappa)=1$, $o(\kappa)=\kappa^{++}$
- measurable cardinal, weakly measurable cardinal
- Ramsey cardinal, strongly Ramsey cardinal, virtually Ramsey cardinal
- Rowbottom cardinal
- Jónsson cardinal
- $\omega_1$-Erdős cardinal and $\gamma$-Erdős cardinals for uncountable $\gamma$
- $0^\sharp$, $j:L\to L$ cardinal
- 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
- ineffable cardinal, weakly ineffable cardinal, and the $n$-ineffable cardinals hierarchy; completely ineffable cardinal
- subtle cardinal
- ethereal cardinal
- superstrongly unfoldable cardinal, strongly uplifting cardinal
- weakly superstrong cardinal
- unfoldable cardinal, strongly unfoldable cardinal
- indescribable cardinal, totally indescribable cardinal
- weakly compact cardinal
- The positive set theory $GPK^+_\infty$
- Mahlo cardinal, $1$-Mahlo, the $\alpha$-Mahlo hierarchy, hyper-Mahlo cardinals
- psuedo uplifting cardinal, 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, weakly inaccessible cardinal
- Morse-Kelley 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
- Zermelo-Fraenkel set theory
- down to the middle attic