Large cardinal

Large cardinals are cardinals typically defined as satisfying certain combinatorial or reflection-type properties. Their existence is asserted by various large cardinal axioms, which are usually unprovable in \( \mathrm{ZFC} \), assuming its consistency. This is because almost all large cardinals, if they exist, are worldly: a worldly cardinal is a \( \kappa \) so that \( V_\kappa \) satisfies ZFC, and thus Gödel's second incompleteness theorem applies.

Due to issues that occur later on, such as the identity crisis and the fact that \( \Omega_\omega \) can have large cardinal properties yet not be worldly, it is typically not possible to compare large cardinals in terms of size. More common is comparing them in terms of consistency strength - an assertion \( A \) has higher consistency strength than \( B \) if \( \mathrm{ZFC} + A \) proves the consistency of \( \mathrm{ZFC} + B \).

Large cardinals near the bottom of the hierarchy (both in terms of size and consistency strength) have some limited usage in ordinal collapsing functions. These are used in ordinal analysis, and convert large ordinals such as the \( \nu \)th uncountable cardinal, \( \Omega_\nu \) into countable ordinals. For analysis of theories such as \( \mathrm{KPi} \), Kripke-Platek set theory with the assertion that every set is contained in an admissible set, \( \mathrm{KPM} \), Kripke-Platek set theory with an admissible reflection principle, and beyond, large cardinals such as weakly inaccessible or weakly Mahlo cardinals are used as an additional "layer of diagonalization".

However, most large cardinals, especially at the level of supercompact cardinals and beyond, likely do not have much proof-theoretic use. Instead, large cardinals are typically developed and used in the literature to measure the consistency strength of other combinatorial statements, such as resurrection axioms, quasi-projective determinacy, Chang's conjecture or the saturation of ideals.

Very large cardinals, at the level of \( 0^\sharp \) and above, imply that there are nonconstructible sets, and thus can not exist within the constructible universe of sets. This has led to the study of inner model theory, in which one attempts to construct canonical models of set theory capable of accommodating large cardinals. The current state of the art is a Woodin cardinal which is a limit of Woodin cardinals, and it is difficult to continue past this point because the well-ordering of the reals in inner models for Woodin cardinals becomes harder and harder to define.

There is a vast range of large cardinal axioms. Below they are listed in ascending order, in terms of consistency strength.

• Worldly cardinals and the worldly hierarchy
• Weakly and strongly inaccessible cardinals
• \(\alpha\)-inaccessible cardinals for \( \alpha > 0 \) and the inaccessible hierarchy
• Reflecting cardinals
• \( \mathrm{Ord} \) is Mahlo
• Uplifting cardinals
• \( \Sigma_n \)-Mahlo cardinals
• Weakly and strongly Mahlo cardinals
• The Mahlo hierarchy, including greatly Mahlo cardinals
• \( \Sigma_n \)-weakly compact cardinals
• Weakly compact cardinals
• \( \Pi^n_m\)-indescribable cardinals
• \( \eta \)-indescribable and \( \eta \)-shrewd cardinals, for \( \eta > \omega \)
• Unfoldable and strongly unfoldable cardinals
• Superstrong unfoldable = strongly uplifting cardinals
• Ethereal, subtle and (weakly) ineffable cardinals
• The \( n \)-subtle, \( n \)-almost ineffable, \( n \)-ineffable and \( n \)-Ramsey hierarchy
• Completely ineffable cardinals
• Weakly Ramsey and \( \omega \)-Ramsey cardinals
• Remarkable = virtually supercompact, virtually measurable and strategic \( \omega \)-Ramsey cardinals
• Virtually extendible and completely remarkable cardinals
• The \( n \)-iterable and the virtually \( n \)-huge* hierarchy
• Virtually rank-into-rank cardinals
• \( \omega \)-Erdős cardinals
• The \( \alpha \)-Erdős and \( \alpha \)-iterable hierarchy
• \( 0^\sharp \)
• \( \omega_1 \)-iterable cardinals
• \( \omega_1 \)-Erdős cardinals
• Almost Ramsey cardinals
• Greatly Erdős cardinals
• Ramsey, Jónsson and Rowbottom cardinals
• Measurable cardinal
• \( 0^\dagger \)
• Mitchell rank
• The tall and strong hierarchies
• Woodin cardinals
• The axiom of (projective) determinacy
• Shelah cardinals
• Superstrong cardinals
• Subcompact cardinals
• Strongly compact cardinals
• Supercompact cardinals
• Woodin for strong compactness cardinals
• The extendible hierarchy
• Vopěnka cardinals = Woodin for supercompactness cardinals
• Shelah for supercompactness cardinals
• Almost high-jump and high-jump cardinals
• Super high-jump and high-jump with unbounded excess closure cardinals
• The huge and \( n \)-superstrong for \( n > 1 \) hierarchy
• \( n \)-fold variants of large cardinals
• \( \mathrm{I}^n_4 \) cardinals
• Rank-into-rank cardinals and \( \omega \)-fold variants of large cardinals
• The Reinhardt hierarchy
• The Berkeley hierarchy