Cardinal: Difference between revisions

Typically, the \(\alpha\)th well-ordered cardinal is denoted by \(\aleph_\alpha\). For example, in the context of choice, the least infinite cardinal is \(\aleph_0\) - in the context of initial ordinals, this is used interchangeably to mean \(\omega\). The next cardinal is \(\aleph_1\), which is used interchangeably with \(\omega_1\).

 

 

A [[Large cardinal|large cardinal axiom]] is often described as a property which states that a cardinal has a certain "largeness" property, such that the existence statement of such a cardinal is unprovable in ZFC.<ref>Bell, J. L. (1985). ''Boolean-Valued Models and Independence Proofs in Set Theory''. Oxford University Press</ref> Examples of large cardinal properties include [[Inaccessible cardinal|inaccessibility]], [[Mahlo cardinal|Mahloness]], and [[Indescribable cardinal|indescribability]].