Cardinal: Difference between revisions
Jump to navigation
Jump to search
no edit summary
(Created page with "Cardinals are an extension of the natural numbers that describe the size of a set. There are two ways to define cardinality: cardinals as initial ordinals, or cardinals as equivalence classes under bijectability. The second is more common in settings without the axiom of choice, since not all sets are necessarily well-orderable.{{citation needed}} The aleph numbers are examples of well-ordered cardinals, and exhaust the infinite well-ordered cardinals.{{citation nedede...") |
RhubarbJayde (talk | contribs) No edit summary |
||
Line 1:
Cardinals are an extension of the natural numbers that describe the size of a set. Unlike ordinals, which describe "length" and order, cardinals are only intended to describe size.
There are two ways to define cardinality: cardinals as initial ordinals, or cardinals as equivalence classes under bijectability. The second is more common in settings without the axiom of choice, since not all sets are necessarily well-orderable.
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 \(\omega_1\).
The aleph numbers are examples of well-ordered cardinals, and exhaust the infinite well-ordered cardinals.{{citation nededed}}▼
▲The aleph numbers are examples of well-ordered cardinals, and exhaust the infinite well-ordered cardinals.
A large cardinal property 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>Maybe "Believing the Axioms II"?</ref> Examples of large cardinal properties include [[Inaccessible cardinal|inaccessibility]], [[Mahlo cardinal|Mahloness]], and [[Indescribable cardinal|indescribability]].▼
▲A [[Large cardinal|large cardinal
| |||