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An initial ordinal used to represent the size of a set.
A cardinal is an initial ordinal. In ZF this represents the size of any well-orderable set; assigning a cardinal to every set is equivalent to the well-ordering principle and hence to the axiom of choice.
wp-cardinalnlab-cardinalopenlogic-set-theoryEach relation below ends at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Require the ordinal to have no equipotent smaller ordinal.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
wp-cardinalnlab-cardinalopenlogic-set-theoryThis is an authored directed relation from the source endpoint to the target endpoint.
A well-order on a set has a unique ordinal order type; its least equipotent ordinal is the set’s cardinal.
Record a genuine theorem implication that is not part of the target definition; these edges may point toward a weaker structure.
wp-cardinalopenlogic-set-theoryEach relation below starts at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
is the first infinite cardinal.
The target is a member or subtype of the broader source class.
wp-aleph-numberopenlogic-set-theoryThis is an authored directed relation from the source endpoint to the target endpoint.
Finite cardinals are the finite initial ordinals.
The target is a member or subtype of the broader source class.
wp-cardinalopenlogic-set-theoryThis is an authored directed relation from the source endpoint to the target endpoint.
Infinite cardinals are the nonfinite initial ordinals.
The target is a member or subtype of the broader source class.
wp-cardinalopenlogic-set-theoryThis is an authored directed relation from the source endpoint to the target endpoint.
Select cardinals satisfying additional closure, compactness, measure, or elementary-embedding properties.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
wp-large-cardinalsep-large-cardinalsThis is an authored directed relation from the source endpoint to the target endpoint.
Limit cardinals are infinite cardinals that are not successors.
The target is a member or subtype of the broader source class.
wp-aleph-numberopenlogic-set-theoryThis is an authored directed relation from the source endpoint to the target endpoint.
For each cardinal , well-order the cardinals above it and take the least one, denoted .
Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.
wp-cardinalopenlogic-set-theory