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A transitive set well-ordered by membership.
Equivalently, an ordinal is a transitive set all of whose elements are transitive; the present formulation follows the standard von Neumann definition.
wp-ordinalnlab-ordinalopenlogic-set-theoryEach relation below ends at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Require the inherited membership relation to be a well-order of the transitive set.
Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.
wp-ordinalnlab-ordinalopenlogic-set-theoryThis is an authored directed relation from the source endpoint to the target endpoint.
Every well-ordered set is uniquely order-isomorphic to one ordinal. This changes to a canonical representative rather than enriching the original carrier.
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-ordinalnlab-ordinalopenlogic-set-theoryEach relation below starts 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.
The ordinals form a proper class rather than a set.
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-ordinalsep-set-theoryThis is an authored directed relation from the source endpoint to the target endpoint.
Countable ordinals are the ordinals admitting an injection into .
The target is a member or subtype of the broader source class.
wp-first-uncountable-ordinalThis is an authored directed relation from the source endpoint to the target endpoint.
Ordinals index the successor and limit stages of the cumulative hierarchy, allowing the rank of a set to be stated precisely.
A mathematical concept supplies part of the formal language, state space, representation, or analytic machinery used by a scientific or mathematical-physics concept. The source is the mathematical predecessor; this does not claim that physical content follows from mathematics alone.
openlogic-set-theorysep-set-theoryThis is an authored directed relation from the source endpoint to the target endpoint.
Finite ordinals are the finite members of the ordinal hierarchy.
The target is a member or subtype of the broader source class.
wp-ordinalwp-von-neumann-ordinalThis is an authored directed relation from the source endpoint to the target endpoint.
is the least uncountable member of the ordinal hierarchy.
The target is a member or subtype of the broader source class.
wp-first-uncountable-ordinalThis is an authored directed relation from the source endpoint to the target endpoint.
Limit ordinals are the nonzero ordinals with no immediate predecessor.
The target is a member or subtype of the broader source class.
wp-ordinalopenlogic-set-theoryThis is an authored directed relation from the source endpoint to the target endpoint.
is the first infinite member of the ordinal hierarchy.
The target is a member or subtype of the broader source class.
wp-ordinalopenlogic-set-theoryThis is an authored directed relation from the source endpoint to the target endpoint.
Define addition, multiplication, and exponentiation recursively on ordinal arguments.
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-ordinal-arithmeticopenlogic-set-theoryThis is an authored directed relation from the source endpoint to the target endpoint.
Successor ordinals are ordinals of the form .
The target is a member or subtype of the broader source class.
wp-ordinalopenlogic-set-theoryThis is an authored directed relation from the source endpoint to the target endpoint.
The well-ordering of every ordinal supports unique definitions from earlier stages.
Record a genuine theorem implication that is not part of the target definition; these edges may point toward a weaker structure.
wp-transfinite-inductionopenlogic-set-theory