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The transfinite enumeration of the infinite initial ordinals.
In ZF, alephs enumerate the well-orderable infinite cardinalities; Choice says every set is well-orderable and hence has an aleph cardinality.
wp-aleph-numberopenlogic-set-theoryEach relation below ends at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Starting from , take successor cardinals and suprema at limit stages.
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-aleph-numberopenlogic-set-theoryThis is an authored directed relation from the source endpoint to the target endpoint.
At limit indices, the aleph hierarchy takes the supremum of earlier alephs.
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-aleph-numberThis is an authored directed relation from the source endpoint to the target endpoint.
is the successor cardinal of .
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-aleph-numberThis is an authored directed relation from the source endpoint to the target endpoint.
Choice ensures every set is well-orderable, so every infinite set has cardinality for some ordinal .
Record a genuine theorem implication that is not part of the target definition; these edges may point toward a weaker structure.
wp-aleph-numbersep-axiom-choiceEach relation below starts at this concept.
No direct relations are authored in this direction.