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- Cardinal_assignment abstract "In set theory, the concept of cardinality is significantly developable without recourse to actually defining cardinal numbers as objects in theory itself (this is in fact a viewpoint taken by Frege; Frege cardinals are basically equivalence classes on the entire universe of sets which are equinumerous). The concepts are developed by defining equinumerosity in terms of functions and the concepts of one-to-one and onto (injectivity and surjectivity); this gives us a pseudo-ordering relationon the whole universe by size. It is not a true ordering because the trichotomy law need not hold: if both and , it is true by the Cantor–Bernstein–Schroeder theorem that i.e. A and B are equinumerous, but they do not have to be literally equal; that at least one case holds turns out to be equivalent to the Axiom of choice.Nevertheless, most of the interesting results on cardinality and its arithmetic can be expressed merely with =c.The goal of a cardinal assignment is to assign to every set A a specific, unique set which is only dependent on the cardinality of A. This is in accordance with Cantor's original vision of a cardinals: to take a set and abstract its elements into canonical "units" and collect these units into another set, such that the only thing special about this set is its size. These would be totally ordered by the relation and =c would be true equality. As Y. N. Moschovakis says, however, this is mostly an exercise in mathematical elegance, and you don't gain much unless you are "allergic to subscripts." However, there are various valuable applications of "real" cardinal numbers in various models of set theory.In modern set theory, we usually use the Von Neumann cardinal assignment which uses the theory of ordinal numbers and the full power of the axioms of choice and replacement. Cardinal assignments do need the full axiom of choice, if we want a decent cardinal arithmetic and an assignment for all sets.".
- Cardinal_assignment wikiPageID "373165".
- Cardinal_assignment wikiPageRevisionID "543735399".
- Cardinal_assignment hasPhotoCollection Cardinal_assignment.
- Cardinal_assignment subject Category:Cardinal_numbers.
- Cardinal_assignment subject Category:Set_theory.
- Cardinal_assignment type Abstraction100002137.
- Cardinal_assignment type CardinalNumber113597585.
- Cardinal_assignment type CardinalNumbers.
- Cardinal_assignment type DefiniteQuantity113576101.
- Cardinal_assignment type Measure100033615.
- Cardinal_assignment type Number113582013.
- Cardinal_assignment comment "In set theory, the concept of cardinality is significantly developable without recourse to actually defining cardinal numbers as objects in theory itself (this is in fact a viewpoint taken by Frege; Frege cardinals are basically equivalence classes on the entire universe of sets which are equinumerous).".
- Cardinal_assignment label "Cardinal assignment".
- Cardinal_assignment label "基数指派".
- Cardinal_assignment sameAs m.020q2d.
- Cardinal_assignment sameAs Q5038627.
- Cardinal_assignment sameAs Q5038627.
- Cardinal_assignment sameAs Cardinal_assignment.
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