en>Trappist the monk: /* References */replace mr template with mr parameter in CS1 templates; using AWB

2014-09-25T13:27:57Z

References: replace mr template with mr parameter in CS1 templates; using AWB

en>Imaginatorium: /* Definition */ move ref by one word

2014-01-30T08:16:10Z

Definition: move ref by one word

← Older revision		Revision as of 10:16, 30 January 2014
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	~~Roberto~~ is ~~the name I love~~ to ~~be recognized with though I commonly do not really like being named like that. My excellent say it'~~s ~~not superb for me but methods I love doing will to bake but I'm thinking~~ on ~~starting something totally new~~. ~~South Carolina is where my brand new home is. Software developing is how My personal support my family. You can believe my~~ [~~http://www.Alexa.com/search?q=website&r=topsites_index&p=bigtop website~~] ~~here: http://circuspartypanama.com<br><br>~~		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 [[Gottlob Frege\|Frege]]; [[Frege cardinal]]s are basically [[equivalence class]]es on the entire [[universe (mathematics)\|universe]] of sets which are [[equinumerous]]). The concepts are developed by defining [[equinumerous\|equinumerosity]] in terms of functions and the concepts of [[injective function\|one-to-one]] and [[surjective function\|onto]] (injectivity and surjectivity); this gives us a pseudo-ordering relation

	~~my weblog~~; [~~http~~://~~circuspartypanama~~.~~com clash~~ of ~~clans hack no survey~~]		:<math>A \leq_c B\quad \iff\quad (\exists f)(f : A \to B\ \mathrm{is\ injective})</math>

			on the whole universe by size. It is not a true ordering because the [[trichotomy (mathematics)\|trichotomy law]] need not hold: if both <math>A \leq_c B</math> and <math>B \leq_c A</math>, it is true by the [[Cantor–Bernstein–Schroeder theorem]] that <math>A =_c B</math> 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 =<sub>c</sub>.

			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 [[Georg Cantor\|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 <math>\leq_c</math> and =<sub>c</sub> 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 [[model theory\|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 [[Axiom of choice\|choice]] and [[Axiom of replacement\|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 without the axiom of choice ==

			Formally, assuming the axiom of choice, cardinality of a set ''X'' is the least ordinal α such that there is a bijection between ''X'' and α. This definition is known as the [[von Neumann cardinal assignment]]. If the axiom of choice is not assumed we need to do something different. The oldest definition of the cardinality of a set ''X'' (implicit in Cantor and explicit in Frege and [[Principia Mathematica]]) is as the set of all sets which are equinumerous with ''X'': this does not work in [[ZFC]] or other related systems of [[axiomatic set theory]] because this collection is too large to be a set, but it does work in [[type theory]] and in [[New Foundations]] and related systems. However, if we restrict from this class to those equinumerous with ''X'' that have the least [[rank (set theory)\|rank]], then it will work (this is a trick due to [[Dana Scott]]: it works because the collection of objects with any given rank is a set).

			==References==
			*Moschovakis, Yiannis N. ''Notes on Set Theory''. New York: Springer-Verlag, 1994.

			[[Category:Set theory]]
			[[Category:Cardinal numbers]]

en>Linas: Category:Free algebraic structures

2012-08-23T03:29:04Z

Category:Free algebraic structures

New page

Roberto is the name I love to be recognized with though I commonly do not really like being named like that. My excellent say it's not superb for me but methods I love doing will to bake but I'm thinking on starting something totally new. South Carolina is where my brand new home is. Software developing is how My personal support my family. You can believe my [http://www.Alexa.com/search?q=website&r=topsites_index&p=bigtop website] here: http://circuspartypanama.com<br><br>

my weblog; [http://circuspartypanama.com clash of clans hack no survey]

Polynomial ring - Revision history

en>Trappist the monk: /* References */replace mr template with mr parameter in CS1 templates; using AWB

en>Imaginatorium: /* Definition */ move ref by one word

en>Linas: Category:Free algebraic structures