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| In [[set theory]] and its applications throughout [[mathematics]], a '''class''' is a collection of [[Set (mathematics)|sets]] (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context. In work on [[Zermelo–Fraenkel set theory]], the notion of class is informal, whereas other set theories, such as [[Von Neumann–Bernays–Gödel set theory]], axiomatize the notion of "proper class", e.g., as entities that are not members of another entity.
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| A class that is not a set (informally in Zermelo–Fraenkel) is called a '''proper class''', and a class that is a set is sometimes called a '''small class'''. For instance, the class of all [[ordinal number]]s, and the class of all sets, are proper classes in many formal systems.
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| Outside set theory, the word "class" is sometimes used synonymously with "set". This usage dates from a historical period where classes and sets were not distinguished as they are in modern set-theoretic terminology. Many discussions of "classes" in the 19th century and earlier are really referring to sets, or perhaps to a more ambiguous concept.
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| == Examples ==
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| The collection of all algebraic objects of a given type will usually be a proper class. Examples include the class of all [[group (mathematics)|group]]s, the class of all [[vector space]]s, and many others. In [[category theory]], a category whose collection of objects forms a proper class (or whose collection of morphisms forms a proper class) is called a [[large category]].
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| The [[surreal number]]s are a proper class of objects that have the properties of a [[field (mathematics)|field]].
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| Within set theory, many collections of sets turn out to be proper classes. Examples include the class of all sets, the class of all ordinal numbers, and the class of all cardinal numbers.
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| One way to prove that a class is proper is to place it in [[bijection]] with the class of all ordinal numbers. This method is used, for example, in the proof that there is no [[free lattice#The complete free lattice|free]] [[complete lattice#Free complete lattices|complete lattice]].
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| == Paradoxes ==
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| The [[naive set theory#Paradoxes|paradoxes of naive set theory]] can be explained in terms of the inconsistent assumption that "all classes are sets". With a rigorous foundation, these paradoxes instead suggest [[proof (mathematics)|proof]]s that certain classes are proper. For example, [[Russell's paradox]] suggests a proof that the class of all sets which do not contain themselves is proper, and the [[Burali-Forti paradox]] suggests that the class of all [[ordinal numbers]] is proper.
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| == Classes in formal set theories ==
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| [[ZF set theory]] does not formalize the notion of classes. They can instead be described in the [[metalanguage]], as equivalence classes of logical formulas. For example, if <math>\mathcal A</math> is a [[structure (mathematical logic)|structure]] interpreting ZF, then the metalanguage expression <math>\{x\mid x=x \}</math> is interpreted in <math>\mathcal A</math> by the collection of all the elements from the domain of <math>\mathcal A</math>; that is, all the sets in <math>\mathcal A</math>. So we can identify the "class of all sets" with the predicate ''x=x'' or any equivalent predicate.
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| Because classes do not have any formal status in the theory of ZF, the axioms of ZF do not immediately apply to classes. However, if an [[inaccessible cardinal]] κ is assumed, then the sets of smaller rank form a model of ZF (a [[Grothendieck universe]]), and its subsets can be thought of as "classes".
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| Another approach is taken by the [[von Neumann–Bernays–Gödel axioms]] (NBG); classes are the basic objects in this theory, and a set is then defined to be a class that is an element of some other class. However, the set existence axioms of NBG are restricted so that they only quantify over sets, rather than over all classes. This causes NBG to be a [[conservative extension]] of ZF.
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| [[Morse–Kelley set theory]] admits proper classes as basic objects, like NBG, but also allows quantification over all proper classes in its set existence axioms. This causes MK to be strictly stronger than both NBG and ZF.
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| In other set theories, such as [[New Foundations]] or the theory of [[semiset]]s, the concept of "proper class" still makes sense (not all classes are sets) but the criterion of sethood is not closed under subsets. For example, any set theory with a universal set has proper classes which are subclasses of sets.
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| ==References==
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| *{{Citation|last1=Jech|first1=Thomas|author1-link=Thomas Jech|title=Set Theory|publisher=[[Springer-Verlag]]|edition=third millennium|location=Berlin, New York|series=Springer Monographs in Mathematics|isbn=978-3-540-44085-7|year=2003}}
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| *{{Citation|authorlink=Azriel Levy|last1=Levy|first1=A.|title=Basic Set Theory|publisher=[[Springer-Verlag]]|location=Berlin, New York |year=1979}}
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| == External links ==
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| *{{MathWorld |title=Set Class |id=SetClass }}
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| {{logic}}
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| {{Set theory}}
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| [[Category:Set theory]]
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Maybe you have started driving traffic to generally or squeeze page and choice can't stay focused and always seem to end up being distracted by other things being carried out?
Use FriendFeed to link everything in concert. FriendFeed ties all of your accounts together. Once you connect your accounts to FriendFeed, so as to when you make a blog post, it automatically is found on Facebook, Twitter, Squidoo, and a sites. Diane puttman is hoping another great lesson in automation. Once this step is completed your BLOG becomes the guts HUB for all of your information you require to store.
On my website I optimize for the keywords, ensuring everything is either the right section. I link back to my website with all of my Social Media presentations too.
1 loyal friend or follower is worthy than of a pregnancy. But does it mean require need A lot of in your list? So. Certainly yes. You need be choosy in allowing people within your circle because majority will possibly not be interested or may possibly add or follow in order to definitely expand their circle and definitely not more than that. It takes place.
Do not post new pictures of yourself a good ex, an exciting new whatever or overtly salacious images so as to inflict additional pain rrn your ex (no mater the amount you think you hate them).4. It is far from recommended posting hourly, self-involved mood updates that is not going to indulge the voyeurism of others, but cheapen the anguish you both feel. In the nutshell, don't twit a twitter.
If to be able to the money, you can pay for advertising campaigns. This is the volume of one strategy that large companies use to bring customers to their websites. Not every of us have the deep pockets to pay for advertising.
Get creative when wanting find cheap MLM prospective customers. Think about all may ways that works then contain twist. Maybe have fun e-mail signature that piques people's curiosity. Take out an ad in a newsletter. Correct busy developing those buyers!
If you liked this short article and you would like to receive extra info concerning Asian Social Media kindly pay a visit to our own website.