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单词 ClassesOfOrdinalsAndEnumeratingFunctions
释义

classes of ordinals and enumerating functions


A class of ordinalsMathworldPlanetmathPlanetmath is just a subclass of the class (http://planetmath.org/Class) 𝐎𝐧 of all ordinals. For every class of ordinals M there is an enumerating function fM defined by transfinite recursion:

fM(α)=min{xMf(β)<x for all β<α},

and we define the order type of M by otype(M)=dom(f). The possible values for this value are either 𝐎𝐧 or some ordinal α. The above functionMathworldPlanetmath simply lists the elements of M in order. Note that it is not necessarily defined for all ordinals, although it is defined for a segment of the ordinals. If α<β then fM(α)<fM(β), so fM is an order isomorphism between otype(M) and M.

For an ordinal κ, we say M is κ-closed if for any NM such that |N|<κ, also supNM.

We say M is κ-unboundedPlanetmathPlanetmath if for any α<κ there is some βM such that α<β.

We say a function f:M𝐎𝐧 is κ-continuousMathworldPlanetmath if M is κ-closed and

f(supN)=sup{f(α)αN}

A function is κ-normal if it is order preserving (α<β implies f(α)<f(β)) and continuous. In particular, the enumerating function of a κ-closed class is always κ-normal.

All these definitions can be easily extended to all ordinals: a class is closed (resp. unbounded) if it is κ-closed (unbounded) for all κ. A function is continuous (resp. normal) if it is κ-continuous (normal) for all κ.

Titleclasses of ordinals and enumerating functions
Canonical nameClassesOfOrdinalsAndEnumeratingFunctions
Date of creation2013-03-22 13:28:55
Last modified on2013-03-22 13:28:55
Ownermathcam (2727)
Last modified bymathcam (2727)
Numerical id14
Authormathcam (2727)
Entry typeDefinition
Classificationmsc 03F15
Classificationmsc 03E10
Definesorder type
Definesenumerating function
Definesclosed
Defineskappa-closed
Definescontinuous
Defineskappa-continuous
Definescontinuous function
Defineskappa-continuous function
Definesclosed class
Defineskappa-closed class
Definesnormal function
Defineskappa-normal function
Definesnormal
Defineskappa-normal
Definesunbounded
Definesunbounded clas
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更新时间:2025/5/3 17:56:22