glossary of set theory 1

Glossary of set theory 1From Wikipedia, the free encyclopedia

Contents

1 Additively indecomposable ordinal 11.1 Multiplicatively indecomposable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 0 22.1 Ordinal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Veblen hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Surreal numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 FefermanSchtte ordinal 63.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 Lvy hierarchy 74.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.2.1 0=0=0 formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.2.2 1-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.2.3 1-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.2.4 1-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.2.5 2-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.2.6 2-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.2.7 2-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.2.8 3-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.2.9 3-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.2.10 3-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.2.11 4-formulas and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

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ii CONTENTS

4.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5 Order type 105.1 Order type of well-orderings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.2 Rational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

6 Ordinal number 126.1 Ordinals extend the natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.2 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

6.2.1 Well-ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.2.2 Denition of an ordinal as an equivalence class . . . . . . . . . . . . . . . . . . . . . . . 156.2.3 Von Neumann denition of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.2.4 Other denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

6.3 Transnite sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.4 Transnite induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

6.4.1 What is transnite induction? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.4.2 Transnite recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.4.3 Successor and limit ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.4.4 Indexing classes of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.4.5 Closed unbounded sets and classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

6.5 Arithmetic of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.6 Ordinals and cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

6.6.1 Initial ordinal of a cardinal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.6.2 Conality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

6.7 Some large countable ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.8 Topology and ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.9 Downward closed sets of ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.10 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.13 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

7 Rational number 227.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.2 Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

7.2.1 Embedding of integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.2.2 Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.2.3 Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

CONTENTS iii

7.2.4 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.2.5 Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.2.6 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.2.7 Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.2.8 Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247.2.9 Exponentiation to integer power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

7.3 Continued fraction representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247.4 Formal construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247.5 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.6 Real numbers and topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.7 p-adic numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

8 Stoneech compactication 298.1 Universal property and functoriality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298.2 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

8.2.1 Construction using products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308.2.2 Construction using the unit interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308.2.3 Construction using ultralters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308.2.4 Construction using C*-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

8.3 The Stoneech compactication of the natural numbers . . . . . . . . . . . . . . . . . . . . . . . 318.3.1 An application: the dual space of the space of bounded sequences of reals . . . . . . . . . 318.3.2 Addition on the Stoneech compactication of the naturals . . . . . . . . . . . . . . . . . 32

8.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

9 Sunower (mathematics) 349.1 lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359.2 lemma for !2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359.3 Sunower lemma and conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

10 Veblen function 3610.1 The Veblen hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

10.1.1 Fundamental sequences for the Veblen hierarchy . . . . . . . . . . . . . . . . . . . . . . 3610.1.2 The function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

10.2 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.2.1 Finitely many variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

iv CONTENTS

10.2.2 Transnitely many variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

11 set 3911.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3911.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3911.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3911.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3911.5 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 41

11.5.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4111.5.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211.5.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Chapter 1

Additively indecomposable ordinal

In set theory, a branch of mathematics, an additively indecomposable ordinal is any ordinal number that is not0 such that for any ; < , we have + < : The class of additively indecomposable ordinals (aka gammanumbers) is denoted H:From the continuity of addition in its right argument, we get that if < and is additively indecomposable, then + = :

Obviously 1 2 H , since 0 + 0 < 1: No nite ordinal other than 1 is in H: Also, ! 2 H , since the sum of two niteordinals is still nite. More generally, every innite cardinal is in H:H is closed and unbounded, so the enumerating function of H is normal. In fact, fH() = !:The derivative f 0H() (which enumerates xed points of fH) is written : Ordinals of this form (that is, xed pointsof fH ) are called epsilon numbers. The number 0 = !!

!

is therefore the rst xed point of the sequence!; !!; !!

!

; : : :

1.1 Multiplicatively indecomposableA similar notion can be dened for multiplication. The multiplicatively indecomposable ordinals (aka delta numbers)are those of the form !! for any ordinal . Every epsilon number is multiplicatively indecomposable; and everymultiplicatively indecomposable ordinal is additively indecomposable. The delta numbers are the same as the primeordinals that are limits.

1.2 See also Ordinal arithmetic

1.3 References Sierpiski, Wacaw (1958), Cardinal and ordinal numbers., Polska Akademia NaukMonograeMatematyczne34, Warsaw: Pastwowe Wydawnictwo Naukowe, MR 0095787

This article incorporates material from Additively indecomposable on PlanetMath, which is licensed under the CreativeCommons Attribution/Share-Alike License.

1

Chapter 2

0

This article is about an ordinal in mathematics. For the physical constant 0, see vacuum permittivity.

In mathematics, the epsilon numbers are a collection of transnite numbers whose dening property is that they arexed points of an exponential map. Consequently, they are not reachable from 0 via a nite series of applicationsof the chosen exponential map and of weaker operations like addition and multiplication. The original epsilonnumbers were introduced by Georg Cantor in the context of ordinal arithmetic; they are the ordinal numbers thatsatisfy the equation

" = !";

in which is the smallest innite ordinal. Any solution to this equation has Cantor normal form " = !" .The least such ordinal is 0 (pronounced epsilon nought or epsilon zero), which can be viewed as the limit obtainedby transnite recursion from a sequence of smaller limit ordinals:

"0 = !!!

= supf!; !!; !!! ; !!!!

; : : : gLarger ordinal xed points of the exponential map are indexed by ordinal subscripts, resulting in "1; "2; : : : ; "!; "!+1; : : : ; ""0 ; : : : ; ""1 ; : : : ; """ ; : : :. The ordinal 0 is still countable, as is any epsilon number whose index is countable (there exist uncountable ordinals,and uncountable epsilon numbers whose index is an uncountable ordinal).The smallest epsilon number 0 is very important in many induction proofs, because for many purposes, transniteinduction is only required up to 0 (as in Gentzens consistency proof and the proof of Goodsteins theorem). Itsuse by Gentzen to prove the consistency of Peano arithmetic, along with Gdels second incompleteness theorem,show that Peano arithmetic cannot prove the well-foundedness of this ordering (it is in fact the least ordinal with thisproperty, and as such, in proof-theoretic ordinal analysis, is used as a measure of the strength of the theory of Peanoarithmetic).Many larger epsilon numbers can be dened using the Veblen function.A more general class of epsilon numbers has been identied by John Horton Conway and Donald Knuth in the surrealnumber system, consisting of all surreals that are xed points of the base exponential map x x.Hessenberg (1906) dened gamma numbers (see additively indecomposable ordinal) to be numbers >0 such that+= whenever 1 such that =whenever 0

2.1. ORDINAL NUMBERS 3

0 = 1 ; +1 = ; = lim sup 1, the mapping 7! is a normal function, so it hasarbitrarily large xed points by the xed-point lemma for normal functions. When = ! , these xed points areprecisely the ordinal epsilon numbers. The smallest of these, , is the supremum of the sequence

0; !0 = 1; !1 = !; !!; !!!

; : : : ; ! "" k; : : :

in which every element is the image of its predecessor under the mapping 7! ! . (The general term is given usingKnuths up-arrow notation; the "" operator is equivalent to tetration.) Just as is dened as the supremum of { k} for natural numbers k, the smallest ordinal epsilon number may also be denoted ! "" ! ; this notation is muchless common than .The next epsilon number after "0 is

"1 = supf"0 + 1; !"0+1; !!"0+1 ; !!!"0+1

; : : : g;

in which the sequence is again constructed by repeated base exponentiation but starts at "0 + 1 instead of at 0.Notice

!"0+1 = !"0 !1 = "0 ! ;

!!"0+1

= !("0!) = (!"0)! = "!0 ;

!!!"0+1

= !"0!

= !"01+!

= !("0"0!) = (!"0)

"0!

= "0"0!

:

A dierent sequence with the same supremum, "1 , is obtained by starting from 0 and exponentiating with base instead:

"1 = supf0; 1; "0; "0"0 ; "0"0"0 ; : : :g;

The epsilon number "+1 indexed by any successor ordinal +1 is constructed similarly, by base exponentiationstarting from " + 1 (or by base " exponentiation starting from 0).

"+1 = supf" + 1; !"+1; !!"+1 ; : : : g = supf0; 1; "; "" ; """ ; : : : g

An epsilon number indexed by a limit ordinal is constructed dierently. The number " is the supremum of the setof epsilon numbers f" ; < g . The rst such number is "! . Whether or not the index is a limit ordinal, " isa xed point not only of base exponentiation but also of base exponentiation for all ordinals 1 < < " .Since the epsilon numbers are an unbounded subclass of the ordinal numbers, they are enumerated using the ordinalnumbers themselves. For any ordinal number , " is the least epsilon number (xed point of the exponential map)not already in the set f" ; < g . It might appear that this is the non-constructive equivalent of the constructivedenition using iterated exponentiation; but the two denitions are equally non-constructive at steps indexed by limitordinals, which represent transnite recursion of a higher order than taking the supremum of an exponential series.The following facts about epsilon numbers are very straightforward to prove:

Although it is quite a large number, "0 is still countable, being a countable union of countable ordinals; in fact," is countable if and only if is countable.

The union (or supremum) of any nonempty set of epsilon numbers is an epsilon number; so for instance

4 CHAPTER 2. 0

"! = supf"0; "1; "2; : : :g

is an epsilon number. Thus, the mapping n 7! "n is a normal function.

Every uncountable cardinal number is an epsilon number.

1 ! "! = ! :

2.2 Veblen hierarchyMain article: Veblen function

The xed points of the epsilon mapping x 7! "x form a normal function, whose xed points form a normal function,whose ; this is known as the Veblen hierarchy (the Veblen functions with base 0() = ). In the notation of theVeblen hierarchy, the epsilon mapping is 1, and its xed points are enumerated by 2.Continuing in this vein, one can dene maps for progressively larger ordinals (including, by this rareed formof transnite recursion, limit ordinals), with progressively larger least xed points (0). The least ordinal notreachable from 0 by this procedurei. e., the least ordinal for which (0)=, or equivalently the rst xed pointof the map ! (0)is the FefermanSchtte ordinal 0. In a set theory where such an ordinal can be provento exist, one has a map that enumerates the xed points 0, 1, 2, ... of ! (0) ; these are all still epsilonnumbers, as they lie in the image of for every 0, including of the map 1 that enumerates epsilon numbers.

2.3 Surreal numbersIn On Numbers and Games, the classic exposition on surreal numbers, John Horton Conway provided a number ofexamples of concepts that had natural extensions from the ordinals to the surreals. One such function is the ! -mapn 7! !n ; this mapping generalises naturally to include all surreal numbers in its domain, which in turn provides anatural generalisation of the Cantor normal form for surreal numbers.It is natural to consider any xed point of this expanded map to be an epsilon number, whether or not it happens tobe strictly an ordinal number. Some examples of non-ordinal epsilon numbers are

"1 = f0; 1; !; !!; : : : j "0 1; !"01; : : :g

and

" 12= f"0 + 1; !"0+1; : : : j "1 1; !"11; : : :g:

There is a natural way to dene "n for every surreal number n, and the map remains order-preserving. Conwaygoes on to dene a broader class of irreducible surreal numbers that includes the epsilon numbers as a particularly-interesting subclass.

2.4 See also Ordinal arithmetic Large countable ordinal

2.5. REFERENCES 5

2.5 References J.H. Conway, On Numbers and Games (1976) Academic Press ISBN 0-12-186350-6 Section XIV.20 of Sierpiski, Wacaw (1965), Cardinal and ordinal numbers (Second revised ed.), PWN Polish Scientic Publishers

2.6 External links Fusible numbers

Chapter 3

FefermanSchtte ordinal

In mathematics, the FefermanSchtte ordinal 0 is a large countable ordinal. It is the proof theoretic ordinal ofseveral mathematical theories, such as arithmetical transnite recursion. It is named after Solomon Feferman andKurt Schtte.It is sometimes said to be the rst impredicative ordinal, though this is controversial, partly because there is nogenerally accepted precise denition of predicative. Sometimes an ordinal is said to be predicative if it is less than0.Unfortunately there is no standard notation for ordinals at and beyond the FefermanSchtte ordinal, so there areseveral ways of representing it, some of which use ordinal collapsing functions: (

) , () or (0)

3.1 DenitionThe FefermanSchtte ordinal can be dened as the smallest ordinal that cannot be obtained by starting with 0 andusing the operations of ordinal addition and the Veblen functions (). That is, it is the smallest such that (0)= .

3.2 References Pohlers, Wolfram (1989), Proof theory, Lecture Notes in Mathematics 1407, Berlin: Springer-Verlag, ISBN3-540-51842-8, MR 1026933

Weaver, Nik (2005), Predicativity beyond Gamma_0, arXiv:math/0509244

6

Chapter 4

Lvy hierarchy

In set theory and mathematical logic, the Lvy hierarchy, introduced by Azriel Lvy in 1965, is a hierarchy offormulas in the formal language of the ZermeloFraenkel set theory, which is typically called just the language ofset theory. This is analogous to the arithmetical hierarchy which provides the classications but for sentences of thelanguage of arithmetic.

4.1 DenitionsIn the language of set theory, atomic formulas are of the form x = y or x y, standing for equality and respectivelyset membership predicates.The rst level of the Levy hierarchy is dened as containing only formulas with no unbounded quantiers, and isdenoted by 0 = 0 = 0 .[1] The next levels are given by nding an equivalent formula in Prenex normal form,and counting the number of changes of quantiers:In the theory ZFC, a formula A is called:[1]

i+1 if A is equivalent to 9x1:::9xnB in ZFC, where B is ii+1 if A is equivalent to 8x1:::8xnB in ZFC, where B is iIf a formula is both i and i , it is called i . As a formula might have several dierent equivalent formulas inPrenex normal form, it might belong to several dierent levels of the hierarchy. In this case, the lowest possible levelis the level of the formula.The Lvy hierarchy is sometimes dened for other theories S. In this case i and i by themselves refer only toformulas that start with a sequence of quantiers with at most i1 alternations, and Si and Si refer to formulasequivalent to i and i formulas in the theory S. So strictly speaking the levels i and i of the Lvy hierarchy forZFC dened above should be denoted by ZFCi and ZFCi .

4.2 Examples

4.2.1 0=0=0 formulas and concepts x = {y, z} x y x is a transitive set x is an ordinal, x is a limit ordinal, x is a successor ordinal x is a nite ordinal The rst countable ordinal .

7

8 CHAPTER 4. LVY HIERARCHY

f is a function. The range and domain of a function. The value of a function on a set.

The product of two sets.

The union of a set.

4.2.2 1-formulas and concepts

x is a well-founded relation on y

x is nite

Ordinal addition and multiplication and exponentiation

The rank of a set

The transitive closure of a set


x is countable

|X||Y |, |X|=|Y |

x is constructible


x is a cardinal

x is a regular cardinal

x is a limit cardinal

x is an inaccessible cardinal.

x is the powerset of y


is -supercompact


the Continuum Hypothesis

there exists an inaccessible cardinal

there exists a measurable cardinal

is an n-huge cardinal


The axiom of constructibility: V = L

4.3. PROPERTIES 9


4.2.9 3-formulas and concepts There is a supercompact cardinal

4.2.10 3-formulas and concepts is an extendible cardinal

4.2.11 4-formulas and concepts There is a extendible cardinal

4.3 PropertiesJech p. 184 Devlin p. 29

4.4 See also arithmetic hierarchy Absoluteness

4.5 References[1] Walicki, Michal (2012). Mathematical Logic, p. 225. World Scientic Publishing Co. Pte. Ltd. ISBN 9789814343862

Devlin, Keith J. (1984). Constructibility. Perspectives in Mathematical Logic. Berlin: Springer-Verlag. pp.2730. Zbl 0542.03029.

Jech, Thomas (2003). Set Theory. Springer Monographs in Mathematics (Third Millennium ed.). Berlin, NewYork: Springer-Verlag. p. 183. ISBN 978-3-540-44085-7. Zbl 1007.03002.

Kanamori, Akihiro (2006). Levy and set theory (PDF). Annals of Pure and Applied Logic 140: 233252.doi:10.1016/j.apal.2005.09.009. Zbl 1089.03004.

Levy, Azriel (1965). A hierarchy of formulas in set theory. Mem. Am. Math. Soc. 57. Zbl 0202.30502.

Chapter 5

Order type

Not to be confused with ordered types.

In mathematics, especially in set theory, two ordered sets X,Y are said to have the same order type just when theyare order isomorphic, that is, when there exists a bijection (each element matches exactly one in the other set) f: XY such that both f and its inverse are strictly increasing (order preserving i.e. the matching elements are also in thecorrect order). In the special case when X is totally ordered, monotonicity of f implies monotonicity of its inverse.For example, the set of integers and the set of even integers have the same order type, because the mapping n 7! 2npreserves the order. But the set of integers and the set of rational numbers (with the standard ordering) are not orderisomorphic, because, even though the sets are of the same size (they are both countably innite), there is no order-preserving bijective mapping between them. To these two order types we may add two more: the set of positiveintegers (which has a least element), and that of negative integers (which has a greatest element). The open interval(0,1) of rationals is order isomorphic to the rationals (since

y =2x 1

1 j2x 1j

provides a strictly increasing bijection from the former to the latter); the half-closed intervals [0,1) and (0,1], and theclosed interval [0,1], are three additional order type examples.Since order-equivalence is an equivalence relation, it partitions the class of all ordered sets into equivalence classes.

5.1 Order type of well-orderingsEvery well-ordered set is order-equivalent to exactly one ordinal number. The ordinal numbers are taken to be thecanonical representatives of their classes, and so the order type of a well-ordered set is usually identied with thecorresponding ordinal. For example, the order type of the natural numbers is .The order type of a well-ordered set V is sometimes expressed as ord(V).[1]

For example, consider the set of even ordinals less than 2+7, which is:

V = {0, 2, 4, 6, ...; , +2, +4, ...; 2, 2+2, 2+4, 2+6}.

Its order type is:

ord(V) = 2+4 = {0, 1, 2, 3, ...; , +1, +2, ...; 2, 2+1, 2+2, 2+3}.

Because there are 2 separate lists of counting and 4 in sequence at the end.

10

5.2. RATIONAL NUMBERS 11

5.2 Rational numbersAny countable totally ordered set can be mapped injectively into the rational numbers in an order-preserving way.Any dense countable totally ordered set with no highest and no lowest element can be mapped bijectively onto therational numbers in an order-preserving way.

5.3 NotationThe order type of the rationals is usually denoted . If a set S has order type , the order type of the dual of S (thereversed order) is denoted .

5.4 See also Well-order

5.5 External links Weisstein, Eric W., Order Type, MathWorld.

5.6 References[1] Ordinal Numbers and Their Arithmetic

Chapter 6

Ordinal number

This article is about the mathematical concept. For number words denoting a position in a sequence (rst, second,third, etc.), see Ordinal number (linguistics).In set theory, an ordinal number, or ordinal, is the order type of a well-ordered set. They are usually identiedwith hereditarily transitive sets. Ordinals are an extension of the natural numbers dierent from integers and fromcardinals. Like other kinds of numbers, ordinals can be added, multiplied, and exponentiated.Ordinals were introduced by Georg Cantor in 1883[1] to accommodate innite sequences and to classify derived sets,which he had previously introduced in 1872 while studying the uniqueness of trigonometric series.[2]

Two sets S and S' have the same cardinality if there is a bijection between them (i.e. there exists a function f that isboth injective and surjective, that is it maps each element x of S to a unique element y = f(x) of S' and each elementy of S' comes from exactly one such element x of S).If a partial order < is dened on set S, and a partial order

6.1. ORDINALS EXTEND THE NATURAL NUMBERS 13

0

12

3

+

1+

2+3

2

3

2+

1

2+2

4

+1

+2

+

+22

34

+

+5

4

5

+4

2

2+3

Representation of the ordinal numbers up to . Each turn of the spiral represents one power of

6.1 Ordinals extend the natural numbersA natural number (which, in this context, includes the number 0) can be used for two purposes: to describe the sizeof a set, or to describe the position of an element in a sequence. When restricted to nite sets these two conceptscoincide; there is only one way to put a nite set into a linear sequence, up to isomorphism. When dealing withinnite sets one has to distinguish between the notion of size, which leads to cardinal numbers, and the notion ofposition, which is generalized by the ordinal numbers described here. This is because, while any set has only one size(its cardinality), there are many nonisomorphic well-orderings of any innite set, as explained below.Whereas the notion of cardinal number is associated with a set with no particular structure on it, the ordinals areintimately linked with the special kind of sets that are called well-ordered (so intimately linked, in fact, that some

14 CHAPTER 6. ORDINAL NUMBER

mathematicians make no distinction between the two concepts). A well-ordered set is a totally ordered set (givenany two elements one denes a smaller and a larger one in a coherent way) in which there is no innite decreasingsequence (however, there may be innite increasing sequences); equivalently, every non-empty subset of the set hasa least element. Ordinals may be used to label the elements of any given well-ordered set (the smallest element beinglabelled 0, the one after that 1, the next one 2, and so on) and to measure the length of the whole set by the leastordinal that is not a label for an element of the set. This length is called the order type of the set.Any ordinal is dened by the set of ordinals that precede it: in fact, the most common denition of ordinals identieseach ordinal as the set of ordinals that precede it. For example, the ordinal 42 is the order type of the ordinals less thanit, i.e., the ordinals from 0 (the smallest of all ordinals) to 41 (the immediate predecessor of 42), and it is generallyidentied as the set {0,1,2,,41}. Conversely, any set (S) of ordinals that is downward-closedmeaning that forany ordinal in S and any ordinal < , is also in Sis (or can be identied with) an ordinal.There are innite ordinals as well: the smallest innite ordinal is , which is the order type of the natural numbers(nite ordinals) and that can even be identied with the set of natural numbers (indeed, the set of natural numbers iswell-orderedas is any set of ordinalsand since it is downward closed it can be identied with the ordinal associatedwith it, which is exactly how is dened).

A graphical matchstick representation of the ordinal . Each stick corresponds to an ordinal of the form m+n where m and nare natural numbers.

Perhaps a clearer intuition of ordinals can be formed by examining a rst few of them: as mentioned above, they startwith the natural numbers, 0, 1, 2, 3, 4, 5, After all natural numbers comes the rst innite ordinal, , and afterthat come +1, +2, +3, and so on. (Exactly what addition means will be dened later on: just consider them asnames.) After all of these come 2 (which is +), 2+1, 2+2, and so on, then 3, and then later on 4. Nowthe set of ordinals formed in this way (the m+n, where m and n are natural numbers) must itself have an ordinalassociated with it: and that is 2. Further on, there will be 3, then 4, and so on, and , then , and muchlater on 0 (epsilon nought) (to give a few examples of relatively smallcountableordinals). This can be continuedindenitely far (indenitely far is exactly what ordinals are good at: basically every time one says and so on whenenumerating ordinals, it denes a larger ordinal). The smallest uncountable ordinal is the set of all countable ordinals,expressed as 1.

6.2. DEFINITIONS 15

6.2 Denitions

6.2.1 Well-ordered sets

Further information: Ordered set

In a well-ordered set, every non-empty subset contains a distinct smallest element. Given the axiom of dependentchoice, this is equivalent to just saying that the set is totally ordered and there is no innite decreasing sequence,something perhaps easier to visualize. In practice, the importance of well-ordering is justied by the possibility ofapplying transnite induction, which says, essentially, that any property that passes on from the predecessors of anelement to that element itself must be true of all elements (of the given well-ordered set). If the states of a computation(computer program or game) can be well-ordered in such a way that each step is followed by a lower step, then thecomputation will terminate.It is inappropriate to distinguish between two well-ordered sets if they only dier in the labeling of their elements, ormore formally: if the elements of the rst set can be paired o the with the elements of the second set such that if oneelement is smaller than another in the rst set, then the partner of the rst element is smaller than the partner of thesecond element in the second set, and vice versa. Such a one-to-one correspondence is called an order isomorphismand the two well-ordered sets are said to be order-isomorphic, or similar (obviously this is an equivalence relation).Provided there exists an order isomorphism between two well-ordered sets, the order isomorphism is unique: thismakes it quite justiable to consider the two sets as essentially identical, and to seek a canonical representative ofthe isomorphism type (class). This is exactly what the ordinals provide, and it also provides a canonical labeling ofthe elements of any well-ordered set.Essentially, an ordinal is intended to be dened as an isomorphism class of well-ordered sets: that is, as an equivalenceclass for the equivalence relation of being order-isomorphic. There is a technical diculty involved, however, inthe fact that the equivalence class is too large to be a set in the usual ZermeloFraenkel (ZF) formalization of settheory. But this is not a serious diculty. The ordinal can be said to be the order type of any set in the class.

6.2.2 Denition of an ordinal as an equivalence class

The original denition of ordinal number, found for example in Principia Mathematica, denes the order type ofa well-ordering as the set of all well-orderings similar (order-isomorphic) to that well-ordering: in other words, anordinal number is genuinely an equivalence class of well-ordered sets. This denition must be abandoned in ZF andrelated systems of axiomatic set theory because these equivalence classes are too large to form a set. However, thisdenition still can be used in type theory and in Quines axiomatic set theory New Foundations and related systems(where it aords a rather surprising alternative solution to the Burali-Forti paradox of the largest ordinal).

6.2.3 Von Neumann denition of ordinals

Rather than dening an ordinal as an equivalence class of well-ordered sets, it will be dened as a particular well-ordered set that (canonically) represents the class. Thus, an ordinal number will be a well-ordered set; and everywell-ordered set will be order-isomorphic to exactly one ordinal number.The standard denition, suggested by John vonNeumann, is: each ordinal is the well-ordered set of all smaller ordinals.In symbols, = [0,).[3][4] Formally:

A set S is an ordinal if and only if S is strictly well-ordered with respect to set membership and everyelement of S is also a subset of S.

Note that the natural numbers are ordinals by this denition. For instance, 2 is an element of 4 = {0, 1, 2, 3}, and 2is equal to {0, 1} and so it is a subset of {0, 1, 2, 3}.It can be shown by transnite induction that every well-ordered set is order-isomorphic to exactly one of these ordinals,that is, there is an order preserving bijective function between them.Furthermore, the elements of every ordinal are ordinals themselves. Given two ordinals S and T, S is an element ofT if and only if S is a proper subset of T. Moreover, either S is an element of T, or T is an element of S, or they are


equal. So every set of ordinals is totally ordered. Further, every set of ordinals is well-ordered. This generalizes thefact that every set of natural numbers is well-ordered.Consequently, every ordinal S is a set having as elements precisely the ordinals smaller than S. For example, every setof ordinals has a supremum, the ordinal obtained by taking the union of all the ordinals in the set. This union existsregardless of the sets size, by the axiom of union.The class of all ordinals is not a set. If it were a set, one could show that it was an ordinal and thus a member of itself,which would contradict its strict ordering by membership. This is the Burali-Forti paradox. The class of all ordinalsis variously called Ord, ON, or "".An ordinal is nite if and only if the opposite order is also well-ordered, which is the case if and only if each of itssubsets has a maximum.

6.2.4 Other denitions

There are other modern formulations of the denition of ordinal. For example, assuming the axiom of regularity, thefollowing are equivalent for a set x:

x is an ordinal,

x is a transitive set, and set membership is trichotomous on x,

x is a transitive set totally ordered by set inclusion,

x is a transitive set of transitive sets.

These denitions cannot be used in non-well-founded set theories. In set theories with urelements, one has to furthermake sure that the denition excludes urelements from appearing in ordinals.

6.3 Transnite sequenceIf is a limit ordinal and X is a set, an -indexed sequence of elements of X is a function from to X. This concept,a transnite sequence or ordinal-indexed sequence, is a generalization of the concept of a sequence. An ordinarysequence corresponds to the case = .

6.4 Transnite inductionMain article: Transnite induction

6.4.1 What is transnite induction?

Transnite induction holds in any well-ordered set, but it is so important in relation to ordinals that it is worth restatinghere.

Any property that passes from the set of ordinals smaller than a given ordinal to itself, is true of allordinals.

That is, if P() is true whenever P() is true for all

6.4. TRANSFINITE INDUCTION 17

6.4.2 Transnite recursionTransnite induction can be used not only to prove things, but also to dene them. Such a denition is normallysaid to be by transnite recursion the proof that the result is well-dened uses transnite induction. Let F denote a(class) function F to be dened on the ordinals. The idea now is that, in dening F() for an unspecied ordinal ,one may assume that F() is already dened for all < and thus give a formula for F() in terms of these F(). Itthen follows by transnite induction that there is one and only one function satisfying the recursion formula up to andincluding .Here is an example of denition by transnite recursion on the ordinals (more will be given later): dene function Fby letting F() be the smallest ordinal not in the set {F() | < }, that is, the set consisting of all F() for < .This denition assumes the F() known in the very process of dening F; this apparent vicious circle is exactly whatdenition by transnite recursion permits. In fact, F(0) makes sense since there is no ordinal < 0, and the set {F()| < 0} is empty. So F(0) is equal to 0 (the smallest ordinal of all). Now that F(0) is known, the denition applied toF(1) makes sense (it is the smallest ordinal not in the singleton set {F(0)} = {0}), and so on (the and so on is exactlytransnite induction). It turns out that this example is not very exciting, since provably F() = for all ordinals ,which can be shown, precisely, by transnite induction.

6.4.3 Successor and limit ordinalsAny nonzero ordinal has the minimum element, zero. It may or may not have a maximum element. For example, 42has maximum 41 and +6 has maximum +5. On the other hand, does not have a maximum since there is nolargest natural number. If an ordinal has a maximum , then it is the next ordinal after , and it is called a successorordinal, namely the successor of , written +1. In the von Neumann denition of ordinals, the successor of is [ fg since its elements are those of and itself.[3]A nonzero ordinal that is not a successor is called a limit ordinal. One justication for this term is that a limit ordinalis indeed the limit in a topological sense of all smaller ordinals (under the order topology).When hj < i is an ordinal-indexed sequence, indexed by a limit and the sequence is increasing, i.e. < whenever < ;its limit is dened the least upper bound of the set fj < g;that is, the smallest ordinal (it alwaysexists) greater than any term of the sequence. In this sense, a limit ordinal is the limit of all smaller ordinals (indexedby itself). Put more directly, it is the supremum of the set of smaller ordinals.Another way of dening a limit ordinal is to say that is a limit ordinal if and only if:

There is an ordinal less than and whenever is an ordinal less than , then there exists an ordinal such that < < .

So in the following sequence:

0, 1, 2, ... , , +1

is a limit ordinal because for any smaller ordinal (in this example, a natural number) there is another ordinal (naturalnumber) larger than it, but still less than .Thus, every ordinal is either zero, or a successor (of a well-dened predecessor), or a limit. This distinction isimportant, because many denitions by transnite induction rely upon it. Very often, when dening a function F bytransnite induction on all ordinals, one denes F(0), and F(+1) assuming F() is dened, and then, for limit ordinals one denes F() as the limit of the F() for all


any set of ordinals is naturally indexed by the ordinals less than some . The same holds, with a slight modication,for classes of ordinals (a collection of ordinals, possibly too large to form a set, dened by some property): any classof ordinals can be indexed by ordinals (and, when the class is unbounded in the class of all ordinals, this puts it inclass-bijection with the class of all ordinals). So the -th element in the class (with the convention that the 0-this the smallest, the 1-th is the next smallest, and so on) can be freely spoken of. Formally, the denition is bytransnite induction: the -th element of the class is dened (provided it has already been dened for all < ), asthe smallest element greater than the -th element for all < .This could be applied, for example, to the class of limit ordinals: the -th ordinal, which is either a limit or zero is! (see ordinal arithmetic for the denition of multiplication of ordinals). Similarly, one can consider additivelyindecomposable ordinals (meaning a nonzero ordinal that is not the sum of two strictly smaller ordinals): the -thadditively indecomposable ordinal is indexed as ! . The technique of indexing classes of ordinals is often useful inthe context of xed points: for example, the -th ordinal such that ! = is written " . These are called the"epsilon numbers".

6.4.5 Closed unbounded sets and classesA classC of ordinals is said to be unbounded, or conal, when given any ordinal , there is a inC such that < (then the class must be a proper class, i.e., it cannot be a set). It is said to be closed when the limit of a sequenceof ordinals in the class is again in the class: or, equivalently, when the indexing (class-)function F is continuous inthe sense that, for a limit ordinal, F () (the -th ordinal in the class) is the limit of all F () for < ; this isalso the same as being closed, in the topological sense, for the order topology (to avoid talking of topology on properclasses, one can demand that the intersection of the class with any given ordinal is closed for the order topology onthat ordinal, this is again equivalent).Of particular importance are those classes of ordinals that are closed and unbounded, sometimes called clubs. Forexample, the class of all limit ordinals is closed and unbounded: this translates the fact that there is always a limitordinal greater than a given ordinal, and that a limit of limit ordinals is a limit ordinal (a fortunate fact if the termi-nology is to make any sense at all!). The class of additively indecomposable ordinals, or the class of " ordinals, orthe class of cardinals, are all closed unbounded; the set of regular cardinals, however, is unbounded but not closed,and any nite set of ordinals is closed but not unbounded.A class is stationary if it has a nonempty intersection with every closed unbounded class. All superclasses of closedunbounded classes are stationary, and stationary classes are unbounded, but there are stationary classes that are notclosed and stationary classes that have no closed unbounded subclass (such as the class of all limit ordinals withcountable conality). Since the intersection of two closed unbounded classes is closed and unbounded, the intersectionof a stationary class and a closed unbounded class is stationary. But the intersection of two stationary classes may beempty, e.g. the class of ordinals with conality with the class of ordinals with uncountable conality.Rather than formulating these denitions for (proper) classes of ordinals, one can formulate them for sets of ordinalsbelow a given ordinal : A subset of a limit ordinal is said to be unbounded (or conal) under provided anyordinal less than is less than some ordinal in the set. More generally, we can call a subset of any ordinal conalin provided every ordinal less than is less than or equal to some ordinal in the set. The subset is said to be closedunder provided it is closed for the order topology in , i.e. a limit of ordinals in the set is either in the set or equalto itself.

6.5 Arithmetic of ordinalsMain article: Ordinal arithmetic

There are three usual operations on ordinals: addition, multiplication, and (ordinal) exponentiation. Each can be de-ned in essentially two dierent ways: either by constructing an explicit well-ordered set that represents the operationor by using transnite recursion. Cantor normal form provides a standardized way of writing ordinals. The so-callednatural arithmetical operations retain commutativity at the expense of continuity.

6.6 Ordinals and cardinals

6.7. SOME LARGE COUNTABLE ORDINALS 19

6.6.1 Initial ordinal of a cardinal

Each ordinal has an associated cardinal, its cardinality, obtained by simply forgetting the order. Any well-orderedset having that ordinal as its order-type has the same cardinality. The smallest ordinal having a given cardinal as itscardinality is called the initial ordinal of that cardinal. Every nite ordinal (natural number) is initial, but most inniteordinals are not initial. The axiom of choice is equivalent to the statement that every set can be well-ordered, i.e.that every cardinal has an initial ordinal. In this case, it is traditional to identify the cardinal number with its initialordinal, and we say that the initial ordinal is a cardinal.Cantor used the cardinality to partition ordinals into classes. He referred to the natural numbers as the rst numberclass, the ordinals with cardinality @0 (the countably innite ordinals) as the second number class and generally,the ordinals with cardinality @n2 as the n-th number class.[5]The -th innite initial ordinal is written ! . Its cardinality is written @ . For example, the cardinality of 0 = is @0 , which is also the cardinality of 2 or 0 (all are countable ordinals). So (assuming the axiom of choice) weidentify with @0 , except that the notation @0 is used when writing cardinals, and when writing ordinals (this isimportant since, for example, @20 = @0 whereas !2 > ! ). Also, !1 is the smallest uncountable ordinal (to see thatit exists, consider the set of equivalence classes of well-orderings of the natural numbers: each such well-orderingdenes a countable ordinal, and !1 is the order type of that set), !2 is the smallest ordinal whose cardinality is greaterthan @1 , and so on, and !! is the limit of the !n for natural numbers n (any limit of cardinals is a cardinal, so thislimit is indeed the rst cardinal after all the !n ).See also Von Neumann cardinal assignment.

6.6.2 Conality

The conality of an ordinal is the smallest ordinal that is the order type of a conal subset of . Notice that anumber of authors dene conality or use it only for limit ordinals. The conality of a set of ordinals or any otherwell-ordered set is the conality of the order type of that set.Thus for a limit ordinal, there exists a -indexed strictly increasing sequence with limit . For example, the conalityof is , because the sequence m (where m ranges over the natural numbers) tends to ; but, more generally,any countable limit ordinal has conality . An uncountable limit ordinal may have either conality as does !! oran uncountable conality.The conality of 0 is 0. And the conality of any successor ordinal is 1. The conality of any limit ordinal is at least! .An ordinal that is equal to its conality is called regular and it is always an initial ordinal. Any limit of regular ordinalsis a limit of initial ordinals and thus is also initial even if it is not regular, which it usually is not. If the Axiom ofChoice, then !+1 is regular for each . In this case, the ordinals 0, 1, ! , !1 , and !2 are regular, whereas 2, 3, !!, and are initial ordinals that are not regular.The conality of any ordinal is a regular ordinal, i.e. the conality of the conality of is the same as the conalityof . So the conality operation is idempotent.

6.7 Some large countable ordinalsFor more details on this topic, see Large countable ordinal.

We have already mentioned (see Cantor normal form) the ordinal 0, which is the smallest satisfying the equation! = , so it is the limit of the sequence 0, 1, ! , !! , !!! , etc. Many ordinals can be dened in such a manneras xed points of certain ordinal functions (the -th ordinal such that ! = is called " , then we could go ontrying to nd the -th ordinal such that " = , and so on, but all the subtlety lies in the and so on). We cantry to do this systematically, but no matter what system is used to dene and construct ordinals, there is always anordinal that lies just above all the ordinals constructed by the system. Perhaps the most important ordinal that limitsa system of construction in this manner is the ChurchKleene ordinal, !CK1 (despite the !1 in the name, this ordinalis countable), which is the smallest ordinal that cannot in any way be represented by a computable function (this canbe made rigorous, of course). Considerably large ordinals can be dened below !CK1 , however, which measure the


proof-theoretic strength of certain formal systems (for example, "0 measures the strength of Peano arithmetic).Large ordinals can also be dened above the Church-Kleene ordinal, which are of interest in various parts of logic.

6.8 Topology and ordinalsFor more details on this topic, see Order topology.

Any ordinal can be made into a topological space in a natural way by endowing it with the order topology. See theTopology and ordinals section of the Order topology article.

6.9 Downward closed sets of ordinalsA set is downward closed if anything less than an element of the set is also in the set. If a set of ordinals is downwardclosed, then that set is an ordinalthe least ordinal not in the set.Examples:

The set of ordinals less than 3 is 3 = { 0, 1, 2 }, the smallest ordinal not less than 3. The set of nite ordinals is innite, the smallest innite ordinal: . The set of countable ordinals is uncountable, the smallest uncountable ordinal: 1.

6.10 See also Counting Ordinal space

6.11 Notes[1] Thorough introductions are given by Levy (1979) and Jech (2003).

[2] Hallett, Michael (1979), Towards a theory ofmathematical research programmes. I, The British Journal for the Philosophyof Science 30 (1): 125, doi:10.1093/bjps/30.1.1, MR 532548. See the footnote on p. 12.

[3] von Neumann 1923

[4] Levy (1979, p. 52) attributes the idea to unpublished work of Zermelo in 1916 and several papers by von Neumann the1920s.

[5] Dauben (1990:97)

6.12 References Cantor, G., (1897), Beitrage zur Begrundung der transnitenMengenlehre. II (tr.: Contributions to the Foundingof the Theory of Transnite Numbers II), Mathematische Annalen 49, 207-246 English translation.

Conway, J. H. and Guy, R. K. Cantors Ordinal Numbers. In The Book of Numbers. New York: Springer-Verlag, pp. 266267 and 274, 1996.

Dauben, Joseph Warren, (1990), Georg Cantor: his mathematics and philosophy of the innite. Chapter 5:The Mathematics of Cantors Grundlagen. ISBN 0-691-02447-2

6.13. EXTERNAL LINKS 21

Hamilton, A. G. (1982), Numbers, Sets, and Axioms : the Apparatus of Mathematics, New York: CambridgeUniversity Press, ISBN 0-521-24509-5 See Ch. 6, Ordinal and cardinal numbers

Kanamori, A., Set Theory from Cantor to Cohen, to appear in: Andrew Irvine and John H. Woods (editors),The Handbook of the Philosophy of Science, volume 4, Mathematics, Cambridge University Press.

Levy, A. (1979), Basic Set Theory, Berlin, New York: Springer-Verlag Reprinted 2002, Dover. ISBN 0-486-42079-5

Jech, Thomas (2003), Set Theory, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag Sierpiski, W. (1965). Cardinal and Ordinal Numbers (2nd ed.). Warszawa: Pastwowe WydawnictwoNaukowe. Also denes ordinal operations in terms of the Cantor Normal Form.

Suppes, P. (1960), Axiomatic Set Theory, D.Van Nostrand Company Inc., ISBN 0-486-61630-4 von Neumann, Johann (1923), Zur Einfhrung der trasniten Zahlen, Acta litterarum ac scientiarum RagiaeUniversitatis Hungaricae Francisco-Josephinae, Sectio scientiarum mathematicarum 1: 199208

von Neumann, John (January 2002) [1923], On the introduction of transnite numbers, in Jean van Hei-jenoort, From Frege to Gdel: A Source Book in Mathematical Logic, 1879-1931 (3rd ed.), Harvard UniversityPress, pp. 346354, ISBN 0-674-32449-8 - English translation of von Neumann 1923.

6.13 External links Hazewinkel, Michiel, ed. (2001), Ordinal number, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Weisstein, Eric W., Ordinal Number, MathWorld. Ordinals at ProvenMath Beitraege zur Begruendung der transniten Mengenlehre Cantors original paper published in MathematischeAnnalen 49(2), 1897

Ordinal calculator GPL'd free software for computing with ordinals and ordinal notations Chapter 4 of Don Monks lecture notes on set theory is an introduction to ordinals.

Chapter 7

Rational number

Rationals redirects here. For other uses, see Rational (disambiguation).

In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers,p and q, with the denominator q not equal to zero.[1] Since q may be equal to 1, every integer is a rational number.The set of all rational numbers is usually denoted by a boldface Q (or blackboard bold Q , Unicode );[2] it was thusdenoted in 1895 by Peano after quoziente, Italian for "quotient".The decimal expansion of a rational number always either terminates after a nite number of digits or begins torepeat the same nite sequence of digits over and over. Moreover, any repeating or terminating decimal representsa rational number. These statements hold true not just for base 10, but also for any other integer base (e.g. binary,hexadecimal).A real number that is not rational is called irrational. Irrational numbers include 2, , e, and . The decimalexpansion of an irrational number continues without repeating. Since the set of rational numbers is countable, andthe set of real numbers is uncountable, almost all real numbers are irrational.[1]

The rational numbers can be formally dened as the equivalence classes of the quotient set (Z (Z \ {0})) / ~, wherethe cartesian product Z (Z \ {0}) is the set of all ordered pairs (m,n) where m and n are integers, n is not 0 (n 0),and "~" is the equivalence relation dened by (m1,n1) ~ (m2,n2) if, and only if, m1n2 m2n1 = 0.In abstract algebra, the rational numbers together with certain operations of addition and multiplication form thearchetypical eld of characteristic zero. As such, it is characterized as having no proper subeld or, alternatively,being the eld of fractions for the ring of integers. Finite extensions of Q are called algebraic number elds, and thealgebraic closure of Q is the eld of algebraic numbers.[3]

In mathematical analysis, the rational numbers form a dense subset of the real numbers. The real numbers can beconstructed from the rational numbers by completion, using Cauchy sequences, Dedekind cuts, or innite decimals.Zero divided by any other integer equals zero; therefore, zero is a rational number (but division by zero is undened).

7.1 TerminologyThe term rational in reference to the set Q refers to the fact that a rational number represents a ratio of two integers.In mathematics, the adjective rational often means that the underlying eld considered is the eld Q of rationalnumbers. Rational polynomial usually, and most correctly, means a polynomial with rational coecients, also calleda polynomial over the rationals. However, rational function does not mean the underlying eld is the rationalnumbers, and a rational algebraic curve is not an algebraic curve with rational coecients.

7.2 ArithmeticSee also: Fraction (mathematics) Arithmetic with fractions

22

7.2. ARITHMETIC 23

7.2.1 Embedding of integersAny integer n can be expressed as the rational number n/1.

7.2.2 Equalityab =

cd if and only if ad = bc:

7.2.3 OrderingWhere both denominators are positive:

ab 0 ^ m1n2 n1m2) _ (n1n2 < 0 ^ m1n2 n1m2):The integers may be considered to be rational numbers by the embedding that maps m to [(m,1)].

26 CHAPTER 7. RATIONAL NUMBER

7.5 Properties

2/1

3/1

4/1

1/1

5/1

6/1

7/1

8/1

1/2 1/3 1/4 1/5 1/6 1/7 1/8

2/2

3/2

4/2

5/2

6/2

8/2

7/2

3/3

2/3

4/3

5/3

6/3

7/3

8/3

2/4

3/4

4/4

5/4

6/4

7/4

8/4

2/5

3/5

4/5

5/5

6/5

7/5

8/5

2/6

3/6

4/6

5/6

6/6

7/6

8/6

2/7

3/7

4/7

5/7

6/7

7/7

8/7

2/8

3/8

4/8

5/8

6/8

7/8

8/8 ...

...

...

...

...

...

...

... ............... ... ...

...

...

A diagram illustrating the countability of the rationals

The set Q, together with the addition and multiplication operations shown above, forms a eld, the eld of fractionsof the integers Z.The rationals are the smallest eld with characteristic zero: every other eld of characteristic zero contains a copy ofQ. The rational numbers are therefore the prime eld for characteristic zero.The algebraic closure of Q, i.e. the eld of roots of rational polynomials, is the algebraic numbers.The set of all rational numbers is countable. Since the set of all real numbers is uncountable, we say that almost allreal numbers are irrational, in the sense of Lebesgue measure, i.e. the set of rational numbers is a null set.The rationals are a densely ordered set: between any two rationals, there sits another one, and, therefore, innitelymany other ones. For example, for any two fractions such that

a

b, ordered lexicographically. Then X is a set. The union of all these sets is the class ofsurreal numbers.A dense totally ordered set without endpoints is a set if and only if it is saturated.

11.3 PropertiesAny setX is universal for totally ordered sets of cardinality at most , meaning that any such set can be embeddedinto X.For any given ordinal , any two sets of cardinality are isomorphic (as ordered sets). An set of cardinality exists if is regular and

40 CHAPTER 11. SET

Hausdor (1907), Untersuchungen ber Ordnungstypen V, Ber. ber die Verhandlungen der Knigl. Schs.Ges. der Wiss. zu Leipzig. Math.-phys. Klasse 59: 105159 English translation in Hausdor (2005)

Hausdor, F. (1914), Grundzge der Mengenlehre, Leipzig: Veit & Co Hausdor, Felix (2005), Plotkin, J. M., ed.,Hausdor on ordered sets, History ofMathematics 25, Providence,RI: American Mathematical Society, ISBN 0-8218-3788-5, MR 2187098

11.5. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 41

11.5 Text and image sources, contributors, and licenses11.5.1 Text

Additively indecomposable ordinal Source: https://en.wikipedia.org/wiki/Additively_indecomposable_ordinal?oldid=617618247Con-tributors: Michael Hardy, Rich Farmbrough, R.e.b., JRSpriggs, CBM, David Eppstein, Hans Adler, Mathemens, Yobot, 777sms andAnonymous: 1

Epsilon numbers (mathematics) Source: https://en.wikipedia.org/wiki/Epsilon_numbers_(mathematics)?oldid=665941340 Contribu-tors: AxelBoldt, Zundark, MarXidad, The Anome, Patrick, Michael Hardy, Stevenj, ThomasStrohmann~enwiki, Fropu, Rich Farm-brough, Paul August, Blotwell, Arthena, Sligocki, Oleg Alexandrov, Linas, Justinlebar, BD2412, Jodihamann, R.e.b., Trovatore, ArthurRubin, SmackBot, Stie, Bluebot, Vanished User 0001, Lambiam, ArglebargleIV, Mr Death, Deadcode, Gen.Sec., JRSpriggs, Sninoy,RobHar, DWorley, Michael K. Edwards, YohanN7, Juqi, Addbot, Yobot, Omnipaedista, VladimirReshetnikov, AlreadyDone, Set theorist,BlissfulSerenity, D.Lazard, KLBot2, LegionMammal978, Feynmaniac and Anonymous: 16

FefermanSchtte ordinal Source: https://en.wikipedia.org/wiki/Feferman%E2%80%93Sch%C3%BCtte_ordinal?oldid=663848943Contributors: Gro-Tsen, R.e.b., JRSpriggs, Headbomb,MystBot, Addbot, Youre dreaming eh?, Yobot, ZroBot, Andyhowlett, DrCeasiumand Anonymous: 2

Lvy hierarchy Source: https://en.wikipedia.org/wiki/L%C3%A9vy_hierarchy?oldid=657387428Contributors: Michael Hardy, Bearcat,Tsirel, Rjwilmsi, R.e.b., JRSpriggs, Snalin, Magioladitis, Minimiscience, Biscuittin, SchreiberBike, Yobot, Xqbot, Tijfo098, Catrincm,Deltahedron, Mark viking and Anonymous: 1

Order type Source: https://en.wikipedia.org/wiki/Order_type?oldid=617119064 Contributors: The Anome, Patrick, Dominus, TobiasBergemann, Dbenbenn, Francis Davey, Arthena, Kazvorpal, Mike Segal, SmackBot, Melchoir, BiT, Mhss, Schaef, Octahedron80, Lam-biam, JRSpriggs, Vaughan Pratt, Pjoef, Iamthedeus, Hccrle, DumZiBoT, Addbot, Yobot, Pelotom, EmausBot, Makecat-bot, JamesWaddington and Anonymous: 11

Ordinal number Source: https://en.wikipedia.org/wiki/Ordinal_number?oldid=668709653 Contributors: AxelBoldt, Mav, Bryan Derk-sen, Zundark, The Anome, Iwnbap, LA2, Christian List, B4hand, Olivier, Stevertigo, Patrick, Michael Hardy, Llywrch, Jketola, Chinju,TakuyaMurata, Karada, Docu, Vargenau, Revolver, Charles Matthews, Dysprosia, Malcohol, Owen, Rogper~enwiki, Hmackiernan, Bald-hur, Adhemar, Fuelbottle, Tobias Bergemann, Giftlite, Markus Krtzsch, var Arnfjr Bjarmason, Lethe, Fropu, Gro-Tsen, That-tommyhall, Jorend, Siroxo, Wmahan, Beland, Joeblakesley, Elroch, 4pq1injbok, Luqui, Silence, Paul August, EmilJ, Babomb, RandallHolmes, Wood Thrush, Robotje, Blotwell, Crust, Jumbuck, Sligocki, SidP, DV8 2XL, Jim Slim, Oleg Alexandrov, Warbola, Linas, MiaowMiaow, Graham87, Grammarbot, Jorunn, Rjwilmsi, Bremen, Salix alba, Mike Segal, R.e.b., FlaBot, Jak123, Chobot, YurikBot, Wave-length, RobotE, Hairy Dude, CanadianCaesar, Archelon, Gaius Cornelius, Trovatore, Crasshopper, DeadEyeArrow, Pooryorick~enwiki,Hirak 99, Closedmouth, Arthur Rubin, PhS, GrinBot~enwiki, Brentt, Nicholas Jackson, SmackBot, Pokipsy76, KocjoBot~enwiki, Blue-bot, AlephNull~enwiki, Jiddisch~enwiki, Dreadstar, Mmehdi.g, Lambiam, Khazar, Minna Sora no Shita, Bjankuloski06en~enwiki, 16@r,Loadmaster, Limaner, Quaeler, Jason.grossman, Joseph Solis in Australia, Easwaran, Zero sharp, Tawkerbot2, JRSpriggs, Vaughan Pratt,CRGreathouse, CBM, Gregbard, FilipeS, HdZ, Pcu123456789, Lyondif02, Odoncaoa, Jj137, Hannes Eder, Shlomi Hillel, JAnDbot,Agol, BrentG, Smartcat, Bongwarrior, Swpb, David Eppstein, Jondaman21, R'n'B, IPonomarev, RockMFR, Ttwo, It IsMeHere, Policron,VolkovBot, Dommedagsprofet, Hotfeba, Je G., LokiClock, PMajer, Alphaios~enwiki, Cremepu222, Wikithesource, Arcfrk, SieBot,Mrw7, J-puppy, TheCatalyst31, ClueBot, DFRussia, DanielDeibler, DragonBot, Hans Adler, StevenDH, Lacce, Against the current,Dthomsen8, Addbot, Dyaa, Mathemens, Unzerlegbarkeit, Luckas-bot, Yobot, Utvik old, THEN WHO WAS PHONE?, KamikazeBot,AnomieBOT, Angry bee, Citation bot, Nexx892, Twri, Xqbot, Freebirth Toad, Capricorn42, RJGray, GrouchoBot, VladimirReshetnikov,SassoBot, Citation bot 1, RedBot, Burritoburritoburrito, TheStrayCat, Raiden09, EmausBot, Fly by Night, Jens Blanck, SporkBot, Clue-Bot NG, Frietjes, Rezabot, Helpful Pixie Bot, BG19bot, Anthony.de.almeida.lopes, Jochen Burghardt, Mark viking, Pop-up casket, JoseBrox, The Horn Blower, Dustin V. S., George8211, Dconman2, Gareld Gareld, Lalaloopsy1234, SoSivr, Eth450, Neposner, MirceaBRT, Divad42, Smwrd, Wilsonator5000 and Anonymous: 139

Rational number Source: https://en.wikipedia.org/wiki/Rational_number?oldid=669894454 Contributors: AxelBoldt, Brion VIBBER,Bryan Derksen, Zundark, Tarquin, Jan Hidders, Andre Engels, XJaM, Christian List, Toby Bartels, PierreAbbat, Roadrunner, FvdP, Stev-ertigo, Patrick, Michael Hardy, Wshun, MartinHarper, Ixfd64, TakuyaMurata, Mdebets, Ciphergoth, AugPi, Andres, Evercat, Panoramix,Pizza Puzzle, Hashar, Hawthorn, Revolver, Charles Matthews, Dcoetzee, Dysprosia, Jitse Niesen, Greenrd, Hyacinth, Thue, McKay,Guppy, Denelson83, Robbot, Romanm,Mayooranathan, Thunderbolt16, Henrygb, Borislav, Fuelbottle, Lupo, Tobias Bergemann, Giftlite,Gene Ward Smith, var Arnfjr Bjarmason, Lethe, Dissident, Fropu, Dratman, Guanaco, Bovlb, Jason Quinn, Jorge Stol, Nayuki,Tagishsimon, Rheun, Ato, Antandrus, MarkSweep, Bob.v.R, Vina, Scott Burley, Ehamberg, Lostchicken, Mormegil, Discospinster,Notinasnaid, Paul August, El C, EmilJ, Deanos, Bobo192, Elipongo, Jung dalglish, Blotwell, Deryck Chan, Obradovic Goran, Jum-buck, Msh210, Alansohn, Ncik~enwiki, Silver86, Wtmitchell, Velella, L33th4x0rguy, Mikeo, Btornado, Oleg Alexandrov, Linas, Stradi-variusTV, Prashanthns, Graham87, BD2412, SixWingedSeraph, Yurik, Zzedar, Jshadias, Josh Parris, Sdornan, Salix alba, The wub,Bhadani, Yamamoto Ichiro, David H Braun (1964), DVdm, YurikBot, Wavelength, Stephenb, Pseudomonas, NawlinWiki, Rick Nor-wood, E rulez, Hennobrandsma, Charlie Wiederhold, Lt-wiki-bot, Gesslein, GrinBot~enwiki, Crystallina, Hydrogen Iodide, Zerida, Un-yoyega, Yamaguchi, Gilliam, Skizzik, Persian Poet Gal, Raymond arritt, Raja Hussain, MalafayaBot, Akanemoto, DHN-bot~enwiki,Can't sleep, clown will eat me, Shunpiker, Grover cleveland, Daqu, Nakon, Dreadstar, NickPenguin, Salamurai, Bidabadi~enwiki, Vina-iwbot~enwiki, Ck lostsword, SashatoBot, Lambiam, Nishkid64, Btritchie, Kuru, CorvetteZ51, Cronholm144, Gobonobo, Jim.belk,Ekrub-ntyh, Loadmaster, Dr Greg, Mets501, Lee Carre, Quaeler, Jazzcello, Majora4, ILikeThings, JForget, CRGreathouse, Wafulz,Penbat, SuperMidget, Cydebot, Worthingtonse, Boardhead, Epbr123, Koeplinger, Martin Hogbin, Marek69, Wmasterj, AbcXyz, Escar-bot, Ju66l3r, AntiVandalBot, Vvidetta, Edokter, Braindrain0000, JAnDbot, MER-C, Smiddle, .anacondabot, Connormah, Bongwarrior,VoABot II, Twsx, WODUP, Avicennasis, Seberle, JoergenB, DerHexer, GermanX, Aschmitz, Tokidoki27, MartinBot, Kostisl, Jarhed,J.delanoy, Katalaveno, Stwitzel, SJP, Policron, CompuChip, Angular, DavidCBryant, Wikieditor06, VolkovBot, Johan1298~enwiki,TallNapoleon, AlnoktaBOT, VasilievVV, TXiKiBoT, Maximillion Pegasus, Dendodge, Broadbot, Hrundi Bakshi, Maxim, Wolfrock,Synthebot, Allan1114, Monty845, AlleborgoBot, LuigiManiac, EmxBot, Omerks, Demmy, Bfpage, SieBot, Legion , Yintan, Ben-togoa, Flyer22, Tiptoety, Macy, OKBot, Diego Grez, Angielaj, Randomblue, Dolphin51, Troy 07, Joe Photon, ClueBot, Rumping,PipepBot, Fyyer, The Thing That Should Not Be, Cli, Jpcs, Dylan620, Paritybit, Jusdafax, Eeekster, Cenarium, Jotterbot, Aitias,Katanada, SoxBot III, Crazy Boris with a red beard, AgnosticPreachersKid, Spitre, Crapme, Zrs 12, NellieBly, Dnvrfantj, Ryan-Cross, HexaChord, Addbot, Proofreader77, ConCompS, Friginator, Ronhjones, TutterMouse, Fieldday-sunday, Skyezx, LaaknorBot,

42 CHAPTER 11. SET

LinkFA-Bot, Jaydec, AgadaUrbanit, Numbo3-bot, Tide rolls, MZaplotnik, Teles, LuK3, Luckas-bot, Yobot, Tempodivalse, Democrat-icLuntz, 9258fahskh917fas, Kingpin13, Materialscientist, Erikekahn, OllieFury, Xelnx, ArthurBot, Xqbot, TinucherianBot II, Capri-corn42, Doctor Rosenberg, Isheden, Hackabhihack, RibotBOT, The Wiki Octopus, Aaron Kauppi, Stlrams22, Lothar von Richthofen,DivineAlpha, Tkuvho, Pinethicket, ShadowRangerRIT, I dream of horses, Adlerbot, MarcelB612, BigDwiki, Jujutacular, Reconsiderthe static, Tim1357, ItsZippy, Jonkerz, Dinamik-bot, Vrenator, JuanGabrielRobalino, Stroppolo, Luhar1997, TjBot, Bento00, TomT0m,EmausBot, Khalidmathematics, Slightsmile, TuHan-Bot, Wikipelli, ReySquared, Alpha Quadrant, Quondum, Mburdis, Wayne Slam, LKensington, Donner60, Chewings72, Pun, Orange Suede Sofa, 28bot, ClueBot NG, This lousy T-shirt, MikuMikuCookie, Dfarrell07,Mpaa, Ichliebepferde, Spel-Punc-Gram, Kasirbot, Widr, WikiPuppies, Calabe1992, Kelsi1122, Sheilds, Darksonn, Ihatechickens214,Mercrutio, Nick white03, Glacialfox, Bodema, Kishugoyal, Teammm, Pratyya Ghosh, Dexbot, FoCuSandLeArN, Mcash001, Scha-ran09, Lugia2453, Epicgenius, Encyclopedia 12, Harry styles5555554, Imperial Marshmallow, AmaryllisGardener, Maxtheaxe1999,Ray Lightyear, Zenibus, JDiala, Wellset, Wikiwonka7777, Moorelife, Programmer 112, Wilson Widyadhana, Anirudh Babu, ChamithN,DemonKiller3527, Engilukol albert and Anonymous: 595

Stoneech compactication Source: https://en.wikipedia.org/wiki/Stone%E2%80%93%C4%8Cech_compactification?oldid=644791321Contributors: AxelBoldt, Toby Bartels, Michael Hardy, Tosha, Giftlite, Fropu, Jason Quinn, DemonThing, DRMacIver, Paul August,EmilJ, Linas, Rjwilmsi, R.e.b., YurikBot, RussBot, Trovatore, Kompik, Commander Keane bot, Chris the speller, Bluebot, Nbarth,Stotr~enwiki, JoeBot, CmdrObot, J L, KennyDC, Jan stary, Nhindman, Tomaxer, AlexeyMuranov, Addbot, Legobot, Luckas-bot, Yobot,AnomieBOT, Citation bot, Xqbot, Freebirth Toad, Omnipaedista, Citation bot 1, Le Docteur, EmausBot, ZroBot, SporkBot, Uni.Liu,Helpful Pixie Bot, Kephir, Mark viking, GreenKeeper17, SoSivr and Anonymous: 23

Sunower (mathematics) Source: https://en.wikipedia.org/wiki/Sunflower_(mathematics)?oldid=671273168Contributors: Michael Hardy,Charles Matthews, David Haslam, R.e.b., Jason22~enwiki, Cydebot, David Eppstein, VolkovBot, Rei-bot, Hans Adler, Addbot, Luckas-bot, Yobot, Citation bot, Trappist the monk, EmausBot, Set theorist, ZroBot and Anonymous: 4

Veblen function Source: https://en.wikipedia.org/wiki/Veblen_function?oldid=664823666 Contributors: Tobias Bergemann, Giftlite,Gro-Tsen, Sligocki, Rjwilmsi, R.e.b., Dicklyon, Zero sharp, JRSpriggs, Headbomb, Synthebot, Cheesefondue, Addbot, Citation bot,ZroBot, ClueBot NG, SuperJedi224 and Anonymous: 6

set Source: https://en.wikipedia.org/wiki/%CE%97_set?oldid=631228645 Contributors: Anthony Appleyard, R.e.b., Magioladitis andYobot

11.5.2 Images File:Acap.svg Source: https://upload.wikimedia.org/wikipedia/commons/5/52/Acap.svg License: Public domain Contributors: Own

work Original artist: F l a n k e r File:CardContin.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/75/CardContin.svg License: Public domain Contrib-

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11.5.3 Content license Creative Commons Attribution-Share Alike 3.0

Additively indecomposable ordinalMultiplicatively indecomposable See also References

0Ordinal numbers Veblen hierarchySurreal numbers See also ReferencesExternal links

FefermanSchtte ordinalDefinitionReferences

Lvy hierarchyDefinitionsExamples 0=0=0 formulas and concepts1-formulas and concepts1-formulas and concepts1-formulas and concepts2-formulas and concepts2-formulas and concepts2-formulas and concepts3-formulas and concepts3-formulas and concepts3-formulas and concepts4-formulas and concepts

Properties See also References

Order typeOrder type of well-orderingsRational numbersNotation See alsoExternal links References

Ordinal numberOrdinals extend the natural numbers Definitions Well-ordered sets Definition of an ordinal as an equivalence class Von Neumann definition of ordinals Other definitions

Transfinite sequenceTransfinite induction What is transfinite induction? Transfinite recursion Successor and limit ordinals Indexing classes of ordinals Closed unbounded sets and classes

Arithmetic of ordinals Ordinals and cardinals Initial ordinal of a cardinal Cofinality

Some large countable ordinals Topology and ordinals Downward closed sets of ordinalsSee alsoNotesReferencesExternal links

Rational numberTerminologyArithmeticEmbedding of integersEqualityOrderingAdditionSubtractionMultiplicationDivisionInverseExponentiation to integer power

Continued fraction representationFormal constructionPropertiesReal numbers and topological propertiesp-adic numbersSee alsoReferencesExternal links

Stoneech compactificationUniversal property and functoriality ConstructionsConstruction using productsConstruction using the unit intervalConstruction using ultrafiltersConstruction using C*-algebras

The Stoneech compactification of the natural numbersAn application: the dual space of the space of bounded sequences of reals Addition on the Stoneech compactification of the naturals

See alsoNotes References External links

Sunflower (mathematics) lemma lemma for 2 Sunflower lemma and conjectureReferences

Veblen functionThe Veblen hierarchy Fundamental sequences for the Veblen hierarchy The function

GeneralizationsFinitely many variablesTransfinitely many variables

References

setDefinitionExamplesPropertiesReferencesText and image sources, contributors, and licensesTextImagesContent license

glossary of set theory 1

Documents

arithmetic of ordinals

limit ordinals

external links

closed sets of ordinals

neumann denition of

indecomposable ordinal

natural numbers

rational numbers