Ordinal number

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Ordinal numbers from 0 to ω ω

Ordinal numbers are mathematical objects that generalize the concept of the position or index of an element in a sequence to well-orders over arbitrary sets . Positions in sequences are understood as natural numbers (expressed linguistically by the ordinalia first , second , third , ... element), which form the finite ordinal numbers. The decisive factor in this generalization is that, as with sequences, there is a smallest position (the ordinal number zero ) and each element (with the exception of a possibly existing last element) has a unique successor. Since total arrangements that fulfill these conditions can still have very different structures, one introduces as an additional condition that there should be a minimal index for every non-empty subset of indices, and so we arrive at well-orders.

Ordinal numbers allow the generalization of the proof methods of complete induction , which are restricted to sequences, to arbitrarily large sets or even real classes , provided that they can be well ordered, by means of the method of transfinite induction .

The description of the size of a set, naively speaking the number of its elements, leads in contrast to the term cardinal number ( one , two , three , ...).

Georg Cantor had the idea of ​​how the two concepts - number as quantity and number as index - can be generalized to infinite sets within set theory ; for while they agree for finite sets, one has to distinguish them for infinite sets. Cardinal numbers are defined as special ordinal numbers. The totality of ordinal numbers, which are usually referred to as or , in modern set theory - just like the totality of cardinal numbers - does not form a set, but a real class .

For many of these considerations (such as transfinite induction and the definition of cardinal numbers as ordinals), the axiom of choice or the equivalent order theorem is necessary.

Ordinal numbers are of particular importance for set theory, in other areas of mathematics other generalized indexes are also used, for example in networks and filters , which are of particular importance for topology and operate over other orders as well-orders. In particular, these generalized indexes, in contrast to ordinals, generalize the concept of convergence, which is important for sequences .

Story of discovery

The modern mathematical concept of ordinal numbers was largely developed by the mathematician Georg Cantor. He found the basic idea while investigating the uniqueness of the representation of real functions by trigonometric series. However, the ordinal number theory ultimately did not prove to be fruitful for these investigations.

It was known from preliminary work by Eduard Heine that the functions which are continuous in the interval have a clear representation as trigonometric series. Cantor showed (1870) that this is correct for every function whose trigonometric series converges everywhere . The question of the existence of other function classes that have this property is not yet answered. Heine's theorem is already correct for functions that are almost everywhere continuous, i.e. those with a finite set of points of discontinuity. The question of uniqueness is equivalent to the question of whether the trigonometric series disappears

on the set also the disappearance of the coefficients and with it. Sets with this property are called sets of type U (from the French unicité - uniqueness) and all other sets - sets of type M ( multiplicité - ambiguity). Finite sets are therefore amounts of type U . By integrating twice, one obtains the Riemann function:

If is linear, then all and are equal . So if one were to prove for a set that the linearity of follows, then the membership of the type U would have been proven. Cantor uses this idea in his article On the Extension of a Theorem from the Theory of Trigonometric Series from 1871 and shows:

"If (p, q) is any interval in which there is only a finite number of points of the set P, then F (x) is linear in this interval ..." (there page 131)

If is infinite, then it has at least one accumulation point . Cantor calls the set of accumulation points of a set a derived set and denotes it with , he denotes the set derived from with etc. ( see main article: Derivation of a set ). If after a finite number of steps a finite set is reached, then Cantor calls the set a -th kind. Cantor states that the linearity of can also be proven in the interval if there are finitely many points in the set , where correctness this statement does not depend on the choice of the natural number . Amounts with a blank multiple dissipation are therefore always of the type U .

In this article, Cantor's considerations do not go beyond finite iteration processes; however, it already contains thought patterns that will later shape the entire set of theory. He assigns a secondary role to the illustration of the real numbers by means of geometric points by defining the real numbers as Cauchy sequences of elements of the set of rational numbers. He denotes the set of these sequences with and defines the usual types of calculation. Cauchy sequences from elements of the set form another set . Theoretically, this process can be continued indefinitely. Cantor understands from now on in point is an element of any amounts , , , ... The design of such minor hierarchies, in which the transition from one stage to the next through border crossings takes place is, later became a frequently used means of introducing new quantitative theoretical concepts. We will see that such a hierarchy can also be seen in the ordinal numbers.

After this work on trigonometric series Cantor's interest in the problem of a simultaneously necessary and sufficient condition for the uniqueness of the expansion of functions in trigonometric series has weakened. The question was later examined very intensively by Paul Du Bois-Reymond , Charles-Jean de La Vallée Poussin , William Henry Young , Arnaud Denjoy , Nina Bari , Raichmann and Dmitri Evgenjewitsch Menschow , but without arriving at a satisfactory result. Cantor himself has dedicated himself to the task of classifying the point sets according to when the process of deriving ends. Sets in which this happens after a finite number of steps is what Cantor calls sets of the first genus. A set is a set of the first genus if and only if the average

is empty. A natural thought here is to make precisely this set the first derivative of the transfinite order for sets of the second genus. Cantor calls it . This is followed by the derivatives

Cantor writes in his article On Infinite, Linear Point Manifolds from 1880:

"By consequent advancement one gains successively the further concepts:
etc.; we see here a dialectical generation of concepts, which leads on and on and remains free of any arbitrariness in itself necessary and consequent. "

In this article, which is less than five pages long, the whole way is mapped out how one can develop a complete transfinite system of ordinal numbers from the natural numbers. In 1883 Cantor defined this series of numbers with two generation principles, the "addition of a unit to an existing number already formed" and the formation of a next larger number as the limit towards which the previously defined numbers strive, namely in the series of articles on infinite, linear point manifolds . He then spoke of ordinal numbers from 1895 and defined them in his two articles, Contributions to the foundation of the transfinite set theory of 1895/97, as order types of well-ordered sets.

The natural numbers as ordered sets

Visualization of the construction of natural numbers according to John von Neumann

Ordinal numbers are a term used in set theory in modern mathematics. In order to define them as a generalization of the natural numbers, it is obvious to embed the natural numbers in a set-theoretical hierarchy. Thereby one explains the empty set for the zero of the natural number sequence. The empty set is therefore the number specially marked in the Peano axiom system of natural numbers and not explicitly defined without a predecessor. Following a suggestion by John von Neumann , each additional number is then defined as the set of numbers that have already been defined:


The quantities , , etc. by the element relation ( ) in good order . For example, the amount has the elements , , , , by be arranged. That's why you write too . So a natural number is less than a number if an element is of. For the entire set of natural numbers are employed: . The set represents a model of the Peano axiom system. Its existence is secured in the Zermelo-Fraenkel set theory by the axiom of infinity .

Motivation and Definition

The theory of ordinal numbers is a theory of abstraction in which the true nature of the set elements is disregarded and only those properties are examined that can be derived from their arrangement. One defines: A bijection from the totally ordered set to the totally ordered set is called an order isomorphism (or a similar mapping) if and for all are equivalent. One says: the sets and are order isomorphic (or similar) and writes if there is an order isomorphism between and . A total of all sets that are isomorphic in order to one another represent an equivalence class called the order type.

One can show that every finite well-ordered set is order isomorphic to exactly one natural number. In addition, for a well-ordered set, the following three statements are equivalent:

  • It is finite.
  • The reverse order is a well order.
  • Every non-empty subset has a largest element.

This provides the basis for the generalization of the natural numbers to ordinal numbers, which are chosen as special well-ordered sets in such a way that each well-ordered set is order isomorphic to exactly one ordinal number. Thus every ordinal number is a special representative of a certain type of order. The following definition improves Cantor's approach and was first given by John von Neumann :

Definition I. (presupposes the axiom of foundation ): A set is called an ordinal number if every element of is also a subset of and is totally ordered with respect to the set inclusion .

Such a set is automatically well-ordered due to the axiom of foundation, which says: Every non-empty set has an element that is disjoint to . According to this definition, the natural numbers are ordinal numbers. For example, an element of and is also a subset. is also an ordinal number, the smallest transfinite ordinal number (greater than any natural number). The Neumann definition has the advantage over the first definition that, from the point of view of basic research, it determines a perfectly defined set-theoretical object within axiomatic set theory . Every well-ordered set is order isomorphic to exactly one ordinal number, which is usually denoted by or .

Remarks and Other Definitions

The use of equivalence classes of all sets with regard to the order isomorphism is problematic from the point of view of modern mathematics because they represent incredibly large objects that, in contrast to the von Neumann ordinal numbers, are genetically and not substantially defined. Their existence is assumed in naive set theory without an explicit justification and can not be justified within ZFC without the use of ordinal numbers.

In every set theory, objects that fulfill the axiom of ordinal numbers are called ordinal numbers: Every well-ordered set (or possibly another well-ordered structure) can be assigned an ordinal number in such a way that any ordinal numbers assigned to two different sets are exactly the same if the two sets are order isomorphic are to each other. In all axiomatic set theory, in order to avoid the introduction of new basic objects, one tries to find suitable objects given by the theory that satisfy the axiom of ordinal numbers. One possibility for this is to build up special set hierarchies (such as the Von Neumann numbers ).

The difficulties associated with dispensing with such hierarchies can be illustrated using the example of general linear orders , for which no suitable set hierarchy is known (2004). In this case, postulating the existence of order types can only be avoided by resorting to rank or level functions. After special objects that would be suitable for introducing ordinal numbers have already been named, the ordinal axiom is eliminated (if at all possible) ( downgraded to a theorem ). Within ZFC, one needs the replacement axiom that Fraenkel 1922 added to the Zermelos axiom system.

How great the set-theoretical strength of the ordinal number axiom is, is indicated by the fact that the axiom of infinity, replacement and, for some, even the axiom of choice must be used to prove the existence of "many" Von Neumann ordinal numbers. The von Neumann definition of ordinal numbers is the most widely used today. Definitions of ordinal numbers based on the formation of equivalence classes can also be found in axiomatic set theory. However, to avoid contradictions, these equivalence classes are only formed with certain restrictions. So z. B. The set of countable ordinal numbers according to Hartogs is structured as follows: It is defined as the set of equivalence classes in the subset of well-ordered elements of , if is. Two subsets are equivalent if they can be mapped to one another in an isomorphic order. is with a well-ordered crowd. This hierarchy can be continued by placing and forming the quantities for . Hartogs' definition does not use representative selection and is sufficient for many applications of ordinals in calculus and topology. Equivalence classes of order isomorphic sets are also formed in set theory with a step-theoretical structure ( Bertrand Russell , Willard Van Orman Quine , Dana Scott , Dieter Klaua, and others). In the AM by Klaua z. B. all sets are elements of universal sets . The ordinal number of the well-ordered set is then the equivalence class of all elements that are too order isomorphic of the smallest universal set that contains order isomorphic sets. In Scott-Potter set theory , which is an example of set theory without a substitution axiom , the Von Neumann ordinal numbers are called pseudo ordinal numbers . The ordinal numbers in this set theory are defined for each well-ordered collection by and the ordinal numbers of the well-ordered sets are called small ordinals . is the collection of small ordinal numbers and - the smallest large ordinal number. There is no collection of all ordinal numbers in Scott Potter set theory. It has already been mentioned that the well-order property of the ordinal numbers in ZF can be derived from the axiom of the foundation. However, it is customary in the set theoretical literature to formulate definitions as independently of the axioms as possible.

In the following, seven alternative definitions of the ordinal numbers are given, all of which in ZF without the foundation axiom to each other and in ZF with the foundation axiom are also equivalent to the definition formulated above. Before that, two terms: A set is called transitive if . In words: In a transitive set , with every element of the set, all elements of in are also contained as elements. From this definition it follows: A set is transitive if and only if . A set is called well-founded if there is one such that and are disjoint.

Definition II by Ernst Zermelo (1915, published 1941)

A set is called an ordinal number if for each the set is an element of or identical to and for each subset of the union of the elements of is an element of or identical to .

Definition III by Ernst Zermelo (1915, published 1930), first published ordinal number definition by John von Neumann (1923)

A well-ordered set is called an ordinal number if every element of is identical to the set of all elements preceding it, i.e. H. if .

Definition IV by Kurt Gödel (1937, published 1941)

A transitive set whose elements are transitive is called ordinal if every non-empty subset of is well-founded.

Definition V by Raphael M. Robinson (1937)

A transitive set , the non-empty subsets of which are well-founded, is called an ordinal number if for each two different elements and of either or holds. (The last property is also called connectivity).

Definition VI by Paul Bernays (1941)

A transitive set is called an ordinal number if all transitive real subsets of elements of .

Definition VII

An irreflexively ordered set is called an ordinal number if it is transitive and well-ordered.

Definition VIII

Ordinal numbers are called the images of the functions for well-ordered sets .

The last definition also shows how one can determine the ordinal number of a well-ordered set. That the functions are well-defined follows from the theorem about transfinite recursion and that their images - called epsilon images - are sets, from the replacement axiom.

Limes and successor numbers

The elements of a Von Neumann ordinal number are themselves ordinal numbers. If one has two ordinals and , then is an element of if and only if is a proper subset of , and it holds that either an element of , or an element of , or is. An irreflexive total order relation is thus defined by the element relation between the elements of an ordinal number . It is even more true: any set of ordinal numbers is well ordered. This generalizes the well-ordering principle that any set of natural numbers is well-ordered, and allows the free application of transfinite induction and the proof method of infinite descent to ordinal numbers.

Each ordinal number has exactly the ordinal numbers as elements that are smaller than . The set-theoretical structure of an ordinal number is completely described by smaller ordinal numbers. Every intersection or union of ordinals is a set of ordinals. Since every set of ordinal numbers is well ordered, every transitive set of ordinal numbers is itself an ordinal number (see Definition VII). It follows that every intersection or union of ordinal numbers is also an ordinal number. The union of all elements of a set of ordinal numbers is called the supremum of and denoted by. It is not difficult to show that for each valid and that it to each one with there. In the sense of this definition, the supremum of the empty set is the empty set, i.e. the ordinal number .

Well-ordered sets are not order isomorphic to any of their initial segments. Therefore, a similar mapping only exists between two ordinal numbers if they are the same. The class of all ordinal numbers is not a set (see also: the Burali-Forti paradox ). If it were a set, then it would be a well-ordered and transitive set - that is, an ordinal number for which applies. Ordinal numbers that contain themselves as an element do not exist, however, because they would have to be order isomorphic to one of their initial segments (namely to ). From the theorem that is a real class, it follows that for any set of ordinals there exist ordinals that are greater than any element of . Among the ordinals that are greater than any element of a set of ordinals, there is always a smallest one. It is called the upper limit of the set and is called .

The smallest ordinal number greater than the ordinal number is called the successor of and is often referred to as. The designation also makes sense outside of transfinite arithmetic (without contradicting it). If has a largest element, then this predecessor is named and denoted by. Not every ordinal number has a predecessor, e.g. B. . One calls an ordinal number that has a predecessor, a successor number or a number of the first kind. An ordinal number is of the first kind if and only if . This is synonymous with what, in turn, follows. The ordinal numbers of the first kind and they are called isolated. A positive ordinal number without a predecessor is called a limit number. A positive ordinal number is a limit number if and only if . Ordinal numbers of the second kind are the Limes numbers and the . Each ordinal number is thus a number either of the first or second type and either limit number or isolated, with the terms limit number and number of the second type as well as isolated number and number of the first type coinciding for positive numbers. The number is the only isolated number of the second kind.

For each ordinal number, the predecessor of is the ordinal number itself. The Limes numbers form a real class, which is denoted by. If an ordinal number of the first type, then there is a finite but no infinite sequence: , , , , starting with one reaches after a finite number downs of an ordinal to its predecessor a number of the second kind, it is even more true. If a transfinite ordinal, then one can build genuinely falling sequences with the first element of any length , but not infinite ones.

Infinite sequences of ordinal numbers always contain infinite, non-decreasing subsequences.

On as a recursive data type

In metamathematics, and especially in proof theory , ordinal numbers are often defined recursively or axiomatically, just as natural numbers can be defined by the Peano axioms. But the purpose of these definitions is not to determine a true class of ordinals, but as long as possible, in contrast to the Ordinalzahldefinitionen of set theory early stretch of find that with from the perspective of metamathematical program approved products can be defined and analyzed. While writing figures are used within metamathematics , the term ordinal number notation introduced by Stephen Cole Kleene is used in set theory and recursion theory . For metamathematics, the properties of the ordinal numbers that follow from the well-ordered system and some arithmetic properties are of particular importance. If one considers the existence of successor and limit numbers as basic properties, then within the theory of recursive data types (inductively defined classes) the following definition for the class of ordinal numbers can be formulated. and let the constructors of the recursive data type ordinal number, - the foundation and - a partial order relation on with the properties:

  • for each chain :
  • for each chain :

One can show by means of structural induction that is a well-ordered class. In the terminology of the theory of recursive data types, the von Neumann ordinal numbers represent an implementation of the recursive data type ordinal number ; H. a model of the above set of axioms.

Arithmetic operations

The arithmetic operations with ordinal numbers are introduced as a generalization of the types of arithmetic known from elementary arithmetic. Under the sum of two ordinal numbers and means the ordinal number of a well-ordered set, which consists of the elements of the two sets, when all elements of the well-ordering from the elements of standing. This corresponds exactly to the idea that we are familiar with from finite numbers, that concatenating two finite sequences of length and one finite sequence of length results. Since one has to distinguish between isolated and limit numbers with the transfinite ordinal numbers, care is taken when introducing the arithmetic operations that these are continuous continuations of the finite arithmetic operations. The continuity of the arithmetic operations with the ordinal numbers can be seen most clearly in the so-called functional introduction of transfinite arithmetic. The functional introduction of ordinal number arithmetic is justified by means of transfinite recursion. Not all of the properties of arithmetic operations known from finite arithmetic can be carried over to infinity. So the addition is generally not commutative. With the help of Cantor's polynomial representation, which is a kind of transfinite place value system, alternative arithmetic operations can be introduced: the so-called natural operations between ordinal numbers, so that none of the rules known from finite arithmetic have to be missed.

Topological properties

Every ordinal number can be made into a topological space due to its total order through the order topology . In this topology the sequence converges to , and the sequence

converges against . Ordinal numbers without a predecessor can always be represented as the limit value of a network of smaller ordinals, for example through the network of all smaller ordinals with their natural order. The size of the smallest such network is called confinality . This can be uncountable, i.e. that is, in general, those ordinals are not the limit of a sequence of smaller ordinals, such as B. the smallest uncountable ordinal number .

The topological spaces and are often mentioned in textbooks as an example of an uncountable topology. For example, in space it holds that the element lies in the closure of the subset , but no sequence in converges against the element . The space satisfies the first, but not the second countability axiom and neither.

The room has exactly one (Hausdorff) compactification , namely . This means that the largest possible, the Stone-Čech compacting , here with the smallest possible, the one-point or Alexandroff compacting , corresponds.

See also


Using von Neumann's definition:

When using order types:

Further sources:

  • K. Schütte: Proof Theory. Springer, 1977, ISBN 0-387-07911-4 .
  • A. Fraenkel, Y. Bar-Hillel: Foundations of set theory. North-Holland Publishing Co., 1958.

Web links

Wiktionary: ordinal number  - explanations of meanings, word origins, synonyms, translations

References and comments

  1. a b S. Natanson, 1977, Chapter X., § 6.
  2. ^ S. Riemann Function (at MathWorld) .
  3. in: Mathematische Annalen 5 (1872) pp. 123-132
  4. 2nd article in: Mathematische Annalen 17 (1880) 357f
  5. in: Mathematische Annalen 21 (1883), p. 576f.
  6. in: Mathematische Annalen 46 (1895) p. 499 and Mathematische Annalen 49 (1897) p. 207
  7. Which is also an advantage for mathematics as a whole, because it gives numerous other mathematical terms a set-theoretical interpretation.
  8. a b August 15, 1923 - Letter from Johann von Neumann to Ernst Zermelo (see H. Meschkowski: Problemgeschichte der neueren Mathematik. BI-Wissenschaftsverlag, 1978, ISBN 3-411-01542-X . XIV.1. And Plate 10 .)
  9. More precisely: not without the use of the so-called cutting back through consideration of rank by Alfred Tarski , which in turn presupposes the presence of already defined ordinals (see Levy, 1979, II.7.7, II.7.13).
  10. s. Bachmann, 1968, § 3.5
  11. For unsound set universes, such a function does not necessarily have to be present.
  12. A. Fraenkel: To the basics of Cantor-Zermeloschen set theory. In: Mathematical Annals. 86, 1922, pp. 230-237
  13. s. Deiser, 2004, 2.6., P. 256 and 3.1, p. 433 and Bachmann, 1968, § 6., § 38., § 42.
  14. s. O. Deiser: Real numbers. The classical continuum and the natural consequences. Springer, 2007, ISBN 978-3-540-45387-1 , pp. 382-386
  15. To prove that there is a set and not a real class, one needs the replacement axiom within ZFC (see Zuckerman, 1974, 5.12)
  16. On ordinal numbers in the sense of Russell and Alfred North Whitehead see: J. Rosser: Logic for Mathematicians. McGraw-Hill Book Company, 1953, ISBN 0-8284-0294-9 , XII.
  17. ^ W. Quine: New Foundations for Mathematical Logic. In: American Mathematical Monthly. 44, 1937, pp. 70-80
  18. D. Scott: Axiomatizing Set Theory. In: Proceedings of Symposia in Pure Mathematics. 13, 2, American Mathematical Society, 1971, pp. 207-214
  19. AM: General set theory - s. Klaua , 1968 a. Klaua, 1969
  20. s. Klaua, 1974 and D. Klaua: An axiomatic set theory with the largest universe and hyperclasses. In: Monthly books for mathematics. 92, 3, 1981, pp. 179-195.
  21. s. Potter, 1994, 6.2
  22. The pseudo ordinal numbers were known to Zermelo and Mirimanoff before 1923. However, they only gained importance within ZFC after von Neumann recognized that the replacement axiom implies the existence of a rank function for all sets and an ordinal number function for all well-ordered sets. Therefore, nowadays, the pseudo ordinal numbers are mainly known through the concept of von Neumann numbers .
  23. Here , with the respect to the smallest element of -relation and , , .
  24. ^ S. Bachmann, 1967, § 4.3 and Deiser, 2004, 2.6, pp. 257-258 and Enderton, 1977, Chapter 7, pp. 182-194.
  25. ^ Definition from Zermelo's estate 1915 according to: M. Hallett, Cantorian set theory and limitation of size, Oxford 1984, pp. 277f. This definition was first published by Paul Bernays, Zermelo's assistant at the time in Zurich: Bernays, A System of Axiomatic Set Theory II, in: Journal of Symbolic Logic 6 (1941), pp. 6 and 10. Zermelo also mentioned the condition resulting from the last condition for the empty subset results in 0.
  26. M. Hallett: Cantorian set theory and limitation of size , Oxford 1984, p. 279, definition from Zermelo's estate from 1915, first published in: E. Zermelo: About limit numbers and quantity ranges , in: Fundamenta Mathematicae 16 (1930) (PDF ; 1.6 MB), 29-47, definition p. 31.
  27. J. v. Neumann: On the introduction of transfinite numbers , 1923, in: Acta scientiarum mathematicarum 1 (1922/23) 199-208, definition pp. 199f., Edited in: J. v. Neumann: Collected Works I, Oxford, London, New York, Paris 1961, pp. 24-34. The same definition also in: J. v. Neumann: About the definition by transfinite induction and related questions of general set theory , 1927, in: Mathematische Annalen 99 (1928) , 373-391, definition p. 378.
  28. The definition goes back to a lecture by Gödel in Vienna in 1937 according to: Bernays, A System of Axiomatic Set Theory II, in: Journal of Symbolic Logic 6 (1941), p. 10.
  29. ^ RM Robinson: The theory of classes. A modification of von Neumann's system , in: Journal of Symbolic Logic 2 (1937), pp. 29-36.
  30. ^ Bernays, A System of Axiomatic Set Theory II, in: Journal of Symbolic Logic 6 (1941), p. 6.
  31. a b c d s. Levy, 1979, II.2.14, II.3.11, II.3.12, II.3.13, II.3.16
  32. A well-order relation can also be defined between well-order types (i.e. between ordinal numbers in the Cantor sense). A well-ordered set is called smaller (or shorter) than a well-ordered set if is order isomorphic to a proper subset of . It should be agreed that in the following, if not expressly stated otherwise, an ordinal number in the von Neumann sense is meant under ordinal number and that the assertions that are made are sentences in ZF or ZFC.
  33. Note that the union (intersection) of elements of a set of transitive sets is transitive.
  34. This claim is independent of the foundation axiom.
  35. If there were no such numbers, then there would be a subset of (a real set).
  36. If there is an ordinal number that is greater than all elements of the set , then it is not a real class, but a well-ordered set and therefore has a smallest element.
  37. a b s. Bachmann, 1967, § 4.1.3., § 4.1.4
  38. The English term for this is strict upper bound.
  39. a b c s. Komjath, Tototik, 2006, 8.2, 8.3, 8.18
  40. s. for example Schütte, 1997 and Deutsch, 1999, pp. 308-309
  41. s. Forster, 2003, 7.1; see. also with I. Phillips: Recursion Theory. In: S. Abramsky, D. Gabbay, T. Maibaum (Eds.): Handbook of Logic in Computer Science. Vol. 1, Oxford University Press, 1992, ISBN 0-19-853735-2 and Stephen A. Cook , Hao Wang , Characterizations of Ordinal Numbers in Set Theory.  ( Page no longer available , search in web archivesInfo: The link was automatically marked as defective. Please check the link according to the instructions and then remove this notice. @1@ 2Template: Toter Link / gdzdoc.sub.uni-goettingen.de   Mathematical Annals, 164, 1, 1966
  42. However, this can not be proven without the axiom of choice , see Kenneth Kunen: Set Theory , North Holland, Amsterdam 1980, pp. 30 and 33
  43. a b c This book is based on a special system of axioms.
  44. Nowhere in this book is the foundation axiom used.