Starting number
The term of the initial number (Engl. Initial number or initial ordinal ) who comes from set theory .
It is directly related to the classification of infinite ordinals to their thickness . In each of the number classes formed in this way, the associated initial number forms the smallest ordinal number within this number class . In this way, initial numbers and alephs are in a reversibly clear relationship ( bijection ).
definition
Let any infinite cardinal number be given . For this one forms within the ordinal number class the associated number class of those ordinal numbers for which is. There is a clearly defined smallest ordinal number in .
This number is called the corresponding initial number or the initial number of the mightiness and denotes it with .
If there is an aleph, say for , then you bet .
properties
The starting numbers have the following properties:
- (1) No initial number is equal to an ordinal number which is really smaller than itself within the ordinal number class .
- (2)
- (3) If one denotes the Hartogs number function , then is always .
- (4) if a limit number is
- (5)
- (6)
- (7) For each starting number there is a with .
- (8) Every starting number is a limit number.
- (9) For each has the order type and thus the power .
- (10) For applies if and only if .
- (11) For applies if and only if .
Remarks
- In addition to the spelling, there is also the spelling
- Some authors interpret the terms aleph and initial number in the same way.
- The above property (1) is in a certain sense characteristic of the initial numbers, so it could be used for definition. If one proceeds in this way, one also has to consider finite initial numbers , i.e. the natural numbers .
- Following Georg Cantor , the set of natural numbers is called the first number class , while the second number class is called. The first number class therefore has power , the second number class has power . The famous continuum problem can therefore be equated with the question whether the second number class, the cardinality of the continuum has .
- In connection with the initial numbers, Felix Hausdorff formulated the Hausdorff sentence named after him .
literature
- PS Alexandroff : Introduction to set theory and the theory of real functions (= university books for mathematics . Volume 23 ). 6th edition. German Science Publishing House , Berlin 1973.
- Heinz-Dieter Ebbinghaus : Introduction to set theory . 3rd, completely revised and expanded edition. BI-Wissenschaftsverlag, Mannheim (among others) 1994, ISBN 3-411-17113-8 .
- Adolf Fraenkel : Introduction to set theory (= The basic doctrines of the mathematical sciences in individual representations . Volume 9 ). 3rd, revised and greatly expanded edition. Springer Verlag , Berlin (among others) 1928.
- Egbert Harzheim : Ordered Sets (= Advances in Mathematics . Volume 7 ). Springer Verlag , New York, NY 2005, ISBN 0-387-24219-8 ( MR2127991 ).
- Karel Hrbacek - Thomas Jech : Introduction to Set Theory (= Monographs and Textbooks in Pure and Applied Mathematics . Volume 85 ). 2nd, revised and expanded edition. Dekker, New York (et al.) 1984, ISBN 0-8247-7074-9 .
- Erich Kamke : Set theory (= Göschen Collection . 999 / 999a). 6th edition. De Gruyter , Berlin 1969.
- Dieter Klaua : General set theory . Akademie-Verlag , Berlin 1964.
- Arnold Oberschelp : General set theory . BI-Wissenschaftsverlag, Mannheim (ua) 1994, ISBN 3-411-17271-1 .
- Wacław Sierpiński : Cardinal and Ordinal Numbers . Panstwowe Wydawnictwo Naukowe, Warszawa 1958. MR0095787
Individual evidence
- ↑ Kamke: p. 174
- ↑ Alexandroff: p. 79
- ↑ Alexandroff: p. 79 ff.
- ↑ Fraenkel: p. 192 ff.
- ↑ Kamke: p. 174 ff.
- ↑ Hrbacek-Jech: p. 132 ff.
- ↑ Oberschelp: p. 189 ff.
- ↑ Sierpiński: p. 391 ff.
- So ↑ consists exactly of the natural numbers .
- ^ Klaua: p. 289
- ↑ Ebbinghaus: p. 134 ff.
- ↑ Hrbacek-Jech: p. 135
- ↑ See Hrbacek-Jech: p. 133
- ↑ Kamke: p. 181
- ↑ Klaua: p. 290
- ↑ Kamke: p. 181