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 . ${\displaystyle {\mathfrak {m}}}$ ${\displaystyle \mathrm {On} }$ ${\displaystyle Z({\mathfrak {m}})}$ ${\displaystyle \eta }$ ${\displaystyle |\eta |={\mathfrak {m}}}$ ${\displaystyle Z({\mathfrak {m}})}$

This number is called the corresponding initial number ${\displaystyle {\mathfrak {m}}}$ or the initial number of the mightiness ${\displaystyle {\mathfrak {m}}}$ and denotes it with . ${\displaystyle \omega ({\mathfrak {m}})}$

If there is an aleph, say for , then you bet . ${\displaystyle {\mathfrak {m}}}$ ${\displaystyle {\mathfrak {m}}=\aleph _{\tau }}$ ${\displaystyle \tau \in \mathrm {On} }$ ${\displaystyle \omega _{\tau }=\omega (\aleph _{\tau })}$

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 . ${\displaystyle \mathrm {On} }$

(2) ${\displaystyle \omega _{0}=\omega }$

(3) If one denotes the Hartogs number function , then is always . ${\displaystyle h}$ ${\displaystyle \omega _{\tau +1}=h(\omega _{\tau })}$

(4) if a limit number is ${\displaystyle \omega _{\lambda }=\sup\{\omega _{\tau }\mid \,\tau <\lambda \}}$ ${\displaystyle \lambda }$

(5) ${\displaystyle |\omega _{\tau }|=\aleph _{\tau }}$

(6) ${\displaystyle Z(\aleph _{\tau })=\{\eta \in \mathrm {On} \mid \omega _{\tau }\leq \eta <\omega _{\tau +1}\}}$

(7) For each starting number there is a with . ${\displaystyle \alpha }$ ${\displaystyle \tau \in \mathrm {On} }$ ${\displaystyle \alpha =\omega _{\tau }}$

(8) Every starting number is a limit number.

(9) For each has the order type and thus the power . ${\displaystyle \tau \in \mathrm {On} }$ ${\displaystyle Z(\aleph _{\tau })}$ ${\displaystyle \omega _{\tau +1}}$ ${\displaystyle \aleph _{\tau +1}}$

(10) For applies if and only if . ${\displaystyle \sigma ,\tau \in \mathrm {On} }$ ${\displaystyle \omega _{\sigma }=\omega _{\tau }}$ ${\displaystyle \sigma =\tau }$

(11) For applies if and only if . ${\displaystyle \sigma ,\tau \in \mathrm {On} }$ ${\displaystyle \omega _{\sigma }<\omega _{\tau }}$ ${\displaystyle \sigma <\tau }$

Remarks

1. In addition to the spelling, there is also the spelling ${\displaystyle \omega _{\tau }}$ ${\displaystyle \Omega _{\tau }.}$
2. Some authors interpret the terms aleph and initial number in the same way.
3. 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 .
4. 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 . ${\displaystyle Z(\aleph _{0})}$ ${\displaystyle \aleph _{0}}$ ${\displaystyle \aleph _{1}}$
5. 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

1. ↑ Kamke: p. 174
2. ↑ Alexandroff: p. 79
3. ↑ Alexandroff: p. 79 ff.
4. ↑ Fraenkel: p. 192 ff.
5. ↑ Kamke: p. 174 ff.
6. ↑ Hrbacek-Jech: p. 132 ff.
7. ↑ Oberschelp: p. 189 ff.
8. ↑ Sierpiński: p. 391 ff.
9. So ↑ consists exactly of the natural numbers . ${\displaystyle \omega _{0}}$
10. ^ Klaua: p. 289
11. ↑ Ebbinghaus: p. 134 ff.
12. ↑ Hrbacek-Jech: p. 135
13. ↑ See Hrbacek-Jech: p. 133
14. ↑ Kamke: p. 181
15. ↑ Klaua: p. 290
16. ↑ Kamke: p. 181