Cantor's normal form

The Cantor normal form is in the mathematical branch of set theory treated, it generalizes the representation of numbers in place value system regarding. A fixed base on ordinals .

Cantor's normal form to the base β

Let it be an ordinal number. Then for every ordinal number there is a uniquely determined natural number and uniquely determined ordinal numbers and so that ${\displaystyle \beta >1}$ ${\displaystyle \alpha >0}$ ${\displaystyle n}$ ${\displaystyle \sigma _{1},\ldots ,\sigma _{n}}$ ${\displaystyle \tau _{1},\ldots ,\tau _{n}}$

${\displaystyle \alpha =\beta ^{\sigma _{1}}\cdot \tau _{1}+\ldots +\beta ^{\sigma _{n}}\cdot \tau _{n}}$ and

${\displaystyle \sigma _{1}>\ldots >\sigma _{n}}$and for . ${\displaystyle 1\leq \tau _{i}<\beta }$ ${\displaystyle i=1,\ldots ,n}$

For proof

The proof is given by means of transfinite induction . Using simple lemmas about Ordinalzahlenarithmetik one provides the least ordinal with . Then there are ordinals and with . Finally, is or one can apply to the induction hypothesis, which also ends the proof. ${\displaystyle \sigma _{1}}$ ${\displaystyle \beta ^{\sigma _{1}+1}>\alpha }$ ${\displaystyle 1\leq \tau _{1}<\beta }$ ${\displaystyle \gamma <\alpha }$ ${\displaystyle \alpha =\beta ^{\sigma _{1}}\cdot \tau _{1}+\gamma }$ ${\displaystyle \gamma =0}$ ${\displaystyle \gamma }$

Remarks

Position of the coefficients

In the above representation of the ordinal number with respect to the base , the coefficients are to the right of the powers . This differs from the usual notation in the place value system in number theory , where the coefficients are often written before the powers. This is not a problem there, since the multiplication in the natural numbers is commutative , but this is not the case for the ordinal number multiplication . For example , where is the smallest infinite ordinal number. The above sentence is even wrong if you put the coefficients before the powers. ${\displaystyle \alpha }$ ${\displaystyle \beta }$ ${\displaystyle \tau _{i}}$ ${\displaystyle \beta ^{\sigma _{i}}}$ ${\displaystyle 2\cdot \omega =\omega \not =\omega +\omega =\omega \cdot 2}$ ${\displaystyle \omega }$

Base ω

If special , the above sentence takes the following form: ${\displaystyle \beta =\omega }$

For every ordinal number there are clearly certain natural numbers and ordinal numbers , so that ${\displaystyle \alpha >0}$ ${\displaystyle n,k_{1},\ldots ,k_{n}}$ ${\displaystyle \sigma _{1}>\ldots >\sigma _{n}}$

${\displaystyle \alpha =\omega ^{\sigma _{1}}\cdot k_{1}+\ldots +\omega ^{\sigma _{n}}\cdot k_{n}}$.

Please note that ordinal numbers must be natural numbers, which are designated with in this formulation . This sentence is also called the Cantor's normal form sentence . It was first proven in 1897 by Cantor for certain ordinal numbers, but the proof could be extended to any ordinal numbers. ${\displaystyle \tau _{i}<\omega }$ ${\displaystyle k_{i}}$

Base 10

If you use the base , you get exactly the usual decimal representation in the place value system for base 10, i.e. for natural numbers. In addition, the sentence also provides representations for larger ordinal numbers, for example or . ${\displaystyle \beta =10}$ ${\displaystyle \alpha <\omega }$ ${\displaystyle \omega =10^{\omega }}$ ${\displaystyle \omega +\omega +37=10^{\omega }\cdot 2+10^{1}\cdot 3+10^{0}\cdot 7}$

Applications

The representations of ordinal numbers as the base are used to define the so-called Hessenberg natural operations . ${\displaystyle \omega }$

Furthermore, they enable a proof of Goodstein's theorem .

Individual evidence

1. ^ Edmund Weitz, Karsten Steffens, Michael Holz: Introduction to Cardinal Arithmetic , Springer Basel AG (2009), ISBN 3-0346-0327-4 , Theorem 1.4.6
2. ↑ Thomas Forster: Logic, Induction and Sets : Cambridge University Press (2003), ISBN 0-521-53361-9 , Chapter 7.1.2: Cantor's normal form theorem
3. ^ Thomas Jech : Set Theory. 3rd millenium edition, revised and expanded, corrected 4th print. Springer, Berlin et al. 2006, ISBN 3-540-44085-2 , Theorem 2.26
4. ^ Joseph G. Rosenstein: Linear orderings , Academic Press (1982), ISBN 0-1259-7680-1 , Theorem 3.46
5. ^ G. Cantor: Contributions to the foundation of the transfinite set theory , Mathematische Annalen (1897), Volume 49, pages 207-246, §19: The normal form of the numbers of the second number class