Natural surgery

The Hessenberg natural operations , named after Gerhard Hessenberg , are mathematical arithmetic operations for ordinal numbers and essentially use Cantor's normal forms of the operands and thus the transfinite arithmetic of ordinal numbers.

Cantor's normal form

The Cantor normal form of an ordinal number has the form of a sum of - powers , the summands of which are ordered according to decreasing size and are all : ${\displaystyle \alpha \neq 0}$ ${\displaystyle \omega }$ ${\displaystyle \neq 0}$

${\displaystyle \alpha =\omega ^{\lambda _{m_{\alpha }}^{(\alpha )}}{\kappa _{m_{\alpha }}^{(\alpha )}}+\cdots +\omega ^{\lambda _{0}^{(\alpha )}}{\kappa _{0}^{(\alpha )}}=\sum _{i\leq m_{\alpha }}\ \omega ^{\lambda _{i}^{(\alpha )}}{\kappa _{i}^{(\alpha )}},}$

where the exponents themselves are ordinal numbers and the coefficients are natural numbers . ${\displaystyle \lambda _{0}^{(\alpha )}<\dots <\lambda _{m_{\alpha }}^{(\alpha )}\leq \alpha }$ ${\displaystyle 0<\kappa _{0}^{(\alpha )},\dots ,\kappa _{m_{\alpha }}^{(\alpha )}<\omega }$

The Cantor normal form of the ordinal number is the sum with the single summand . ${\displaystyle \alpha =0}$ ${\displaystyle 0=\omega ^{0}0}$

Natural sum

The natural sum of two ordinal numbers is determined by their Cantor normal form. This Cantor normal form of results from the Cantor normal forms of and by formally combining their two summands to a new sum, adding the coefficients of summands with the same -potency, and finally arranging these summands again according to decreasing -potencies. ${\displaystyle \alpha \oplus \beta }$ ${\displaystyle \alpha \oplus \beta }$ ${\displaystyle \alpha }$ ${\displaystyle \beta }$ ${\displaystyle \omega }$ ${\displaystyle \omega }$

This natural addition is not only associative and genuinely monotonic, like the ordinary addition of ordinals, it is also commutative . And the ordinal number 0 is again a neutral element, even with natural addition. ${\displaystyle \oplus }$

Natural product

Similarly, the natural product of two ordinal numbers is determined by its Cantor normal form. This Cantor normal form of results from the Cantor normal forms of and by formally multiplying these two sums out, understanding the formal product of two summands and as summands . It is important that the -exponent of this summand contains the natural sum of the -exponents of its formal factors. ${\displaystyle \alpha \odot \beta }$ ${\displaystyle \alpha \odot \beta }$ ${\displaystyle \alpha }$ ${\displaystyle \beta }$ ${\displaystyle \omega ^{\lambda _{i}^{(\alpha )}}{\kappa _{i}^{(\alpha )}}}$ ${\displaystyle \omega ^{\lambda _{j}^{(\beta )}}{\kappa _{j}^{(\beta )}}}$ ${\displaystyle \omega ^{({\lambda _{i}^{(\alpha )}\oplus \lambda _{j}^{(\beta )}})}{\kappa _{i}^{(\alpha )}\cdot \kappa _{j}^{(\beta )}}}$ ${\displaystyle \omega }$ ${\displaystyle \omega }$

Finally, all of these summands are again arranged according to descending powers and combined as a sum. ${\displaystyle \omega }$

This natural multiplication , like the usual multiplication of ordinal numbers, is associative and strictly monotonic in the case of natural multiplication by a factor . It has a zero element and a neutral element. In addition, it is also commutative and (completely) distributive with respect to natural addition. ${\displaystyle \odot }$ ${\displaystyle \neq 0}$ ${\displaystyle \alpha =0}$ ${\displaystyle \alpha =1}$

The ordinal numbers thus form an ordered commutative ring with one element with regard to Hessenberg's natural operations and their usual well-order .

Examples

It is and always is for ordinals and natural numbers . ${\displaystyle \omega \oplus 1=\omega +1}$ ${\displaystyle \alpha \oplus n=\alpha +n}$ ${\displaystyle \alpha }$ ${\displaystyle n<\omega }$

It is and thus different both from and from . ${\displaystyle (\omega +2)\oplus (\omega +1)=\omega \cdot 2+3}$ ${\displaystyle (\omega +2)+(\omega +1)=\omega \cdot 2+1}$ ${\displaystyle (\omega +1)+(\omega +2)=\omega \cdot 2+2}$

And it is different both from and from and again . ${\displaystyle (\omega +1)\odot (\omega +2)=\omega ^{2}+\omega \cdot 3+2}$ ${\displaystyle (\omega +1)\cdot (\omega +2)=\omega ^{2}+\omega \cdot 2+1}$ ${\displaystyle (\omega +2)\cdot (\omega +1)=\omega ^{2}+\omega +2}$

literature

• Heinz Bachmann: Transfinite numbers. Springer, 1967.
• Dieter Klaua : General set theory . Akademie-Verlag, Berlin 1964.
• Wacław Sierpiński : Cardinal and ordinal numbers. Warsaw 1965, OCLC 500328243 .
• Kazimierz Kuratowski , Andrzej Mostowski : Set theory. North-Holland, 1976, OCLC 251672808 .
• Felix Hausdorff : Fundamentals of set theory. 1914. Chelsea Publishing, New York 1949, OCLC 888238 .