Hessenberg theorem (set theory)

The von Hessenberg theorem , named after the German mathematician Gerhard Hessenberg , is a mathematical theorem from the field of set theory , more precisely the theory of cardinal numbers . It essentially states that an infinite cardinal number is equal to its square in so-called cardinal number arithmetic .

Formulation of the sentence

• For every ordinal number is equal to the Cartesian product . ${\displaystyle \alpha }$ ${\displaystyle \aleph _{\alpha }}$ ${\displaystyle \aleph _{\alpha }\times \aleph _{\alpha }}$

The -th stands for an infinite cardinal number, see Aleph function . This theorem applies in ZF, i.e. in Zermelo-Fraenkel set theory without the axiom of choice . ${\displaystyle \aleph _{\alpha }}$ ${\displaystyle \alpha }$

Inferences

If, in addition to ZF, one assumes the axiom of choice, that is, one works in ZFC, which is done in this section, one can draw further conclusions:

• Every infinite quantity is equal to the Cartesian product , because with the axiom of choice every quantity is equal to one . For finite sets this theorem is known to be wrong. ${\displaystyle X}$ ${\displaystyle X\times X}$ ${\displaystyle \aleph _{\alpha }}$
• If an infinite cardinal number and a natural number, then is . According to the axiom of choice, every infinite cardinal number is a and according to Hessenberg's theorem it follows , the rest then follows by induction . ${\displaystyle \kappa }$ ${\displaystyle n\in \mathbb {N} \setminus \{0\}}$ ${\displaystyle \kappa ^{n}=\kappa }$ ${\displaystyle \aleph _{\alpha }}$ ${\displaystyle \kappa ^{2}=\kappa }$
• If and are infinite cardinal numbers, then we have . This follows immediately from the obvious inequalities ${\displaystyle \kappa }$ ${\displaystyle \lambda }$ ${\displaystyle \kappa +\lambda =\kappa \cdot \lambda =\max\{\kappa ,\lambda \}}$

${\displaystyle \max\{\kappa ,\lambda \}\leq \kappa +\lambda \leq \kappa \cdot \lambda \leq \max\{\kappa ,\lambda \}\cdot \max\{\kappa ,\lambda \}=\max\{\kappa ,\lambda \}}$,

where the equation is again Hessenberg's theorem. The addition and multiplication, as defined in cardinal number arithmetic , are the same and trivial for infinite cardinal numbers.

Individual evidence

1. ↑ Heinz-Dieter Ebbinghaus : Introduction to set theory , Spektrum Verlag 2003, ISBN 3-8274-1411-3 , Chapter IX, sentence 1.11