Beth function

The Beth function , named after the second letter of the Hebrew alphabet and also written as, is an enumeration of certain infinite cardinal numbers used in set theory , more precisely in the theory of cardinal numbers. ${\displaystyle \beth }$

definition

The Beth function assigns a cardinal number recursively defined as follows to each ordinal number : ${\displaystyle \alpha }$ ${\displaystyle \beth _{\alpha }}$

• ${\displaystyle \beth _{0}=\aleph _{0}}$, where is the smallest infinite cardinal number, see Aleph function . ${\displaystyle \aleph _{0}}$
• ${\displaystyle \beth _{\alpha +1}=2^{\beth _{\alpha }}}$for successor ordinal numbers . The right side stands for the power of cardinal numbers . ${\displaystyle \alpha +1}$
• ${\displaystyle \beth _{\lambda }=\sup _{\alpha <\lambda }\beth _{\alpha }}$for Limes ordinal numbers . ${\displaystyle \lambda }$

Remarks

The continuum hypothesis is synonymous with , because by definition it is the power of the power set of a countable set and is therefore equal to the continuum . The generalized continuum hypothesis is equivalent to , that is, for all ordinal numbers . ${\displaystyle \aleph _{1}=\beth _{1}}$ ${\displaystyle \beth _{1}}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \aleph =\beth }$ ${\displaystyle \aleph _{\alpha }=\beth _{\alpha }}$ ${\displaystyle \alpha }$

A Limes cardinal number is called a strong Limes if for all cardinal numbers . A cardinal number is a strong Limes cardinal number if and only if for a Limes ordinal number . ${\displaystyle \kappa }$ ${\displaystyle \mu ^{\lambda }<\kappa }$ ${\displaystyle \lambda ,\mu <\kappa }$ ${\displaystyle \kappa }$ ${\displaystyle \kappa =\beth _{\xi }}$ ${\displaystyle \xi }$

It always applies to all ordinal numbers . One can show that there must be fixed points , i.e. ordinal numbers for which applies. The smallest fixed point is the limit of the sequence , which is informally represented as . Likewise, strongly unreachable cardinal numbers are fixed points of the Beth function. ${\displaystyle \alpha \leq \aleph _{\alpha }\leq \beth _{\alpha }}$ ${\displaystyle \alpha }$ ${\displaystyle \alpha }$ ${\displaystyle \alpha =\beth _{\alpha }}$ ${\displaystyle \beth _{0},\beth _{\beth _{0}},\beth _{\beth _{\beth _{0}}},\ldots }$ ${\displaystyle \beth _{\beth _{{}_{\ddots }}}}$

Individual evidence

1. ^ Thomas Jech : Set Theory. 3rd millennium edition, revised and expanded. Springer, Berlin a. a. 2003, ISBN 3-540-44085-2 , Chapter I.5, p. 55.
2. ^ W. Wistar Comfort, Stylianos Negrepontis: The Theory of Ultrafilters (= The basic teachings of the mathematical sciences in individual representations . Vol. 211). Springer, Berlin et al. 1974, ISBN 3-540-06604-7 , Lemma 1.23.