# 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.