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.
definition
The Beth function assigns a cardinal number recursively defined as follows to each ordinal number :
- , where is the smallest infinite cardinal number, see Aleph function .
- for successor ordinal numbers . The right side stands for the power of cardinal numbers .
- for Limes ordinal numbers .
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 .
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 .
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.
Individual evidence
- ^ 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.
- ^ 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.