Numbers bet ( ) in mathematics - the cardinal numbers characterizing the power of an infinite set. The sequence of infinite cardinal numbers is usually written as where Named after the second letter of the Hebrew alphabet ( beta ).
Numbers Beth are part of the hierarchy of alefs ( ). They start the same way: and location among alefs depends on the continuum hypothesis . If we accept the continuum hypothesis, then (power of the continuum ), the reverse is also true. If we accept a more powerful generalized continuum hypothesis , then both hierarchies completely coincide: for any index If we accept that the continuum hypothesis is incorrect, then there are many which are not .
Content
Definition
General definition for finite indexes:
For infinite indices:
Examples
- bet-zero ( ) - equal to .
- bet one ( ) - a lot of continuum .
- bet two ( ) - 2 c , for example, a boolean of real numbers.
- bet omega ( ) - the smallest innumerable strong cardinal limit.
Literature
- Forster TE Set Theory with the Universal Set: Exploring an Untyped Universe, Oxford University Press , 1995 - Beth number is defined on page 5.
- Bell, John Lane. Models and Ultraproducts: An Introduction. - reprint of 1974. - Dover Publications , 2006. - ISBN 0-486-44979-3 . See pages 6 and 204-205 for beth numbers.
- Roitman, Judith. Introduction to Modern Set Theory. - Virginia Commonwealth University , 2011. - ISBN 978-0-9824062-4-3 . See page 109 for beth numbers.