Bet number

Numbers bet ( $\beth$ ${\ displaystyle \ beth}$ ${\ displaystyle \ beth}$ ) in mathematics - the cardinal numbers characterizing the power of an infinite set. The sequence of infinite cardinal numbers is usually written as $\beth _{0},\ \beth _{1},\ \beth _{2},\ \beth _{3},\ \dots$ ${\ displaystyle \ beth _ {0}, \ \ beth _ {1}, \ \ beth _ {2}, \ \ beth _ {3}, \ \ dots}$ ${\ displaystyle \ beth _ {0}, \ \ beth _ {1}, \ \ beth _ {2}, \ \ beth _ {3}, \ \ dots}$ where $\beth$ ${\ displaystyle \ beth}$ ${\ displaystyle \ beth}$ Named after the second letter of the Hebrew alphabet ( beta ).

Numbers Beth are part of the hierarchy of alefs ( $\aleph _{0},\ \aleph _{1},\ \dots$ ${\ displaystyle \ aleph _ {0}, \ \ aleph _ {1}, \ \ dots}$ ${\ displaystyle \ aleph _ {0}, \ \ aleph _ {1}, \ \ dots}$ ). They start the same way: $\beth _{0}=\aleph _{0},$ ${\ displaystyle \ beth _ {0} = \ aleph _ {0},}$ ${\ displaystyle \ beth _ {0} = \ aleph _ {0},}$ and location $\beth _{1}$ ${\ displaystyle \ beth _ {1}}$ $\ beth_1$ among alefs depends on the continuum hypothesis . If we accept the continuum hypothesis, then $\beth _{1}=\aleph _{1}$ ${\ displaystyle \ beth _ {1} = \ aleph _ {1}}$ ${\ displaystyle \ beth _ {1} = \ aleph _ {1}}$ (power of the continuum ), the reverse is also true. If we accept a more powerful generalized continuum hypothesis , then both hierarchies completely coincide: $\beth _{\alpha }=\aleph _{\alpha }$ ${\ displaystyle \ beth _ {\ alpha} = \ aleph _ {\ alpha}}$ ${\ displaystyle \ beth _ {\ alpha} = \ aleph _ {\ alpha}}$ for any index $\alpha .$ ${\ displaystyle \ alpha.}$ $\ alpha.$ If we accept that the continuum hypothesis is incorrect, then there are many $\aleph$ ${\ displaystyle \ aleph}$ $\ aleph$ which are not $\beth$ ${\ displaystyle \ beth}$ ${\ displaystyle \ beth}$ .

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Definition

General definition for finite indexes:

\beth _{0}=\aleph _{0}

{\ displaystyle \ beth _ {0} = \ aleph _ {0}}

\beth _{\alpha +1}=2^{\beth _{\alpha }}

{\ displaystyle \ beth _ {\ alpha +1} = 2 ^ {\ beth _ {\ alpha}}}

For infinite indices:

\beth _{\lambda }=\sup\{\beth _{\alpha }:\alpha <\lambda \}.

{\ displaystyle \ beth _ {\ lambda} = \ sup \ {\ beth _ {\ alpha}: \ alpha <\ lambda \}.}

Examples

bet-zero ( ${\displaystyle \beth _{0}}$ ${\ displaystyle {\ displaystyle \ beth _ {0}}}$ ) - equal to ${\displaystyle \aleph _{0}}$ ${\ displaystyle {\ displaystyle \ aleph _ {0}}}$ .
bet one ( $\beth _{1}$ ${\ displaystyle \ beth _ {1}}$ ) - a lot of continuum .
bet two ( $\beth _{2}$ ${\ displaystyle \ beth _ {2}}$ ) - 2 ^c , for example, a boolean of real numbers.
bet omega ( $\beth _{\omega }$ ${\ displaystyle \ beth _ {\ omega}}$ ) - 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.

Links

Cardinals and ordinals (Unc.) . https://www.math.wisc.edu/ .

Bet number

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Definition

Literature

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