In mathematics, particularly in set theory, the beth numbers are a certain sequence of infinite cardinal numbers (also known as transfinite numbers), conventionally written , where is the second Hebrew letter (beth). The beth numbers are related to the aleph numbers (), but unless the generalized continuum hypothesis is true, there are numbers indexed by that are not indexed by .

Definition

Beth numbers are defined by transfinite recursion:

where is an ordinal and is a limit ordinal.[1]

The cardinal is the cardinality of any countably infinite set such as the set of natural numbers, so that .

Let be an ordinal, and be a set with cardinality . Then,

Given this definition,

are respectively the cardinalities of

so that the second beth number is equal to , the cardinality of the continuum (the cardinality of the set of the real numbers), and the third beth number is the cardinality of the power set of the continuum.

Because of Cantor's theorem, each set in the preceding sequence has cardinality strictly greater than the one preceding it. For infinite limit ordinals, λ, the corresponding beth number is defined to be the supremum of the beth numbers for all ordinals strictly smaller than λ:

One can also show that the von Neumann universes have cardinality .

Relation to the aleph numbers

Assuming the axiom of choice, infinite cardinalities are linearly ordered; no two cardinalities can fail to be comparable. Thus, since by definition no infinite cardinalities are between and , it follows that

Repeating this argument (see transfinite induction) yields for all ordinals .

The continuum hypothesis is equivalent to

The generalized continuum hypothesis says the sequence of beth numbers thus defined is the same as the sequence of aleph numbers, i.e., for all ordinals .

Specific cardinals

Beth null

Since this is defined to be , or aleph null, sets with cardinality include:

Beth one

Main article: cardinality of the continuum

Sets with cardinality include:

Beth two

(pronounced beth two) is also referred to as 2c (pronounced two to the power of c).

Sets with cardinality include:

Beth omega

(pronounced beth omega) is the smallest uncountable strong limit cardinal.

Generalization

The more general symbol , for ordinals α and cardinals κ, is occasionally used. It is defined by:

if λ is a limit ordinal.

So

In Zermelo–Fraenkel set theory (ZF), for any cardinals κ and μ, there is an ordinal α such that:

And in ZF, for any cardinal κ and ordinals α and β:

Consequently, in ZF absent ur-elements with or without the axiom of choice, for any cardinals κ and μ, the equality

holds for all sufficiently large ordinals β. That is, there is an ordinal α such that the equality holds for every ordinal βα.

This also holds in Zermelo–Fraenkel set theory with ur-elements (with or without the axiom of choice), provided that the ur-elements form a set which is equinumerous with a pure set (a set whose transitive closure contains no ur-elements). If the axiom of choice holds, then any set of ur-elements is equinumerous with a pure set.

Borel determinacy

Borel determinacy is implied by the existence of all beths of countable index.[5]

See also

References

  1. ^ Jech, Thomas (2002). Set Theory (3rd Millennium ed, rev. and expanded. Corrected 4th printing 2006 ed.). Springer. p. 55. ISBN 978-3-540-44085-7.
  2. ^ a b Soltanifar, Mohsen (2023). "A Classification of Elements of Function Space F(R,R)". Mathematics. 11 (17): 3715. doi:10.3390/math11173715.
  3. ^ Soltanifar, Mohsen (2021). "A Generalization of the Hausdorff Dimension Theorem for Deterministic Fractals". Mathematics. 9 (13): 1546. doi:10.3390/math9131546.
  4. ^ Soltanifar, Mohsen (2022). "The Second Generalization of the Hausdorff Dimension Theorem for Random Fractals". Mathematics. 10 (5): 706. doi:10.3390/math10050706.
  5. ^ Leinster, Tom (23 July 2021). "Borel Determinacy Does Not Require Replacement". The n-Category Café. The University of Texas at Austin. Retrieved 25 August 2021.

Bibliography