Beth number

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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[edit]

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,

  • denotes the power set of (i.e., the set of all subsets of ),
  • the set denotes the set of all functions from to {0,1},
  • the cardinal is the result of cardinal exponentiation, and
  • is the cardinality of the power set of .[2]

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),[2] 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[edit]

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[edit]

Beth null[edit]

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

Beth one[edit]

Sets with cardinality include:

Beth two[edit]

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

Sets with cardinality include:

  • The power set of the set of real numbers, so it is the number of subsets of the real line, or the number of sets of real numbers
  • The power set of the power set of the set of natural numbers
  • The set of all functions from R to R (RR)
  • The set of all functions from Rm to Rn
  • The power set of the set of all functions from the set of natural numbers to itself, so it is the number of sets of sequences of natural numbers
  • The Stone–Čech compactifications of R, Q, and N
  • The set of deterministic fractals in Rn [3]
  • The set of random fractals in Rn [4]

Beth omega[edit]

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

Generalization[edit]

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[edit]

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

See also[edit]

References[edit]

  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 "beth numbers". planetmath.org. Retrieved 2020-09-05.
  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[edit]