Beth Two

In mathematics, particularly in set theory, the beth numbers are a certain sequence of infinite cardinal numbers (also known as transfinite numbers), conventionally written $\beth_0,\ \beth_1,\ \beth_2,\ \beth_3,\ \dots$, where $\beth$ is the second Hebrew letter ( beth). The beth numbers are related to the aleph numbers ($\aleph_0,\ \aleph_1,\ \dots$), but unless the generalized continuum hypothesis is true, there are numbers indexed by $\aleph$ that are not indexed by $\beth$.

Definition

Beth numbers are defined by transfinite recursion:

* $\beth_0=\aleph_0,$
* $\beth_=2^,$
* $\beth_=\sup\,$
where $\alpha$ is an ordinal and $\lambda$ is a limit ordinal.

The cardinal $\beth_0=\aleph_0$ is the cardinality of any countably infinite set such as the set $\mathbb$ of natural numbers, so that $\beth_0=, \mathbb,$.

Let $\alpha$ be an ordinal, and $A_\alpha$ be a set with cardinality $\beth_\alpha=, A_\alpha,$. Then,
*$\mathcal(A_\alpha)$ denotes the power set of $A_\alpha$ (i.e., the set of all subsets of $A_\alpha$),
*the set $2^ \subset \mathcal(A_\alpha \times 2)$ denotes the set of all functions from $A_\alpha$ to ,
*the cardinal $2^$ is the result of cardinal exponentiation, and
*$\beth_=2^=, 2^, =, \mathcal(A_\alpha),$ is the cardinality of the power set of $A_\alpha$.

Given this definition,

:$\beth_0,\ \beth_1,\ \beth_2,\ \beth_3,\ \dots$

are respectively the cardinalities of

:$\mathbb,\ \mathcal(\mathbb),\ \mathcal(\mathcal(\mathbb)),\ \mathcal(\mathcal(\mathcal(\mathbb))),\ \dots.$

so that the second beth number $\beth_1$ is equal to $\mathfrak c$, the cardinality of the continuum (the cardinality of the set of the real numbers), and the third beth number $\beth_2$ 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:

:$\beth_=\sup\.$

One can also show that the von Neumann universes $V_$ have cardinality $\beth_$.

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 $\aleph_0$ and $\aleph_1$, it follows that
:$\beth_1 \ge \aleph_1.$
Repeating this argument (see transfinite induction) yields
$\beth_\alpha \ge \aleph_\alpha$
for all ordinals $\alpha$.

The continuum hypothesis is equivalent to
:$\beth_1=\aleph_1.$

The generalized continuum hypothesis says the sequence of beth numbers thus defined is the same as the sequence of aleph numbers, i.e.,
$\beth_\alpha = \aleph_\alpha$
for all ordinals $\alpha$.

Specific cardinals

Beth null

Since this is defined to be $\aleph_0$, or aleph null, sets with cardinality $\beth_0$ include:

*the natural numbers N
*the rational numbers Q
*the algebraic numbers
*the computable numbers and computable sets
*the set of finite sets of integers
*the set of finite multisets of integers
*the set of finite sequences of integers

Beth one

Sets with cardinality $\beth_1$ include:

*the transcendental numbers
*the irrational numbers
*the real numbers R
*the complex numbers C
*the uncomputable real numbers
*Euclidean space