See also: Aleph

In Mathematical, a aleph is an element of a succession of numbers used to represent the cardinal (i.e. to some extent size) of the infinite units. They are named according to the symbol which notes them, א, the letter aleph of the Hebrew alphabet.

The cardinal of the natural whole is Aleph-zéro ($\backslash \; aleph\_0$); under certain conditions, the following cardinal is Aleph-un ($\backslash \; aleph\_1$), then $\backslash \; aleph\_2$, and so on. In this way, it is possible to define a cardinal number $\backslash \; aleph\_\; \backslash \; alpha$ for all ordinal Nombre α.

The concept goes back to Georg Cantor, which defined the concept of cardinality and realized that unit infinite distinct could have different cardinals.

The aleph numbers differ from the Infini ($\backslash \; infty$) that one finds in algebra; the aleph measure overall sizes while the infinite one is generally defined like the limit of the real Droite or an extreme of the completed real Droite. While some aleph are larger than others, $\backslash \; infty$ represents only $\backslash \; infty$.

Aleph-zéro

See also: Aleph-zéro

Aleph-zéro ($\backslash \; aleph\_0$) is by definition the cardinal of the whole of the natural whole . If the Axiome of the choice is checked, it is also about the smallest infinite cardinal. A unit has $\backslash \; aleph\_0$ for cardinal if and only if it is infinite and countable.

Aleph-un

See also: Aleph-un

$\backslash \; aleph\_1$ is the cardinal of the whole of the ordinal numbers countable (a unit itself indénombrable). If the axiom of the choice is used, $\backslash \; aleph\_1$ is the smallest cardinal who follows $\backslash \; aleph\_0$.

Continuous

The cardinal of the whole of the real numbers is $2^\; \{\backslash \; aleph\_0\}$. Within the framework of the set theory of Zermelo-Fraenkel provided with the Axiom of the choice, the Hypothèse of continuous the is equivalent to the identity $2^\; \{\backslash \; aleph\_0\}\; =\; \backslash \; aleph\_1.$; in the absence of this assumption, the place of $2^\; \{\backslash \; aleph\_0\}$ in the hierarchy of the aleph is not defined with certainty.

Aleph-ω

Conventionally, smallest ordinal infinite is noted ω and the cardinal $\backslash \; aleph\_\; \backslash \; omega$ is the upper limit of

$\backslash \; left\; \backslash \; \{\backslash ,\; \backslash \; aleph\_n:\; N\; \backslash \; in\; \backslash \; left\; \backslash \; \{\backslash ,\; 0,1,2,\; \backslash \; dowries\; \backslash ,\; \backslash \; right\; \backslash \}\; \backslash ,\; \backslash \; right\; \backslash \}.$

Aleph-α

For an arbitrary ordinal number α, aleph-α can be defined using the function cardinal successor which assigns with any cardinal number ρ the cardinal who follows it immediately ρ⁺.

It is possible then to define the aleph numbers in the following way:

$\backslash \; aleph\_\; \{0\}\; =\; \backslash \; omega$

$\backslash \; aleph\_\; \{\backslash \; alpha+1\}\; =\; \backslash \; aleph\_\; \{\backslash \; alpha\}\; ^+$

and for λ, ordinal limits infinite:

Fixed points

For very ordinal α:

$\backslash \; alpha\; \backslash \; Leq\; \backslash \; aleph\_\; \backslash \; alpha.$

In much of case, $\backslash \; aleph\_\; \{\backslash \; alpha\}$ is strictly higher than α. Certain ordinal limits are fixed points of the function aleph. The first cardinal of this type is the limit of the continuation

$\backslash \; aleph\_0,\; \backslash \; aleph\_\; \{\backslash \; aleph\_0\},\; \backslash \; aleph\_\; \{\backslash \; aleph\_\; \{\backslash \; aleph\_0\}\},\; \backslash \; ldots$

All inaccessible Cardinal is also a fixed point of the function aleph.

See too

Internal bonds

• Beth (number)
• Georg Cantor
• Together countable
• Together infinite
• Assumption of continuous the
• Cardinal number
• transfinite Number

External bonds

• Aleph-0 ( MathWorld )
• '' Aleph numbers '' ( PlanetMath )

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