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Aleph-0


The set theory symbol aleph_0 refers to a set having the same cardinal number as the "small" infinite set of integers. The symbol aleph_0 is often pronounced "aleph-null" rather than "aleph-zero," probably because Null is the word for "zero" in Georg Cantor's native language of German. It is sometimes also pronounced "aleph-zero" or "aleph-naught," the latter of which is also spelled "aleph-nought."

The algebraic numbers also have cardinality aleph_0. Rather surprising properties satisfied by aleph_0 include

aleph_0^r=aleph_0  for r>0
(1)
raleph_0=aleph_0  for r!=0
(2)
aleph_0+f=aleph_0,
(3)

where f is any finite set. However,

 2^(aleph_0)=aleph_0^(aleph_0)=c,
(4)

where c is the continuum.

Renteln and Dundes (2005) give the following humorous mathematical analog of the "99 bottles of beer on the wall" drinking song, which refers to its property that aleph_0-1=aleph_0: "Aleph-null bottles of beer on the wall, Aleph-null bottles of beer, Take one down, and pass it around, Aleph-null bottles of beer on the wall" (repeat).

Incidentally, another variant runs, "99 bottles of beer on the wall, 99 bottles of beer, Take one down, Put it back up, 99 bottles of beer on the wall" (repeat).


See also

Aleph-1, Cardinal Number, Continuum, Continuum Hypothesis, Countably Infinite, Finite, Infinite, Transfinite Number, Uncountably Infinite

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References

Renteln, P. and Dundes, A. "Foolproof: A Sampling of Mathematical Folk Humor." Notices Amer. Math. Soc. 52, 24-34, 2005.

Referenced on Wolfram|Alpha

Aleph-0

Cite this as:

Weisstein, Eric W. "Aleph-0." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Aleph-0.html

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