Aleph-0 (original) (raw)
Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of Mathematics Geometry History and Terminology Number Theory Probability and Statistics Recreational Mathematics Topology
Alphabetical Index New in MathWorld
The set theory symbol 
refers to a set having the samecardinal number as the "small" infinite set of integers. The symbol 
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 
. Rather surprising properties satisfied by 
include
where 
is any finite set. However,
 |
(4) |
|---|
where 
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-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
Explore with Wolfram|Alpha
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
Cite this as:
Weisstein, Eric W. "Aleph-0." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Aleph-0.html