The set theory symbol 
refers to a set having the same
cardinal 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
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
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).
