Aleph-0

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).

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