<Li> "Aleph - null bottles of beer on the wall". Aleph - null is the size of the set of all natural numbers, and is the smallest infinity and the only countable one; therefore, even if an infinite aleph - null of bottles fall, the same amount remains . </Li> <Li> "Aleph - one / two / three / etc. bottles of beer on the wall". Aleph - one, two, three, etc. are uncountable infinite sets, which are larger than countable ones; therefore, if only a countable infinity of bottles fall, an uncountable number remains . </Li> <P> Other versions in Byrd's collection involve concepts including geometric progressions, differentials, Euler's identity, complex numbers, summation notation, the Cantor set, the Fibonacci sequence, and the continuum hypothesis, among others . </P> <P> The computer scientist Donald Knuth proved that the song has a complexity of O (log ⁡ N) (\ displaystyle O (\ log N)) in his in - joke - article "The Complexity of Songs". </P>

How many bottles of beer on the wall