From: Mark Mandel

Subject: bottles of beer

I learned it as "happen to fall". Actually, I first understood it as "hop into fall", and only later got it parsed right. This would

have been at summer camp in upstate NY (upstate being defined for me then as anywhere north of Westchester

County), in the late 50s or early 60s. "Take one down" I learned later.

Then there are the higher mathematics of beer:

Aleph-null bottles of beer on the wall,

Aleph-null bottles of beer,

If one of those bottles should happen to fall,

Aleph-null bottles of beer on the wall.

(How many integers are there? An infinite number; specifically, Georg Cantor's aleph-null [written with the Hebrew letter

aleph and a subscript zero]. How many non-zero integers are there? Just as many. Therefore "Aleph-null bottles of beer

on the wall".)

Aleph-null bottles of beer on the wall,

Aleph-null bottles of beer,

If aleph-null bottles should happen to fall,

Aleph-null bottles of beer on the wall.

(How many even integers are there? Well, if you list all the integers (all aleph-null of them) and then replace each one by

doubling it, you have a list of all the even integers, and it's the same length as the original list. You can also derive this list

by striking out all the odd integers (all aleph-null of them) in the original list, i.e., taking away aleph-null from aleph-null.

Therefore...)

And, to switch to the other form of the lyric:

Aleph-null bottles of beer on the wall,

Aleph-null bottles of beer,

Take aleph-null bottles down aleph-null times,

Aleph-null bottles of beer on the wall.

(How many prime numbers are there? Aleph-null, apparently. Starting with the list of integers, take away all the powers

of 2 except 2 itself (there are aleph-null of them: 2^2, 2^3, 2^4...). Then take away all the powers of 3 (aleph-null of

them), then all the powers of 5, of 7, and of each of the aleph-null primes in turn. What's left? Why, all the prime numbers

themselves, as well as every number that's a multiple of two or more different primes. Q.E.D.)

And now I'd better sober up and get back to work.

Mark A. Mandel : Senior Linguist : mark[AT SYMBOL GOES HERE]dragonsys.com

Dragon Systems, Inc. : speech recognition : +1 617 796-0267

320 Nevada St., Newton, MA 02160, USA : http://www.dragonsys.com/

Personal home page: http://world.std.com/~mam/