Date:    Tue, 5 May 1998 11:29:32 -0500
<br>From:    Mark Mandel <Mark[AT SYMBOL GOES HERE]DRAGONSYS.COM>
<br>Subject: bottles of beer
<br>
<br>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
<br>have been at summer camp in upstate NY (upstate being defined for me then as anywhere north of Westchester
<br>County), in the late 50s or early 60s. "Take one down" I learned later.
<br>
<br>Then there are the higher mathematics of beer:
<br>
<br>        Aleph-null bottles of beer on the wall,
<br>        Aleph-null bottles of beer,
<br>        If one of those bottles should happen to fall,
<br>        Aleph-null bottles of beer on the wall.
<br>
<br>(How many integers are there? An infinite number; specifically, Georg Cantor's aleph-null [written with the Hebrew letter
<br>aleph and a subscript zero]. How many non-zero integers are there? Just as many. Therefore "Aleph-null bottles of beer
<br>on the wall".)
<br>
<br>        Aleph-null bottles of beer on the wall,
<br>        Aleph-null bottles of beer,
<br>        If aleph-null bottles should happen to fall,
<br>        Aleph-null bottles of beer on the wall.
<br>
<br>(How many even integers are there? Well, if you list all the integers (all aleph-null of them) and then replace each one by
<br>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
<br>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.
<br>Therefore...)
<br>
<br>And, to switch to the other form of the lyric:
<br>
<br>        Aleph-null bottles of beer on the wall,
<br>        Aleph-null bottles of beer,
<br>        Take aleph-null bottles down aleph-null times,
<br>        Aleph-null bottles of beer on the wall.
<br>
<br>(How many prime numbers are there? Aleph-null, apparently. Starting with the list of integers, take away all the powers
<br>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
<br>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
<br>themselves, as well as every number that's a multiple of two or more different primes. Q.E.D.)
<br>
<br>And now I'd better sober up and get back to work.
<br>
<br>       Mark A. Mandel : Senior Linguist : mark[AT SYMBOL GOES HERE]dragonsys.com
<br>    Dragon Systems, Inc. : speech recognition : +1 617 796-0267
<br> 320 Nevada St., Newton, MA 02160, USA : http://www.dragonsys.com/
<br>           Personal home page: http://world.std.com/~mam/
<br>