Newsgroups: sci.lang,sci.math
Path: cantaloupe.srv.cs.cmu.edu!das-news2.harvard.edu!news2.near.net!news.mathworks.com!europa.eng.gtefsd.com!howland.reston.ans.net!vixen.cso.uiuc.edu!uchinews!dent.uchicago.edu!wald
From: wald@dent.uchicago.edu (Kevin Wald)
Subject: Re: Procrustean linguistics (was: Ferengi Language)
Message-ID: <1994Nov4.200603.6085@midway.uchicago.edu>
Followup-To: sci.math 
Sender: news@uchinews.uchicago.edu (News System)
Organization: Dept. of Mathematics
References: <372f0l$p0d@mother.usf.edu> <38v0su$fcn@tardis.trl.OZ.AU> <smryanCyJ38I.ACz@netcom.com> <39c9rh$h5j@mother.usf.edu>
Date: Fri, 4 Nov 1994 20:06:03 GMT
Lines: 66
Xref: glinda.oz.cs.cmu.edu sci.lang:32614 sci.math:85331

In article <39c9rh$h5j@mother.usf.edu> millert@grad.csee.usf.edu (Timothy Miller) writes:
>S M Ryan (smryan@netcom.com) wrote:
>: : the one who believes that the cardinality of possible sentences is
>: : beyond aleph-0, beyond aleph-1, .... aleph-aleph (wicked grin)
>: : and more. And was  thanked by the party hacks for this illuminating 
>
>: Don't you mean aleph-omega?
>
>No.  And even if you were close to correct, it would be alpha-omega.  But 
>aleph-0, aleph-1 actually means something.
>
>However, I'm not sure what that is.  Could someone tell me?

Actually, S. M. Ryan is right, and T. Miller is somewhat off. However,
to explain why, and what aleph_0 and aleph_1 are, takes a little while.
All those who don't want to hear about transfinite set theory, bail
out now.

Lowercase omega is the name given to the first infinite ordinal; that
is, if we "count" transfinitely in the usual way in set theory, omega
is the first ordinal we reach after all the finite counting numbers
(0, 1, 2, etc.). (After omega, we then hit omega+1, omega+2, . . .
omega+omega = omega*2, omega*2+1, . . . omega*3, and a large collection
of other ordinals, most of them much stranger.)

Note that I haven't yet defined what an ordinal is. In set theory we
define the ordinals to be certain sets, in such a way that each
ordinal is the set of all ordinals less than it. (How we actually
make this definition is somewhat technical.) Thus, 0 is {}, the empty
set, 1 is {0} = {{}}, 2 is {0,1} = {{},{{}}}, omega is 
{ x | x is a nonnegative integer }, and so forth.

Now, if we have the Axiom of Choice, we can show that for any set x,
there is at least one ordinal w of the same size as x (that is, for
some w, there is a one-to-one and onto function from w to x). This
gives us a convenient way to name the "sizes" of sets: For any x,
the cardinality of x is the least ordinal of the same size of x.
Those ordinals which are the least of a given size are called
cardinals. Thus, omega is a cardinal (since all the ordinals less than
it only contain finitely many elements, and omega itself contains
infitely many elements), but omega+1 isn't, since it turns out
to be of the same size as omega.

We can now use the ordinals to give names to all of the infinite
cardinals: The nth infinite cardinal is designated aleph_n. Thus,
omega, the smallest infinite cardinal, is aleph_0, the next cardinal
up is aleph_1, and so forth. The set of integers and the set of 
rational numbers each have cardinality aleph_0 (that is, they are
each the same size as omega); the Continuum Hypothesis, which can
neither be proven nor disproven in ordinary ZFC set theory, claims
that the set of real numbers has cardinality aleph_1.

Aleph_omega, then, is the least cardinal greater than all of aleph_n
for n a nonnegative integer. This may seem rather large, but as
cardinals go, it's small potatoes. Note that we could also call
this cardinal aleph_aleph_0, since aleph_0 = omega (this may have been
what whoever made the original "aleph-aleph" remark was referring to),
but in most cases we wouldn't, because we want to think of omega as
wearing its ordinal hat, not its cardinal hat. (Mitre?)

Followups to sci.math.

Kevin Wald                  |   O Joy, thou beam from Heaven's wire-
			    |   Less, daughter of Elysium;
wald@math.uchicago.edu      |   Thy shrine we enter, drunk with fire,
			    |   Imbibing gin-and-cesium.
