Re: Is zero even or odd?
From: Dave Seaman (dseaman_at_no.such.host)
Date: 12/29/04
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Date: Wed, 29 Dec 2004 01:24:38 +0000 (UTC)
On Tue, 28 Dec 2004 18:49:00 -0600, Matthew Russotto wrote:
> In article <cqs0gk$gpp$1@mailhub227.itcs.purdue.edu>,
> Dave Seaman <dseaman@no.such.host> wrote:
>>On Tue, 28 Dec 2004 09:46:05 -0600, Matthew Russotto wrote:
>>> In article <cqqp1n$rgb$2@mailhub227.itcs.purdue.edu>,
>>> Dave Seaman <dseaman@no.such.host> wrote:
>>>>On Mon, 27 Dec 2004 20:58:35 -0600, Matthew Russotto wrote:
>>>>> In article <t4rss0duo9eho2urcsibtq302e3s3edqkr@4ax.com>,
>>>>> vonroach <hadrainc@earthlink.net> wrote:
>>>>>>On Fri, 24 Dec 2004 19:46:25 GMT, "Nicholas O. Lindan" <see@sig.com>
>>>>>>wrote:
>>>>>>
>>>>> Well, 0^0 is a mess. But lim x->0 0^x is well defined.
>>>>No, it isn't. That limit does not exist.
>>> Most certainly does. It's zero.
>>Wrong. That limit cannot exist because 0^x is undefined for all x < 0.
> Doesn't matter; you can find a delta for every epsilon.
Not true. In fact, you cannot find a single delta that works for *any*
epsilon > 0. I'll let you choose any epsilon you like. You still lose.
>>>>But 0^0 does exist and has nothing to do with limits.
>>> 0^0 can be defined by convention, of course, as is 0 factorial.
>>I consider it to be something more than a mere convention. In Suppes:
>>_Axiomatic Set Theory_, it's a *theorem* that m^0 = 1 for every cardinal
>>m. Since 0 is a cardinal, the corollary is that 0^0 = 1. Specifically,
>>it represents the cardinality of the set of mappings from the empty set
>>to itself.
>>A corollary is the very antithesis of a "convention."
> If there was a proof, one which doesn't depend on a contrived meaning
> for exponentiation, I'd agree with you. But then, there wouldn't have
> been such a controversy if there was such a proof.
This is not a contrived meaning of exponentiation. It's the standard
definition for a^b where a and b are cardinal numbers, and this is a
necessary first step before extending the definition to more general
situations.
Consider 2^3, for example. It's the cardinality of the set of mappings
from the set { 0, 1, 2 } to the set { 0, 1 }, which is 8.
Besides, there is also the definition from algebra, in which x^n is
defined whenever x is a member of a monoid M and n is a natural number.
In particular, x^0 = e, the identity in M.
-- Dave Seaman Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. <http://www.commoncouragepress.com/index.cfm?action=book&bookid=228>
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