From: Technotranscendence (neptune@mars.superlink.net)
Date: Sat Oct 12 2002 - 07:13:53 MDT
On Saturday, October 12, 2002 5:41 AM Lee Corbin lcorbin@tsoft.com
wrote:
>> Infinity is a great thing, it exists between any two
>> points in the continuum between those two points.
Well, one order of infinity does.
>> Here's a funny fact, the constant function f(x)=1
>> integrated over the naturals evaluates to equal to two.
>
> How d'ya get that? I think it's zero.
I get zero too. In fact, measure theory leads to the zero result for
anything smaller than integrating over the reals.
>> That leads into indefiniteness, like, what is the smallest real
number,
>> (iota), whatnot.
>
> There isn't a smallest real number. Or there is.
I don't know what Ross is talking about here. There isn't a smallest
real number in standard analysis. The reals are dense (as are the
rationals), meaning as Ross seems to get, between any two of them,
there's another.
In nonstandard analysis, there is an infinitesimal, but these are
superadded onto the reals. I'm not familiar enough with nonstandard
analysis (hyperreals) to know if the idea is more than a parlor trick.:)
Cheers!
Dan
See more of my writings at:
http://uweb.superlink.net/neptune/
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