Date: Feb 27, 2013 6:59 PM
Author: Scott Fluhrer
Subject: Re: Problems with Infinity?


"Frederick Williams" <freddywilliams@btinternet.com> wrote in message
news:512CF70E.589D832E@btinternet.com...
> Don Kuenz wrote:
>>
>> Joseph Nebus <nebusj-@-rpi-.edu> wrote:

>> >
>> > A couple recent posts on James Nicoll's LiveJournal ---
>> >
>> > http://james-nicoll.livejournal.com/4145868.html
>> > http://james-nicoll.livejournal.com/4194844.html
>> >
>> > --- have left me aware that at least two Heinlein novels (_The Number
>> > Of The Beast_ and _Time Enough For Love_) contain mentions dismissing
>> > the Cantorian idea of there being different-size infinities, and that
>> > at least one Christopher Anvil story in which a journey to hyperspace
>> > reveals that the rationals and the integers *don't* have the same
>> > cardinality.

>>
>> A question for the group, if you please.
>>
>> Let's say a Mobius strip goes to infinity "feedback style" (in layman's
>> terms) while a line goes to two separate but equal infinities "linear
>> style." How many different infinities does that make according to
>> Cantor? One, two, or three?

>
> If you're interested in infinity, here is something that I posted
> recently:
>
> A few years ago Zdislav V. Kovarik made a post listing a dozen or more
> meaning of the word "infinity" as used in different branches of
> mathematics. I'm hoping that he won't mind me reposting it:
>
>
> There is a long list of "infinities (with no claim to exhaustiveness):
> infinity of the one-point compactification of N,
> infinity of the one-point compactification of R,
> infinity of the two-point compactification of R,
> infinity of the one-point compactification of C,
> infinities of the projective extension of the plane,
> infinity of Lebesgue-type integration theory,
> infinities of the non-standard extension of R,
> infinities of the theory of ordinal numbers,
> infinities of the theory of cardinal numbers,
> infinity adjoined to normed spaces, whose neighborhoods are
> complements of relatively compact sets,
> infinity adjoined to normed spaces, whose neighborhoods are
> complements of bounded sets,
> infinity around absolute G-delta non-compact metric spaces,
> infinity in the theory of convex optimization,
> etc.;
>
> each of these has a clear definition and a set of well-defined rules
> for handling it.


In case you're maintaining a list, there's also "the point at infinity" that
comes up with you're dealing with Elliptic Curve groups.

--
poncho