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