
Re: Problems with Infinity?
Posted:
Feb 26, 2013 12:55 PM


Don Kuenz wrote: > > Joseph Nebus <nebusj@rpi.edu> wrote: > > > > A couple recent posts on James Nicoll's LiveJournal  > > > > http://jamesnicoll.livejournal.com/4145868.html > > http://jamesnicoll.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 differentsize 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 onepoint compactification of N, infinity of the onepoint compactification of R, infinity of the twopoint compactification of R, infinity of the onepoint compactification of C, infinities of the projective extension of the plane, infinity of Lebesguetype integration theory, infinities of the nonstandard 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 Gdelta noncompact metric spaces, infinity in the theory of convex optimization, etc.;
each of these has a clear definition and a set of welldefined rules for handling it.
And the winner is... the really, really real infinity imagined by inexperienced debaters of foundations of mathematics; this one has the advantage that it need not be defined ("it's just there, don't you see?") and the user can switch from one set of rules to another, without warning, and without worrying about consistency, for the purpose of scoring points in idle and uneducated (at least on one side) debates.  When a true genius appears in the world, you may know him by this sign, that the dunces are all in confederacy against him. Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting

