From: Peter Percival Subject: Re: x = oo? Date: Fri, 19 May 2000 22:07:07 +0100 Newsgroups: sci.math Summary: [missing] John Prussing wrote: > > Under what conditions (if any) is it permissible to consider oo (infinity) > to be a number rather than a limit? In reply to a question "Is infinity an odd or even number?" on 8/8/99 Professor Kovarik produced this list: 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.; > > My memory is a bit hazy here, but I recall attending a lecture some years > ago on (perhaps?) non-standard calculus. I believe that oo was (due to > some recent advances in logic?) defined to be a number, oo - 1 to be > a very large finite number, and its reciprocal to be an infinitesimal. > > Does this sound at all familiar? Is there some valid theoretical basis > for defining oo to be a number? > > That seems to violate (or at least rewrite) the old rules, e.g., > that for f(x) = 1/x, one should not say that f(0) = oo. > -- > =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= > John E. Prussing > Dept. of Aeronautical & Astronautical Engineering > University of Illinois at Urbana-Champaign > http://www.uiuc.edu/~prussing > =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=