🚀 go-pugleaf

David Madore (madore@clipper.ens.fr) wrote:
: Here is a proof that all real numbers are interesting:
: Let I be the set of interesting real numbers.
:   * I is not empty because 0, 1, e and pi are certainly in I.
:   * Any number sufficiently close to a given interesting number is
: interesting.
: (For example, a number which has the same first billion decimals as pi
: is
: clearly interesting.) Therefore I is open.
:   * A number which is the limit of a convergent sequence of interesting
: numbers
: is certainly interesting. Therefore I is closed.

: But R is connected. Therefore I=R, and all real numbers are interesting.

ROTFL :-) 

Since your argument for I being open might be questioned by some people,
an alternative proof would be:

1) All natural numbers are interesting because a smallest uninteresting
   natural number cannot exist.
2) Sums and products, as well as additive and multiplicative inverses 
   of interesting numbers are clearly interesting, hence all rational 
   numbers are interesting.
3) The limit of a sequence of convergent sequence of interesting numbers 
   is interesting, hence all real numbers are interesting. 

However, I guess both of us used the axiom that algebra and topology are
interesting, which might also be questioned by some people :-) 
 
--
Ulrich Lange                       Dept. of Chemical Engineering
                                   University of Alberta
lange@gpu.srv.ualberta.ca          Edmonton, Alberta, T6G 2G6, Canada