I forgot to say that infinity is both absolutely necessary for a useful (or rich?) mathematics, and the source of almost all its problems. This makes for an interesting "paradox" which can be discussed at some length. Mathematicians have been propelled into a world almost against their will, like physicists wondering why the rules of physics have to be so incredibly complicated and almost beyond comprehension. Steve Gray Fred lunnon wrote:
On 10/25/07, Eugene Salamin <gene_salamin@yahoo.com> wrote:
... 1. You won't have time to cover two topics, i.e. both proofs and infinity, so arrange with the interviewers which one they will start with. My guess is that infinity is the easier of the two. There is so much that can be discussed: the countable and uncountable cardinals, Cantor diagonalization, the ordinals, the asymptotic infinity as in "let x approach infinity".
Also complex infinity, distinct from the last --- although you could argue that "+oo" and "-oo" are shorthand for the direction of approach to the same limit. Some computer systems (CDC 6600 hardware, Maple CAS software) have attempted to introduce infinity as a floating-point value, though with little apparent success.
Also the projective prime at infinity (line in plane geometry). Both these have the useful property that they can be re-interpreted as finite objects in some distinct space (see Riemann sphere, vanishing points, etc).
WFL
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