First I would establish the need for infinity. Point out that the alternative is to declare a greatest legitimate number. That implies that the basic axiom that every integer has a unique successor needs to be modified in a complicated way. Also the rules for addition, etc., need to be modified too. Then I would point out the usefulness of infinity, for example using the fact that the value of sqrt(2) can not be written down with complete precision with a finite number of digits. Then mention that certain important values can't even be written in closed form, such as sin(40 degrees) etc. So far as the cardinality of the evens vs. the integers, don't speak of "just as many" or "equal in number", but mention only that they can be put in 1:1 correspondence, as if that were an odd idea used only by mathematicians. That should stop any arguments. Next, and perhaps finally, say that the reals cannot be put 1:1 with the integers, etc. I would not mention anything geometric, given the absence of illustrations, especially the hard-to-visualize point and line at infinity. (I "get it" only by not trying to picture it.) 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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