infinity is not a number
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Even that is questionable. If you want 1/infinity = 0 then you have problems, but if you think of it as infinitesimally small but non-zero, then you can generate an ordered field that extends R. In that field you're treating infinity as a number and using naive manipulations.
Interesting, and a good point. I haven’t internalized the hyperreals fully (which is what I think you’re referring to). They’ve always been of recreational interest to me, but I never gained an intuition of the notion of “standard part” or nilpotent differentials.
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I was actually talking about something more elementary than the hyperreals: you just take rational functions evaluated at infinity, which form an ordered field (but not a complete one, obviously) in a natural way.
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that’s true.. I’m going to have to think about that in the morning!
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