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pycyn@hidden.email scripsit:
> Nor, I think,
> in the numerical disiplines: arithmetic is incomplete and undecidable,
> analysis is at least complete, so, if integers were real numbers, we could
> complete arithmetic by going through analysis (only slightly specious
> argument).
Umm, rather specious. Saying that the integers are reals does not mean that
integer operations are the same as real operations: obviously they aren't.
> We do -- outside of computer work and very fussy projects -- tend to use the
> same notation for the lot (as the rationals get totally absorbed into the
> reals, while still being very different things -- sets of ordered pairs, vs.
> ordered pairs of sets, for example).
Well, if you take the view that a natural number n *is* the set of sets
of cardinality n (Frege) or that it *is* a Zermelo set, or whatever, then
you may have trouble identifying the rest of the numeric tower with these
particular sets. But I don't take that viewpoint: I'm an unabashed Platonist.
> Nice, if contradictory looking. Certainly more intelligible (and so
> convincing) than the usual one through the calculus.
Here's a calculus-based proof to chew on:
xy = (x + x + ... + x) y times
x^2 = (x + x + ... + x) x times
Take the derivative of both sides:
2x = (1 + 1 + 1 + ... + 1) x times
2x = x
2 = 1
--
John Cowan jcowan@hidden.email
"You need a change: try Canada" "You need a change: try China"
--fortune cookies opened by a couple that I know