Paul Rubin wrote:
.........
That is very straightforward if you don't mind a handwave. Let S be
some arbitrary subset of the reals, and let f(x)=0 if x is in S, and 1
otherwise (this is a discontinuous function if S is nonempty). How
many different such f's can there be? Obviously one for every
possible subset of the reals. The cardinality of such f's is the
power set of the reals, i.e. much larger than the set of reals.
On the other hand, let g be some arbitrary continuous function on the
reals. Let H be the image of Q (the set of rationals) under g. That
is, H = {g(x) such that x is rational}. Since g is continuous, it is
completely determined by H, which is a countable set. So the
cardinality is RxN which is the same as the cardinality of R.
ok so probably true then
If true that makes calculus (and hence all of our understanding of
such "natural" concepts) pretty small and perhaps non-applicable.
No really, it is just set theory, which is a pretty bogus subject in
some sense. There aren't many discontinuous functions in nature.
There is a philosophy of mathematics (intuitionism) that says
classical set theory is wrong and in fact there are NO discontinuous
functions. They have their own mathematical axioms which allow
developing calculus in a way that all functions are continuous.
so does this render all the discreteness implied by quantum theory unreliable?
or is it that we just cannot see(measure) the continuity that really happens?
Certainly there are people like Wolfram who seem to think we're in some kind of
giant calculating engine where state transitions are discontinuous.
On the other hand R Kalman (of Bucy and Kalman filter fame) likened
study of continuous linear dynamical systems to "a man searching for
a lost ring under the only light in a dark street" ie we search
where we can see. Because such systems are tractable doesn't make
them natural or essential or applicable in a generic sense.
Really, I think the alternative he was thinking of may have been
something like nonlinear PDE's, a horribly messy subject from a
practical point of view, but still basically free of set-theoretic
monstrosities. The Banach-Tarski paradox has nothing to do with nature.
My memory of his seminar was that he was concerned about our failure to model
even the simplest of systems with non-linearity and/or discreteness. I seem to
recall that was about the time that chaotic behaviours were starting to appear
in the control literature and they certainly depend on non-linearity.
--
Robin Becker
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