Re: An uncountable countable set

Ross A. Finlayson wrote:
Tony Orlow wrote:

For what it's worth, and I know this doesn't add a lot of credibility to
Ross in your eyes, coming from me, but I think Ross has a genuine
intuition that isn't far off with respect to what's controversial in
modern math. Sure, he gets repetitive and I don't agree with everything
he says, but his cryptic "Well order the reals", which I actually
haven't seen too much of lately, is a direct reference to his EF
(Equivalence Function, yes?) between the naturals and the reals in
[0,1). The reals viewed as discrete infinitesimals map to the
hypernaturals, anyway, and his EF is a special case of my IFR. So, to
answer your question, I think Ross makes some sense. But, of course,
coming from me, that probably doesn't mean much. :)

TOE-Knee

Hi,

What is this IFR, "inverse function rule"? I've heard you mention it.
Is it just general EF?

Ross

Hey Ross!

The Inverse Function Rule uses infinite-case induction to finely order infinite sets of reals mapped from a standard set, N. Where there is a bijection between N and a set S using f(n)=s, there is a mapping from S to N using g(s)=n, where g(f(x))=f(g(x)) (inverse functions for the bijection). The size of the set S over the interval [a,b] is given by floor(g(b)-g(a)+1). (I think I wrote that correctly). This works for all finite sets of reals. The number of square roots, for instance, between 1 and 100 is floor(100^2-1^2+1), 10000 square roots, from sqrt(1) to sqrt(10000). IFR can easily be used to show that the evens are half as numerous as the naturals, and other interesting "facts".

EF is the special case of IFR mapping the naturals in [0,oo) to the reals in [0,1), using the mapping function f(n)=n/oo. Isn't that how you define the equivalency function? Given this mapping, we can say g(s)=s*oo, so that over the entire real line, we have oo^2 reals, oo in each unit interval, over oo unit intervals. Does that sound about right?

Tony
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