Re: An uncountable countable set

Tony Orlow wrote:
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

Isn't there symmetry about the origin thus it's 2 times oo^2?

Obviously half of the integers are even.

What are cases against use or validity of IFR? How do you address
those?

Ross

.