Re: Well Ordering the Reals

David R Tribble said:
> Daryl McCullough wrote:
> > Actually, I think Tony is thinking (in his web page) of an enumeration
> > of a dense subset of the reals. That is, for any two distinct reals r_1
> > and r_2 there is a real r in his set such that r is between r_1 and r_2.
> > He seems to think that this well-orders the reals, when it actually just
> > well-orders a countable subset.
>
> It looks like his ordering produces an infinite binary tree, with
> 2^k reals at each level k, corresponding to the 2^k nodes at
> level k in the binary tree. But as was mentioned previously,
> although the tree has an uncountably infinite number of paths,
> it has only a countably infinite number of nodes.
Oh god, please, not THAT again. There are half as many paths as nodes in any
binary tree. Besides, it is the paths that correspond to the bit strings, so if
you see that as an uncountable set, then so be it.
>
> So it appears [*] that Tony's ordering only produces (orders) a
> countably infinite subset of the reals, as we have all been saying.
Not with countably infinite numbers of bits, then it produces uncountably
infinite numbers of strings/paths.
>
> [*] Actually, I can't figure out what his 'x' and 'x^oo' are really
> supposed to represent as real numbers.
0<x<1 =>

x^-oo=oo
x^oo=0
x^0=1
x^1=x
>
>

--
Smiles,

Tony
http://www.people.cornell.edu/pages/aeo6/WellOrder/
.

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