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Topic: Uncountable Diagonal Problem
Replies: 52   Last Post: Jan 6, 2013 2:43 PM

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 ross.finlayson@gmail.com Posts: 2,720 Registered: 2/15/09
Re: Uncountable Diagonal Problem
Posted: Dec 30, 2012 9:59 PM

On Dec 30, 6:27 pm, forbisga...@gmail.com wrote:
> On Sunday, December 30, 2012 4:24:27 PM UTC-8, Ross A. Finlayson wrote:
> > So, the mega-sequences of the nested interval endpoints would end with
> > side-by-side endpoints?  Or, does any ordinally-indexed sequence of
> > all of a segment of reals necessarily contain duplicates?

>
> I'm having trouble interpretting this.
>
> Given any CIxT where x is some base, the sequence continuing with
> (x-1) at all leaf nodes beyond some node y is equivalent to
> the numeric successor node for y followed by 0 at all leaf nodes
> continuing from it.
>
> Is that what you said or were thinking?

Here, for the complete infinite binary tree, or CI2T, CIBT, that
basically being the Cantor space {0,1}^oo, or all binary sequences:
that naturally represents real numbers between zero and one, with
duplicates where .01(1)... = .10(0)....

Here, I was wondering that for nested intervals in the transfinite,
that the interval wouldn't be empty for two different endpoints (else
it could be for a countable ordinal), that there would be a duplicate,
i.e., that the well-ordering of the reals could be onto yet not 1-1.

Of course I think EF is a function with range [0,1], that well-orders
the unit interval of reals, with the caveat as above that it follows
from an expanded definition of real number, while of course that it is
standardly modeled by real functions.

So, are nested intervals in the transfinite: empty?

Regards,

Ross Finlayson