Re: Well Ordering the Reals

William Hughes said:
>
> Tony Orlow wrote:
> > Matt Gutting said:
> > > >> BINGO! The whole question is "what is this additional specification"
> > > > The places of the bits, either relative or absolute.
> > >
> > > You're arguing circularly. "The places of the bits, either relative or absolute"
> > > must be determined by some indexing set. "The places of the bits" are precisely
> > > what determines "whichever is to the left of the other". But that's precisely
> > > what we're asking.
> > >
> >
> > Okay, sure. I just said this in another post, that as I think about it now, it
> > seems that the general approach is essentially to define multiple digital
> > points. With normal finite digital numbers, all significant digits are within a
> > finite number of steps of THE digital point at bit 0, and that's how we know
> > their values. I guess what my system really boils down to is defining multiple
> > digital points, at locations infinitely far apart in the string, with finite
> > neighborhoods.
>
> What is a your definition of a finite neighborhood?
The set of points within a finite number of bit positions of a "limit" digital
point, such as 0 or log2(N).
>
> >When I say 1:000...000 is N, that colon is a digital point at log2(N).
>
> So how far is the last 0 in 1:000...000 from the digital point?
It's the log2(N)th bit to the left of the root digital point. Of course, it's
the first to the right of the log2(N) digital point.
>
>
> >If I say the point is at N, then I have 2^N as a value. If I say the
> > point is at log2(N), and have 1:111...111.111...111, I have 10N/9. Like I said
>
>
> In 1:111...111.111...111 Do the ellipses represent finite or infinite
> gaps?
Infinite.
>
>
> how far is the forth 1 from the digital point?
The one before the first ellipses? log2(N)-3 to the left.
> how far is the fifth 1 from the digital point?
The fifth displayed? It's the third bit, so 3 places
> how far is the seventh 1 from the digital point?
1
> how far is the last 1 from the digital point?
log2(N) to the right.

I hope I got your questions right, and you were referring to the bits numbered
as displayed, not implied in the string.
>
> -William Hughes
>
>

--
Smiles,

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

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