Re: Well Ordering the Reals

William Hughes said:
>
> Tony Orlow wrote:
> > 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).
>
> Ok so all points are a finite number of bit positions
> of the digit point?
We define the values of bits within a finite distance of the defined limit
points, such as 0, log2(N), N etc. If we need to define bits infintiely far
from all of our points, we need to define another limit point with a uniqu
formula on 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.
>
> So log2(N) is finite?
No. That's a limit point, infinitely far from the 0 point, which is the normal
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.
>
> So not all the points are within a finite number of bit positions?
No, of course not, but we can only specifically define bits within a finite
distance of a defined limit point, which is expressed as a formula on N. We can
define as many lmit points as we need, each infinitely far from every other.
>
>
> -William Hughes
>
>

--
Smiles,

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

Relevant Pages

  • Re: infinity
    ... William Hughes wrote: ... > Tony Orlow wrote: ... >> One can speak of infinite sets UP TO any arbitrary value. ... It exists at every iteration of the proof, ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... > Tony Orlow wrote: ... >> William Hughes said: ... > (where the ellipses represent an infinite number of zeros) which limit ... > point is the 1 within a finite distance of. ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... > William Hughes said: ... >> Tony Orlow wrote: ... > We define the values of bits within a finite distance of the defined limit ... (where the ellipses represent an infinite number of zeros) ...
    (sci.math)
  • Re: infinity
    ... > Tony Orlow wrote: ... >> William Hughes said: ... >>> is no mention of order in the definitons of I or O. ... >>> are balls labelled with infinite integers. ...
    (sci.math)
  • Re: Logarithm of transfinite numbers
    ... Tony Orlow wrote: ... sum can represent an infinite value. ... the sum over all finite bits is finite. ... this because every next point is still a finite distance from the ...
    (sci.math)
Loading