Re: Well Ordering the Reals


Tony Orlow wrote:
> 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.

So given

0:0...010...0

(where the ellipses represent an infinite number of zeros) which limit
point is the 1 within a finite distance of.

-William Hughes

.

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
    ... > Tony Orlow wrote: ... >> William Hughes said: ... We define the values of bits within a finite distance of the defined limit ... >> Infinite. ...
    (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