Re: Well Ordering the Reals


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?

> >
> > >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?

>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?


-William Hughes

.

Relevant Pages

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