Re: Well Ordering the Reals

In article <MPG.1df92539b90a515d98a7dd@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

> David R Tribble said:
> > Tony Orlow wrote:
> > >> 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.
> > >>
> > >> When I say 1:000...000 is N, that colon is a digital point at
> > >> log2(N).
> > >
> >
> > William Hughes wrote:
> > > What is a your definition of a finite neighborhood? So how far is
> > > the last 0 in 1:000...000 from the digital point?
> >
> > More questions for Tony: When you write x = 0:100...000, and you
> > say that the "..." represents an infinite number of zero bits, you
> > appear to be definining a number with an infinite number of low
> > zero bits "...000" and three extra high bits "100" positioned past
> > the last infinitely-positioned low zero bit.
> >
> > It looks x is composed of oo+3 bits. Or does the "..." represent
> > only oo-3 bits? What is 3 bits shy of infinity?
>
> Another infinite number, but I think you are misunderstanding. When I
> say 1:000...000 is N and has log2(N) digits, I mean the 0's, not the
> 1. The whole string between the 0 point and the log2(N) point is
> log2(N) bits.

But starting at the leftmost 0 of 1:000...000 one can count off N zeros
to the right, and counting from the rightmost 0, one can count off N
zeros to the left without ever overlapping or covering the same ground,
so that in TO's arithmetic, log2(N) must satisfy log2(N) >= N + N.

I must say that I prefer standard arithmetic to the odiocies of TO's
T-errible number system.
.

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