Re: Well Ordering the Reals

Virgil said:
> In article <MPG.1df91e0d5247c6dd98a7dc@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
>
> > William Hughes said:
>
> > > 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.
>
> And how does TO determine what formula to use?
By looking at the formulaic expression we wish to represent, and the terms used
in it.
> Are TO's T-errible numbers well ordered?
Probably not in the traditional sense, since we don't have the predecessor
disonctinuities required for the uncountably infinite set to avoid infinite
descending chains.

> Are the "limit points" of TO's Terrible numbering system well ordered?
Is the set of algebraic formulas well ordered? I rather think it's an
uncountably infinite set, is it not?

> Is any part of TO's T-errible numbering sysem well anything?

Are you feeling well?

>
> > > So log2(N) is finite?
> > No. That's a limit point, infinitely far from the 0 point, which is the
> > normal
> > digital point.
>
> Just how infinitely far from 0 is it?
log2(N) bits.
>
> Are there any "limit points" between it and 0?
Sure, an infinite number. For instance, log3(N), log4(N), log5(N) etc. That
alone is a countably infinite set, and not the entire set of limit points
between 0 and log2(N).
>
> How many "limit points" does TO need to cover all of his T-errible
> numbers? Is it more than the number of finite naturals?
Yes, obviously, if the family of limit points defined by logn(N) is but a
subset and is countably infinite.
>
> > > > 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.
>
> How many "limit points" are needed to cover everything, and how does one
> find a necessaary limit point if one is needed between two given limit
> points?
I would imagine it's an uncountably infinite set of possible limit points. If
you have a point between two limit points, but infinitely far from each, then
you require a limit point for that point. How you express this limit point
follows from the requirements for the bit specification.
>
> TO's whole T-errible system is really terrible.
Why do you say that? I have answered every question you've posed, and the only
place I "failed" was accomodating the adics in my system, which isn't a failure
at all. What is it you hate so much about me or my ideas? Is it just because I
smell like a cat?
>

--
Smiles,

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

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