Re: Well Ordering the Reals

In article <MPG.1df62e22f9ea958f98a781@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

> Virgil said:
> > In article <MPG.1df54787db39f44598a77d@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> > Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
> >
> > > Virgil said:
> > >
> > > > How many digits are indicated by "..."? Is it always the same amount?
> > > >
> > > > What about 1:000...000
> > > > versus 1:000...000...000
> > > > versus 1:000...000...000...000?
> > > >
> > > No, "..." is generic, but when comparing two strings the ellipses are
> > > considered to represent the same set of bit positions, and the strings
> > > are
> > > assumed to be of same length, or a substring defined whioch is the length
> > > of
> > > the other string. Then it works fine.
> >
> >
> > But suppose it is impossible to represent the two strings this way?

> Then the numbers are not well-formed.


That does not prevent them from being "numbers" in TO's dream world.
So apparently TO's "number system" contains numbers which defy
comparison for size.
> >
> > TO is assuming that whenever two strings are to be compared this is
> > possible, but without some mechanism to compare the realtive
> > positions of arbitrary digits, this is not possible.

> If you do not specify the number of bits, then you don't know what number you
> are talking about in the first place

Then precisely how does one specify the "number of bits"? That has been
my question all along!



> but it is not necessary to know the
> exact
> number in order to compare it with another number. It is only necessary to
> determine which has the most significant 1 bit where the other has a 0.
> Depending on how you define your numbers, this may not always be
> possible, but such numbers are not well-formed.

One issue is how TO defines HIS numbers, as they are not numbers by
anyone else's standards. And the second is, how does TO know that his
"numbers" are orderable if there are such ill formed numbers that defy
comparisons?

> >
> > So that TO must provide a mechanism to compare the relative position of
> > any two digits in any two strings if his imaginings are to be viable.

> The digit positions we define in the infinite string are going to be
> formulaic
> expressions of N, like N/2 or log2(N), not finitely defined positions, unless
> you are talking about finite values. But, given these anchor points, we can
> then deal with any finite number of bits clustered round those points.

Since N is a variable to TO, digit positions which are variable can
hardly serve a "anchor points".

The issue is whether there is a mechanism to compare sizes of any two
given "TO-numbers" or not. Apparently not.

Since there is such a method for natural numbers, TO-numbers are
unnatural.
> >
> > TO has not done that, and give no evidence of being able to do that.
> No, of course not.
> >
.

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