Re: Well Ordering the Reals

Virgil said:
> 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!
I answered it. Maybe you haven't gotten to that post yet. One defines the
number of bits as a formula using N, such as log2(N) bits in 1:000...000
denotes N, log2(N)-1 bits denotes N/2, N/2 bits denotes sqrt(N), N bits denotes
2^N, etc. etc.
>
>
>
> > 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?
Those numbers are not part of the set. Can I say that the normal binary system
doesn't work because you can't tell me what 10100.10100.00100 is? You would say
that's not a number in that system. Well, ...010101 is not a T-riffic number.
It could be 1:010....0101 or 0:10101....0101, and we have no idea of the number
of bits, so it is not specified correctly. That doesn't make 1:0101...0101 ill-
defined. It's 4N/3 (really (4N-1)/3, since there is another 1/3 not included,
to the right of the binary point).
>
> > >
> > > 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".

In a relative sense, indeed they can, my friend!

>
> The issue is whether there is a mechanism to compare sizes of any two
> given "TO-numbers" or not. Apparently not.
If you give two actual T-riffic numbers, thay can always be compared.
>
> Since there is such a method for natural numbers, TO-numbers are
> unnatural.

Oh they are not only natural, but truly organic. But since you live on Tang and
potroast-in-a-tube, you wouldn't understand that.

> > >
> > > TO has not done that, and give no evidence of being able to do that.
> > No, of course not.
> > >
>

--
Smiles,

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

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