Re: Well Ordering the Reals

Virgil said:
> In article <MPG.1dff9bce5e943ea898a82a@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
>
> > Virgil said:
> > > In article <MPG.1dfe65ec152d14598a819@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> > > Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
> > >
> > > > Virgil said:
> > >
> > > > > Density is not a matter of scale. An ordered set is dense if and
> > > > > only if between any two members there is another member.
> > > >
> > > > It really is. If you stand infinitely back and look at the number
> > > > line, the integers look dense on that scale. It's a matter of
> > > > relatively infinitesimal distance between elements of the set, is
> > > > all.
> > >
> > > "Dense" has a mathematical meaning in the context of ordered sets. In
> > > mathematical discussions such meanings take priority over any
> > > non-mathmatical meanings. And I would like to see TO try to stand
> > > infinitely far back from any line.
> >
> > Dense can have more than one meaning, which you aptly demonstrate every day.
>
> "Dense" has only one meaning in the context of ordered sets, though TO
> seems to be too dense to comprehend that fact..

Ooh you got me back! No tap-backs!!

> >
> > >
> > > > > Here is one more question: Imagine a "TO-number" with zeros from
> > > > > its left end rightward to the furthest extent covered by some known
> > > > > internal "limit point', and all 1's from there on rightward. What
> > > > > is its successor?
> > > > A 1 where the rightmost 0 is, and 0's from there rightward,
> > >
> > >
> > > But according to TO's own descriptions, there cannot be any such
> > > "rightmost" zero. To the right of any limit point (except the right end
> > > point) there is an unending sequence of digits each finitely far from
> > > that limit point, so there cannot be a "last" one.
> >
> > Since we are talking about whole numbers, bits to the right of the 0 point
> > are
> > all zeroes and ignored. So, I am referring to the rightmost 0 that is left of
> > the 0 point.
>
> TO is relying on the non-existent again. He is referring to the
> rightmost zero in an unending sequence of ever more rightward zeros,
> those which are a finite number of places to the right of the left end
> of TO's T-errible two-ended infinite string of digits.
Okay, I don;t think I understood what you were saying. There always seems to be
this underlying assumption of finiteness buried in statements about strings.
So, okay, this is like the example I think Dave Tribble offered, but in mirror
image. So, let's see. So, we didn't specify the limit point, so we'll call it
N, for N bits between it and the 0 point. Now, you want to say that the set of
bits finitely far from the top are 0's, and the rest 1's? Okay. We have N bits,
and the last aleph_0 are 0's (I don't have any number of finite naturals, so
I'll use yours for now). So we have N-aleph_0 1's, which is 2^(N-aleph_0)-1. An
interesting number, but probably not very useful, with aleph_0. Still, if you
want to consider sets of finite naturals, you might as well call it something,
I suppose.
> >
> > >
> > > > just like
> > > > regular binary naturals would be.
> > >
> > >
> > > Not at all like regular binaries (with only finitely many digits in
> > > each) would be.
> > Exactly as they would be. To increment any positive integer, find the
> > rightmost
> > 0 and invert it and every bit to its right.
>
> But, as TO constructs them, there can be TO-strings staring at the left
> with 0, and having an endless string of 0's followed by infintiely many
> 1's.
Well, yes, and you would have your first 0 at bit N-aleph_0, and invert the
string from there rightward to increment it.
>
>
> > > > Carry the increment across all 1's to the first 0,
> > >
> > >
> > > Which can not exist in TO's construction!
>
> > Of course it can. Please give an example where you think it doesn't work, so
> > I
> > can analyze your misconception.
>
> I have: A string starting at the left with a zero at every finite
> natural distance from the left end, and 1's everywhere else.
>
> There cannot be a rightmost 0 in such a number because there cannot be
> a largest finite natural.

That's right, and that's why it doesn't really make that much sense to talk
about it, but if you want to, you can throw in your aleph_0. I haven't found
much use for it, personally.

>
> > >
> > > How can one locate the rightmost digit among a set of digits for which
> > > there is no rightmost?
> > To the left of the 0 point. These are whole numbers we're incrementing right?
>
> These are TO-numbers, which makes them wholly illusional. But as TO
> descxribed them, they have a leftmost digit, a rightmost digit,
> uncountably many digits in between in a sequential order. Have I
> misreprestened these TO-numbers in any way?
Yes, a little. They need to have a variable most significant bit for infinite
values and a variable least significant bit for infinitesimals, but all digital
number systems really have an infinite unending string of bits, even if most
are generally ignored.
>
> > Nope. I am saying the rightmost 0 to the left of the 0 point.
>
> Which does not exist (see above), because each 0 is followed by another
> 0, but all of them are followed by uncountalby many 1's.
That's okay. If you want to talk about the set of finite naturals, you have to
declare some limit point for that, which makes it seem like an infinity, and
yet as I said to Dave Tribble, it occurs to me that limit points don't NEED to
be infinitely far apart.
>
> It is TO's system, not mine, so he is the one responsible for this
> anomaly.
>
Yup.

--
Smiles,

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

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