Re: Well Ordering the Reals

In article <MPG.1e00eacf5e99a95f98a84c@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

> Virgil said:
> > In article <MPG.1dff9bce5e943ea898a82a@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> > Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

> > > > > > 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.


TO carefully avoids the issue to blather on about inconsequentials.

In TO's system of "whole numbers", there is a most significant digit and
a least significant digit and presumably an uncountable sequence of
digits between. TO's "limit points" include both end digits.

I can imagine a TO-number which has 0's in every position a finite
number of places from the most significant place and 1's elsewhere.
Since there is no least significant 0 or most sigificant 1 in such a
TO-number, how does TO suggest finding its successor?

That is my question. Try answering it, TO!


> > 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
> > infinitely 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.

But there is always another less significant 0 after each 0 so there
isn't any end to them ever.

> > 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.

So that To's "solution" is a crock!

>
> >
> > > >
> > > > 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
> > described 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.

At least for standard systems of natural numbers, there is a most
significant bit at a finite position, so that the set of zeros to the
left of all non-zero digits always has a least significant postion.

This is not true for TO-wholes or TO-naturals, and that creates the
anomaly that TO refuses to address,
> >
> > > 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.

If they are not infinitely far apart, you don't need them at all.
> >
> > It is TO's system, not mine, so he is the one responsible for this
> > anomaly.
> >
> Yup.

But TO cannot make it go away without admitting that his "number system"
sucks.
.

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