Re: Well Ordering the Reals
- From: Virgil <ITSnetNOTcom#virgil@xxxxxxxxxxx>
- Date: Thu, 01 Dec 2005 12:43:56 -0700
In article <MPG.1df8ce131350e65b98a7c3@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
> Virgil said:
> > In article <MPG.1df791438f589a9398a7a5@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> > Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
> >
> > > Virgil said:
> > > > In article
> > > > <MPG.1df62a0df73e2afc98a780@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> > > > Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
> > > >
> > > > > Virgil said:
> > > >
> > > > > > But in TO's "infinite sequences of digits with two ends"
> > > > > > how does one tell which digit is most significant? TO keeps
> > > > > > glossing over this critical issue.
> > > > > As in all regular digital systems, the leftmost nonzero digit
> > > > > is most significant. You knew that.
> > > >
> > > > But I do not know what test TO uses to determine which digit is
> > > > leftmost. If the digit positions were to be indexed by the
> > > > finite naturals, I could tell easily, but TO insists that his
> > > > index set is larger than that.
> > > >
> > > > And there is no standard for telling which of two indices is
> > > > larger for non-finite pseudo-naturals that is not essentially
> > > > circular.
> > >
> > > Virgil, we are in a world which used to be absolute and very
> > > finite, but we are moving into a world of relativity and infinite
> > > extremes. It is not necessary to know at which absolute location
> > > you are in the infinite universe, to measure the length of
> > > something finite. That's a relative measure. The fact is that all
> > > finite digital numbers have an infinite string of implied leading
> > > and trailing zeroes, but this doesn't cause a problem, since we
> > > can simply disregard them. The same holds true for these infinite
> > > strings. You may consider any point to be the digital point. It's
> > > just a matter of scale. Whether you are working with
> > > infinitesimals or infinities, it doesn't matter. If you want, you
> > > can think of N as 1.000... instead of 1:000...000, and then just
> > > shift up at the end log2(N) digit places. It's all relative. The
> > > absolute is realtive to an arbitrarily chosen origin.
> >
> > In all of which TO carefully avoids the problem of comparing two or
> > more such strings. How does one chose this "arbitrarily chosen
> > origin" so as to be a valid point of origin for both strings?
>
> Define "valid point of origin" as if any arbitrary point can't serve
> as origin.
>
> >
> > Without some absolute, rather than merely relative, indexing, one
> > can never be sure of being able to find a relevant 'relative
> > origin' appropriately.
>
> Define "appropriate relative origin". What do you mean by
> appropriate?
>
> > >
> > > >
> > > > > > Given two such TO-strings, each with a single 1 and the
> > > > > > rest zeros, how
> > > > > > does on test them to see which , if either, is larger? What
> > > > > > objects is TO using to index the digit positions?
> > > >
> > > > > Actually, one need not index the entire string of bits.
> > > > > Knowing that all bits are the same except some set of
> > > > > corresponding bits is sufficient to order any pair of
> > > > > T-riffic numbers.
> > > >
> > > > And how, precisely, does one determine that?
> >
> >
> > > One defines their numbers that way. I can say ....10110..... is
> > > larger than .....10011...., assuming the ellipses are equal.
> >
> > How does one determine whether the ellipses are equal to start
> > with?
> Determine? You define them as such. Otherwise there is no comparison.
> These numbers don't come from nowhere. You construct them such that
> they have a common digital point.=
> >
> > And if they are not given as equal, what then?
> They may be given as equal length strings which differ in bit
> pattern, in which case if there is no endpoint defined to the left,
> there may be no comparison possible, since one string may have a 1
> where the other has a zero, and vice versa.
> >
> > TO's "number" system has huge holes in it.
> Well, there IS that Twilight Zone in the middle, where the dots are.
> ;)
> >
> >
> > > The ellipses cannot be equal value in the two strings unless they
> > > occupy the same bits, so that's implied by their being equal.
> >
> > And if they are not equal?
> Then they are unequal.
> > >
> > > The ellipses can also denote repeating bits, such as
> > > 1:010101....0101.010101...0101, but that's the full spec,
> > > denoting the real value 4N/3. The example I give above concerns
> > > PART of a full T-riffic number.
> >
> > But what if the ellipses are not equal?
> Then they are not the same.
> > > >
> > > > > The number with a 1 in the most significant differing bit is
> > > > > larger.
> > > >
> > > > And how, precisely, does one determine that?
> >
> > > You look at the corresponding bits that differ.
> >
> > As there are infinitely many before and infinitely many after each
> > 1, precisely how does TO suggest one measures the sizes of those
> > infinities so as to be able to compare the position values of the
> > two 1s?
> By using formulaic expressions on N, as I have said.
> >
> > This is an issue that TO continues to gloss over, but which shows
> > the flaws in TO's pseudonumbers.
> wrong
> > > >
> > > >
> > > > > >
> > > > > > >
> > > > > > > Does that do it for you? If not, why not?
> > > > > >
> > > > > > It does not explain how one tells which of two digit
> > > > > > different positions is the more significant.
> > > >
> > > > > Whichever is to the left of the other, as usual.
> > > >
> > > > And how, precisely, does one determine that?
> >
> > > You look at the corresponding bits that differ.
> >
> > As there are infinitely many before and infinitely many after each
> > 1, precisely how does TO suggest one measures the sizes of those
> > infinities so as to be able to compare the position values of the
> > two 1s?
> Why do you ask the same questions over and over?
Because over and over TO does not answer them, and because without those
answers, TO-numbers make no sense!
> > > >
> > > > TO keeps going in circles on this issue, apparently because he
> > > > does not have a legitimate answer to my questions.
> >
> > > Most of your questions are inane
> >
> > My questions can all be easily and completely answered for binary
> > representations of all finite natural numbers, so why does TO have
> > to hand wave and avoid giving actual answers for TO-numbers?
> Is that what I'm doing?
Yes!
> >
> > Because TO-numbers are self-contradictory. There is no legitimate
> > way to determine between two of his "numbers", each having a single
> > 1 precededed and followed by infinitely many 0s, which, if either,
> > is larger.
> >
> > > but since you asked actual questions here, I answered.
> >
> > What TO provided are not answers but are mere evasions.
> Oh bug off.
Another, though more obvious, evasion of the real issue.
TO claims to have a number system in which it is always possible to
compare two numbers for size, but has produced no effective mechanism
for carrying out such comparisons.
There are simple and effective methods for comparing standard (finite)
naturals for size. Thus the set of standard naturals is clearly ordered,
and even are well ordered.
TOnumbers, at least so far, are not known to be ordered, as there is no
mechanism by which to determine order in general. The best that can be
said of the set of TO-numbers is that it is partially ordered.
.
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- Re: Well Ordering the Reals
- From: Tony Orlow
- Re: Well Ordering the Reals
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