Re: how to list all of the real numbers

On Aug 23, 7:04 pm, "Ross A. Finlayson" <r...@xxxxxxxxxxxxxxx> wrote:
On Aug 23, 1:45 pm, Jonathan Hoyle <jonho...@xxxxxxx> wrote:

It is not. You will always run into a degenerate interval at some
limit ordinal prior to c, thus undoing your "proof".

Then, if it is a degenerate interval, that is, containing only a given
point, with both "endpoints" of the "interval" being identical, there
is no line to be drawn between them that is not coincident.

That is correct.


So, then for some ordinal X less than O = c, any f(X+1) of the left
or right endpoint sequence (that being a(X) or b(X)) of the nested
intervals has exacly that value of that point, else there is no f(X+1).

No, not necessarily. You are assuming that the failure occurs at some
successor ordinal "X+1". You are forgetting the possibility that it
will fail at some limit ordinal. Your proof holds for the former case
but not for the latter.


That's kind of the point, in allusion to that trichotomy of cardinals
does not hold where the reals are a set.

I don't understand this sentence. The trichotomy of cardinals states
that given any two cardinals k1 and k2, exactly one of the following
three cases holds: k1<k2, k1=k2, or k1>k2. Only the Axiom of Choice
is required for this trichotomy to hold true. It has nothing to do
with the reals.


So, and here I think you refer to the paragraph and a half or so above
illustrating that for any given infinite cardinal C > Aleph_1 that C >
c and C < c, as opposed to illustrating that the reals between zero
and one are listable in basically only one way, so that makes the
point instead of breaking the point.

I don't following this at all. I know at least your conclusion is
false, as there are an uncountable number of ways that the reals can
be ordered.


It was nearly two years ago, and yes it was in that discussion where
your outline was proven to be faulty. I could go over your logic flaw
again, but it was detailed pretty well at the time, and you might just
simply re-read that posted topic to refresh your memory.

Yes I should re-read it.

I think you would do well to. Despite the fact that it shows your
proof to be in error, I think it was an example of your best work
mathematically and logically. With a little effort, your construction
could be used as a proof for the existence of limit ordinals.


Incorrect. Even in Geometry, points on a line cannot be consecutive.
Euclid's very first postulate prevents this: "Between any two distinct
points a line can be drawn." This would not be possible if the points
are adjacent; therefore, no two points can ever be adjacent in
geometry.

Then, there would be some consideration of the definition of distinct
points, which I would call definitely distinct, and indefinitely
distinct points.

I don't know what you mean by "definitely distinct" and "indefinitely
distinct". But according to Euclid's 1st postulate, if any two points
*are not the same*, then a line can be drawn between them. Any
concept of "adjacent points" is provably inconsistent in geometry.


The continuum of real numbers, where any numbers between zero and one
are real numbers else they wouldn't be between zero and one, including
the nilpotent infinitesimal iota and iota-sums and iota-multiples,
such that infinitely many iota's in multiple equal exactly one, leads
to reformulation of the infinitesimal analysis as infinitesimal
analysis, where that is nonstandard and not necessarily Non-Standard.
<snip>

This run-on sentence is ill-defined and not even well thought out.
You need to break this into steps. By doing so, the errors will
become more apparent.


Browsing some history books and reading etcetera it seems that much
work in the notion of the infinite series and so on, particularly that
which is used by the analyst today, was done in a framework (or lack
thereof) of the intuitive and not necessarily intuitionistic (having
both constructionist/constructivist and intuitionistic philosophies)
where the framework that yielded a or the large part of analytic
results today was built around a more pragmatic than formalist
approach.

Yes, much of early analysis lacked the rigor it has today. Even many
of Euler's proofs were playgrounds of inconsistencies and unwarranted
assumptions. When Weierstrass and Bolzano came upon the scene, they
added the rigorous approach to analysis so that it would be on as
firm a foundation as Geometry had always been.

Here, there is the impulse function, the unit impulse
function integrating to one. Where the domain has an infinite value,
say, oo, infinity, then you would have that the codomain contains that
value. Here, the key feature of that value is that d(0) = oo, yet,
actually I say that d(0) = oo/2.

On the Extended Real line, the equation oo/2 yelds oo. oo/2 is not
separate number. It is an algebraic formula taking oo and dividing it
by 2, yielding back oo. You can see why this must be by considering
that the inverse of oo is 0. If oo/2 is a different number than oo,
then its inverse must be as well. However, the inverse of oo/2 is
simply 2*0 = 0.


The point is that there is a symmetry between the infinity of the domain
and the infinity of the codomain.

No, on the extended real line, there is a symmetry between oo and the
number 0. And since 0 cannot be changed by multiplying it or dividing
it by another non-zero number, the same must be true for its inverse.

Then, where there is the consideration that 1/x for
x = 0, 1/0 = oo, 1/oo = 0, yet that e^oo would be considered a
discontinuity, in the asymptotic analysis reveals that 2^x < 3^x < ...
and log x < x < e^x etcetera, preserving trichotomy.

On the extended real line, the algebra of the real numbers remains the
same, with the following additions (I think I have them right):

1. oo + x = oo, for all finite x
2. oo - x = oo, for all finite x
3. oo * x = oo, for all non-zero x
4. oo / x = oo, for all finite x
5. x / oo = 0, for all finite x
6. oo ^ x = oo, for positive, finite x
7. oo ^ x = 0, for negative finite x
8. 0 ^ x = oo, for negative finite x

oo - oo, oo / oo, 0/0, etc. all remained undefined as expected. Some
others defined 0 * oo to be 0, whilst others left it undefined. (It
was never defined to be 1, since that would be inconsistent.)

As an aside, you can extend the real line with two infinities, +oo and
-oo. This has a very different algebraic set, and it makes the real
line topologically equivalent to a line segment, not a circle. You
lose the inverse of 0 and the two infinities, but it allows you to
define x^+oo and x^-oo, which you cannot do in the above construction.


Regardless, I fail to see the point you are trying to make about this
function. The existence (or non-existence) Dirac delta's
axiomitization is completely irrelevant to the topology or cardinality
of R.

No, they're inextricably related.

That's kind of the point, the set of real numbers must basically
fulfill all the properties of the continuum of real numbers. As has
been known for thousands of years in for example the paradoxes of
Zeno, where Zeno does arrive and the limit is and can only be the sum,
and for no finite input is the inductive argument of differentiation
complete, there are a wide variety of correct and truth-preserving
identities about the infinite that generally appall today's
formalist. Form follows function.

I do not follow. Zeno's "paradox" is nothing more than an invalid
argument, making false assumptions about the nature of infinity. How
does an unsound argument invalidate a mathematical proof?


Where these tools work and how exactly they do, in the real numbers,
as functions and operations and so on of real numbers, the real
numbers are richer than their general axiomatization would imply.

What do you mean by "richer"? Do you mean to include infinitesimals
into R? Doing so will make R no longer complete, and a basic
definition of R is one that makes it a complete ordered field.


Line the real numbers up, map them to the naturals via the equivalency
function as one-sided points, in base one, two, three, or infinity.

Can you please give me an example of this "equivalency function"? For
instance, which finite natural number maps to pi?


Jonathan Hoyle
http://www.jonhoyle.com

.

Relevant Pages

  • Re: Calculus XOR Probability
    ... If a quantitative set is mapped in ascending order from the naturals, ... number of reals on the line. ... to the subsequent logic that claims such a set cannot have infinite values. ... standard orderings, since sets in general don't come with little tags ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... If a quantitative set is mapped in ascending order from the naturals, with each increment in the domain, the range increases by some amount. ... Like it's the number of unit intervals, and the number of reals in the unit interval. ... You are using a form of infinite induction, making a claim for an infinite set based on all finite initial segments of it. ... don't have a definition for an arbitrary set of its "standard ordering" ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... If a quantitative set is mapped in ascending order from the naturals, with each increment in the domain, the range increases by some amount. ... you had said that the existence ... Like it's the number of unit intervals, and the number of reals in the unit interval. ... You are using a form of infinite induction, making a claim for an infinite set based on all finite initial segments of it. ...
    (sci.math)
  • Re: Dial 999 for the real number line
    ... length n as initial strings followed by the expansion of pi after the nth ... long decimal expansion followed by an infinite expansion. ... part can be as long as we like, we have an infinite sequence of initial ... dense in the reals. ...
    (sci.math)
  • Re: Dial 999 for the real number line
    ... Keep in mind that whatever system of axioms you come up with will ... proofs of uncountablility of the reals, but that is not going to happen. ... I have a hang-up, it's about infinite sequences. ... My point is that mathematical existence has nothing to do with physical ...
    (sci.math)