Re: how to list all of the real numbers

On Aug 23, 1:45 pm, Jonathan Hoyle <jonho...@xxxxxxx> wrote:
On Aug 13, 3:32 pm, "Ross A. Finlayson" <r...@xxxxxxxxxxxxxxx> wrote:



Well-order the reals, i.e., biject the set of real numbers to an
initial segment of an ordinal. Then, apply "Cantor's first" or nested
intervals as it is sometimes called, that each sucessive element
brackets the remaining interval. Then, besides whether c = Aleph_1,
Aleph_2, ..., where it is said to be consistently equal to each of
those with the undecideable continuum hypothesis, yet none of them,
besides that, consider: does there not always exist each real number
of a segment of the continuum in the remaining bracket/interval?

That is where, if some initial segment of the ordinal O equivalent to
c led to a degenerate interval, for example [0,0] containing only the
point/scalar/number 0, then there is a question as to whether the
continuum's cardinality is thus simply equivalent to a lesser cardinal
than O's.

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. 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).

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

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.

Similarly to how other considerations lead to that the real
numbers have a cardinality greater than any given ordinal, it leads to
an argument that they then as well have a cardinality less than any
given ordinal to which they are equivalent. This was discussed
further some year and so ago in "On well-ordering(s) of the reals,
infinity."

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.

Then, where a real may be a mark of a point, consider whether Hardy
would accept that each mark of a point, or simply point, would
represent a real number. Then, there is a question: how can there be
marked each point on the unit interval? Cantor's nested intervals
rreesult would have that any attempt to stipple or pock the line into
existence would fail. Yet, drawing from mark to mark has that any
point so marked is only as a result of a generative process that each
new point in the course of the line is immediately adjacent to: the
previous point.

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.

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.
Infinity in numbers does actually exist. Then, where natural laws are
expected to correspond to the behavior of mathematically defined
entities, and furthermore be defined as mathematical entities, as in
for example the universe containin itself and is infinite and
construing physical objects as mathematical objects and funcitons
between them as physical objects, eg force fields, the universe
itself is an example of infinite set and powerset identical, the
physical universe. So, in continuum analysis of physical objects,
with the known incongruencies of a sort in the large and small
(paradoxes as it were, effects), there should be considered that there
are appreciable analytic effects in the meso-scale, observable
reality, in basically the notion of the polydimensional point, and
real numbers as beads on a string, structured at once as the complete
ordered field.


It was mentioned that the reason that rigor/soundness in analysis is a
perceived requirement is for the differentialists, or analysts. Yet,
consider, some of the most powerful tools of the analyst such as the
impulse function (delta of Dirac), which has a value of infinity at
zero yet zero elsewhere yet integrates over the domain evaluating to
one, not a real function, and in the geometric context that the area
under the point width at infinity equals one. That's much more
directly reconcilable with the notion of the differential of the
constant function, the sum of which over infinitely many points
between zero and one equals the constant, than that instead those
notions are not sound. Those tools are very regularly used.

The Dirac delta function is not a function of the Reals, since its
value at 0 is infinite. It is a function of the Extended Reals, that
is the reals plus an ideal point called "infinity", using the
traditional lemniscate as its symbol. The Extended Reals (denoted
Roo) are topologically equivalent to the circle, and has many nice
properties, but some statements true in R become false in Roo and vice-
versa. (For example, in R the functions e^x and 1/x are continuous
and discontinuous respectively, but the reverse is true in Roo.)
Moreover, some basic arithmetic niceties, such as R being closed over
subtraction, fails in Roo.


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. 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. The point is that there is a
symmetry between the infinity of the domain and the infinity of the
codomain. Infinity is basically considered a constant, when it can be
cancelled simply. 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.

Even within Roo, the Dirac delta function must be formally axiomitized
since it is not provable from the axioms of Roo alone.


It's a valuable and correct tool, beautiful in that way: formalize
it.

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.

Jonathan Hoylehttp://www.jonhoyle.com

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.

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.
Thus, where the continuum of real numbers has more features than a
given axiom set adequate for other work, and much of it, supports, one
would be remiss to suggest otherwise.

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.

Ross

--
Finlayson Consulting





.

Relevant Pages

  • Re: a few questions for you was Re: 150 years without Gauss
    ... Let's clarify who thinks "that the limit for n to infinity of a sum of n ... because this would mutate his irrational reals into rationals. ... of numbers and the quite different world of the genuine continuum. ... Bachmann introduced the reals by means of intervals whose size ...
    (sci.math)
  • Re: Is continuum completely filled up?
    ... piece of continuum. ... this view there is no actual infinity. ... Accept infinity and the reals like ... How did Cantor make sure that the numbers he used in DA2 were actually ...
    (sci.math)
  • Re: Very basic mistakes
    ... According to my reasoning, infinity, continuum, irrational ... I appreciate the tacit substitution of genuine reals by ... You have neither defined IR+ (that's not a standard notation) ...
    (sci.math)
  • Re: how to list all of the real numbers
    ... does not hold where the reals are a set. ... could be used as a proof for the existence of limit ordinals. ... concept of "adjacent points" is provably inconsistent in geometry. ... and the infinity of the codomain. ...
    (sci.math)
  • Re: Epistemology 201: The Science of Science
    ... that there are details of infinity that it does not capture. ... as between 0 and 2, when one obvious has more reals than the other as ... just because there is the same Cantorian cardinality. ... This is why Lester calls you people mathematikers. ...
    (sci.math)