Re: how to list all of the real numbers

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".

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.

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.

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.

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

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 Hoyle
http://www.jonhoyle.com

.

Relevant Pages

  • Re: how to list all of the real numbers
    ... those with the undecideable continuum hypothesis, yet none of them, ... continuum's cardinality is thus simply equivalent to a lesser cardinal ... does not hold where the reals are a set. ... Infinity in numbers does actually exist. ...
    (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)
  • 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.cognitive)
  • 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.physics)