Re: how to list all of the real numbers

On Aug 11, 10:47 pm, lwal...@xxxxxxxxx wrote:
On Aug 11, 7:29 pm, Virgil <vir...@xxxxxxxxxxx> wrote:

Most mathematicians will consider systems of infinitesimals possible,
but unless their own work makes such infinitesimals useful, they are not
liable to spend much energy or time on them.

"Crank" #1: Why accept the Axiom of Infinity?
Standard Mathematician: Because it is useful.
"Crank" #2: Why reject infinitesimals?
Standard Mathematician: Because they are not useful.

So far in both this thread and in "An Inconvenient Truth" the
consensus
among standard mathematicians is that the infinite sets are "useful,"
(in response to HdB's claims that the Axiom of Infinity is not useful)
while infinitesimals are not "useful," which is why the former is part
of standard (or classical) analysis while the latter is relegated to
Nonstandard Analysis.

The difference? The best argument I've seen is that it's not the
infinite sets themselves that are "useful," but the complete ordered
field that is important to the solution of differential equations. It
was stated that a complete ordered field cannot contain
infinitesimals,
for the set of infinitesimals are a set with no least upper bound. In
NSA differential equations become difference equations (with
infinitesimal differences), while integrals become sums of infinitely
many infinitesimals. And since differential equations and integrals
are easier to work with than (even finite) difference equations and
summations, infinitesimals are to be rejected. But one must assume
the Axiom of Infinity in order to prove that a complete ordered field
even exists.

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. 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."

Browsing Hardy's "A course of pure mathematics", I consider this:

The 'real number' $x$, with which we have been concerned in the two
preceding chapters, may be regarded from many different points of
view. It may be regarded as a pure number, destitute of geometrical
significance, or a geometrical significance may be attached to it in
at least three different ways. It may be regarded as _the measure of
a length_, viz. the length $A_0 P$ along the line $\Lambda$ of Ch. I.
It may be regarded as _the mark of a point_, viz. the point $P$ whose
distance from $A_0$ is $x$. Or, it may be regarded as _the measure of
a displacement_ or _change of position_ on the line $\Lambda$.

(Here, underscores delimit the author's emphasis and dollar signs
indicate TeX linear math mode, and "viz." is an abbreviation of
"videlicet.")

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.

( Consider as an aside: There is a notion that where the real numbers
exist, that synthetically there exists a, say, canonical infinite-
dimensional orthogonal vector basis. Then, consider a unit length
segment from the origin, a vector. Where it demands two non-zero
components, each is less than 1, three, each further less, until for
the unit length vector to be uniquely and only expressible in terms of
components that span the space, each is, or almost all are,
infinitesimal. )

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.

Ross

--
Finlayson Consulting

.

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