Re: how to list all of the real numbers

On Aug 24, 6:17 pm, Virgil <vir...@xxxxxxxxxxx> wrote:
In article <1187994742.323515.123...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Ross A. Finlayson" <r...@xxxxxxxxxxxxxxx> wrote:

If both of O < c and O > c hold, cardinals are non-trichotomous or the
reals aren't a set.

If 0 < c and c < 0, cardinals are trichotomous and the reals are a set,
and anything else anyone wants to claim is also true.



No, if that is so then where the reals are a set cardinals wouldn't be
trichotomous, because the ordering properties don't hold, not directly
via violation of excluded middle, to inconsistency and Russell is the
pope.

That the cardinality of the continuum would be consistently equivalent
to a wide variety of cardinals, but not any particular one, is well-
known and swept under the rug.


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.

Euclid's postulates have long been accepted to define geometry.

At least Euclidean geometry, but the are others. But in all but special
or finite geometries, between any two points a line exists.



I would like to learn more about geometry, there is much in it.




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.

No, it's readable.

Word by word readable but as a whole imparsible.


No, it's a clear statement, easily diagrammed and grammatically
correct, suitable for the audience.



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.

I think there is more than one approach.

At least two: there is the rigorous approach and the Ross approach.


Gauss' notion of limit adopted by Cauchy and Weierstrass is surely a
powerful and useful tool, in correct application.

I argue that the inductive argument that the limit is not the sum, of
the partial sums or terms of infinite series, would lead to geometric
anomalies for example as is considered in the arguments of Zeno.
Then, to complete the induction as it were, particularly in a theory
where the natural integers over which induction procedes are infinite
in extent and for some elements character, where "the limit is the
sum", corresponding to completion of a variety of easily specified
processes, has that the limit is the sum.

Among the variety of examples presented here, consider the notion that
for no finite input or at no finite stage of the weak or strong form
of induction is the limit actually the sum, that the limit is exactly
the sum in continuum analysis leads to a reasonable notion that there
is a completion of sorts of the infinite induction.

There are a huge amount of "correct" applications of the notion, or
notation, the use of "infinity" in applied analysis, about symmetry
and asymptotics, that give correct analytical results from often the
relative description of, for example, infinitesimal patches and
instantaneous rates of change to parameters.

...

There are a wide variety of reasonable and useful considerations of
infinity in the numbers.

With none of which is Ross competent.


Competence (competency?) is an acquired trait, apparently so is
politeness.



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.

Since the Dirac delta is not a real function, how is it related?

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

...

You would have that there would always be statements about the real
numbers that, although true, you could never prove. I, instead, would
have that there is some true set of real numbers about which all facts
are provable.

Goedel's work suggests otherwise.

Yes, it would, but not in a theory that doesn't see Goedelian
incompleteness as a consequence of inconsequence. (Consider Cohen's
method as discussed.)

Consider some more the equivalency function, its range includes zero
and one and for each element of the naturals a distinct value. So, it
is dense in the unit interval, in the sense that between any two
distinct real numbers (in the sense of definitely distinct, not in
succession but as closure to multiplication with definite distinct
inverses) there would exist an element of the codomain of the
function, so the codomain is dense in the unit interval, yet, there is
a least positive value EF(1).

Such a simple construction would surely be apparent to many.

Ross

--
Finlayson Consulting

.

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