Matheology § 265

[WM]

Matheology § 265

Abstract. This paper examines the possibilities of extending Cantor’s
two arguments on the uncountable nature of the set of real numbers to
one of its proper denumerable subsets: the set of rational numbers.
The paper proves that, unless certain restrictive conditions are
satisfied, both extensions are possible. It is therefore indispensable
to prove that those conditions are in fact satisfied in Cantor’s
theory of transfinite sets. Otherwise that theory would be
inconsistent.

We have just proved [...] the alternatives of Cantor 1874-argument on
the cardinality of the real numbers can be applied to the set Q of
rational numbers, except the last one, that applies only if the common
limit of the sequences of left and right endpoints of the QP-intervals
is rational. Evidently, if Cantor’s 1874-argument could be extended to
the rational numbers we would have a contradiction: the set Q would
and would not be denumerable. Accordingly, in order to ensure the
impossibility of that contradiction, each of the following points have
to be proved: Whatsoever be the rational interval (a, b) and
whatsoever be the reordering of <q_i>, it must hold:
(1) The number of QP-intervals can never be finite.
(2) The sequences of endpoints <a_i> and <b_i> can never have
different limits.
(3) The common limit of <a_i> and <b_i> can never be rational.

[...] Until those proofs be given, Cantor’s 1874-argument should be
suspended, and the possibility of a contradiction involving the
foundation of (infinitist) set theory should be considered.

[Antonio Leon Sanchez: "Cantor versus Cantor" (2010)]
http://arxiv.org/PS_cache/arxiv/pdf/1001/1001.2874v3.pdf

On the other hand, the proof can feign the uncountability of a
countable set. If, for instance, the alternating harmonic sequence

omega_n = (-1)^n/n --> 0

is taken [...], yielding the intervals (-1, 1/2), (-1/3, 1/4), ... we
find that its limit 0 does not belong to the sequence, although the
set of numbers involved, |N u {0}, is obviously denumerable [...]

The alternating harmonic sequence does not, of course, contain all
real numbers, but this simple example demonstrates that Cantor's first
proof is not conclusive. Based upon this proof /alone/, the
uncountability of this and every other alternating convergent sequence
must be claimed. Only from some other information we know their
countability (as well as that of Q), but how can we exclude that some
other information, not yet available, in future will show the
countability of T or ||R?

[W. Mueckenheim: "On Cantor's important proofs" (2003)]
http://arxiv.org/pdf/math.GM/0306200

Regards, WM

[AMiews]

"WM" <muec...@rz.fh-augsburg.de> wrote in message
news:d9386d18-1923-4a6d...@l5g2000vbn.googlegroups.com...

>Matheology � 265

>
> Based upon this proof /alone/, the
>uncountability of this and every other alternating convergent sequence
>must be claimed.

not so;

During the Christmas holidays, Cantor visited Berlin and showed his work to
his former professor Karl Weierstrass. On December 25, Cantor wrote Dedekind
about his decision to publish:

Although I did not yet wish to publish the subject I recently for the first
time discussed with you, I have nevertheless unexpectedly been caused to do
so. I communicated my results to Mr. Weierstrass on the 22nd; � on the 23rd
I had the pleasure of a visit from him, at which I could communicate the
proofs to him. He was of the opinion that I must publish the thing at least
in so far as it concerns the algebraic numbers. So I wrote a short paper
with the title: "On a property of the set of real algebraic numbers," and
sent it to Professor Borchardt[31] to be considered for the Journal f�r Math
[Crelle's Journal].[32]

In a letter to Philip Jourdain, Cantor provided more details of Weierstrass'
reaction:
With Mr. Weierstrass I had good relations. � Of the conception of
enumerability [countability] of which he heard from me at Berlin on
Christmas holydays 1873, he became at first quite amazed, but [after] one or
two days passed over, it became his own and helped him to an unexpected
development of his wonderful theory of functions.[33]

Weierstrass probably urged Cantor to publish because he found the
countability of the set of algebraic numbers both surprising and useful.[34]
On December 27, Cantor told Dedekind more about his article, and mentioned
its quick acceptance (only four days after submission):[35]

The restriction which I have imposed on the published version of my
investigations is caused in part by local [Berlin] circumstances (about
which I shall perhaps later speak with you orally) and in part because I
believe that it is important to apply my ideas at first to a single case
(such as that of the real algebraic numbers) �[32]

As Mr. Borchardt has already responded to me today, he will have the
kindness to include this article soon in the Math. Journal.[36]
Cantor gave two reasons for restricting his article: "local circumstances"
and the importance of applying "my ideas at first to a single case." Cantor
never told Dedekind what the "local circumstances" were.[37] This has led to
a controversy: Who influenced Cantor so that his article emphasizes the
countability of the set of algebraic numbers rather than the uncountability
of the set of real numbers? This controversy is also fueled by Cantor's
earlier letters, which indicate that he was most interested in the set of
real numbers.

Cantor biographer Joseph Dauben argues that "local circumstances" refers to
the influence of Leopold Kronecker, Weierstrass' colleague at the University
of Berlin. Dauben states that publishing in Crelle's Journal could be
difficult because Kronecker, a member of the journal's editorial board, had
a restricted view of what was acceptable in mathematics.[38] Dauben argues
that to avoid publication problems,[39] Cantor wrote his article to
emphasize the countability of the set of real algebraic numbers.

Dauben uses examples from Cantor's article to show Kronecker's
influence.[40] For example, Cantor did not prove the existence of the limits
used in the proof of his second theorem.[41] Cantor did this despite using
Dedekind's version of the proof. In his private notes, Dedekind wrote:

� [my] version is carried over almost word-for-word in Cantor's article
(Crelle's Journal, 77); of course my use of "the principle of continuity" is
avoided at the relevant place �[42]

The "principle of continuity" requires a general theory of the irrationals,
such as Cantor's or Dedekind's construction of the real numbers from the
rationals. Kronecker accepted neither theory.[43]

In his history of set theory, Jos� Ferreir�s analyzes the situation in
Berlin and arrives at a different conclusion. Ferreir�s emphasizes
Weierstrass' influence: Weierstrass was interested in the countability of
the set of real algebraic numbers because he could use it to build
interesting functions.[44] Also, Ferreir�s suspects that in 1873 Weierstrass
might not have accepted the idea that infinite sets can have different
sizes. The following year, Weierstrass "stated that two 'infinitely great
magnitudes' are not comparable and can always be regarded as equal."[45]
Weierstrass' opinion on infinite sets may have led him to advise Cantor to
omit his remark on the essential difference between the collections of real
numbers and real algebraic numbers.[46] (This remark appears above in "The
article.") Cantor mentions Weierstrass' advice in his December 27 letter:

The remark on the essential difference of the collections, which I could
have very well included, was omitted on the advice of Mr. Weierstrass; but
[he also advised that I] could add it later as a marginal note during
proofreading.[47]

Ferreir�s' strongest statement about the "local circumstances" mentions both
Kronecker and Weierstrass: "Had Cantor emphasized it [the uncountability
result], as he had in the correspondence with Dedekind, there is no doubt
that Kronecker and Weierstrass would have reacted negatively."[48] Ferreir�s
also mentions another aspect of the local situation: Cantor, thinking of his
future career in mathematics, desired to maintain good relations with the
Berlin mathematicians.[49] This desire could have motivated Cantor to create
an article that appealed to Weierstrass' interests, and did not antagonize
Kronecker.[50]

>Regards, WM

[Graham Cooper]

On May 12, 7:48 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> Matheology § 265
>
> Abstract. This paper examines the possibilities of extending Cantor’s
> two arguments on the uncountable nature of the set of real numbers to
> one of its proper denumerable subsets: the set of rational numbers.

There is Real Mathematical Calculus

f(r):R->R

and there is LOGIC.

A:[0|1| -> B:[0|1]

--------------------------

Cantor's muffles the 2 arrows together into an infinite
sum of logical conditionals and pulls out hyper-contradictions
from his bellybutton.

Any REAL MATHEMATICS is based on equivalent FUNCTIONS
defined over N!

f(n):N -> R

In CALCULUS you are allowed to use f(r):R -> R
but underneath you must have some definable domain

f( g(n):N ) -> R

where g(n) COMPUTES the real domain of the function
via integer arithmetic.

LOGIC and f(r):R DO NOT MIX!

f(r):R is just a CALCULUS SHORTHAND!


NEWTON would hit CANTOR with a STICK if he saw the mess
that CANTOR made of his CALCULUS!


Herc
--
ALL NEW www.BLOCKPROLOG.com

[Graham Cooper]

On May 12, 7:48 pm, WM <mueck...@rz.fh-augsburg.de> wrote:

> On the other hand, the proof can feign the uncountability of a
> countable set. If, for instance, the alternating harmonic sequence
>
>         omega_n = (-1)^n/n --> 0
>
> is taken [...], yielding the intervals (-1, 1/2),  (-1/3, 1/4), ... we

The infinite sum is reducible to a finite equation in P.A.

Cantor's infinite sum is not.


IF SET1 has 1 - then MYSET skips 1 ./
or
IF SET1 skips 1 - then MYSET has 1
AND
IF SET2 has 2 - then MYSET skips 2 ./
or
IF SET2 skips 2 - then MYSET has 2
AND
IF SET3 has 3 - then MYSET skips 3
or
IF SET3 skips 3 - then MYSET has 3 ./
AND
...



Herc
--
www.BLoCKPROLOG.com

[fom]

On 5/14/2013 7:10 PM, Graham Cooper wrote:
>
> NEWTON would hit CANTOR with a STICK if he saw the mess
> that CANTOR made of his CALCULUS!
>

Newton's calculus?

Why isn't it Leibniz' calculus?

[Virgil]

In article
<f2be21fc-d7d2-4ccb...@qz2g2000pbb.googlegroups.com>,

Graham Cooper <graham...@gmail.com> wrote:

> NEWTON would hit CANTOR with a STICK if he saw the mess
> that CANTOR made of his CALCULUS!

Newton's calculus, without the benefits of any proper theory of limits,
was based only on a sort of intuition, not the logic offered by a proper
theory of limits.

At least Cantor had a proper theory of limits with which to do the
Fourier analysis which led to uncountability.
--

[Graham Cooper]

On May 15, 12:34 pm, Virgil <vir...@ligriv.com> wrote:
> In article
> <f2be21fc-d7d2-4ccb-aec7-15ced6d60...@qz2g2000pbb.googlegroups.com>,

>  Graham Cooper <grahamcoop...@gmail.com> wrote:
>
> > NEWTON would hit CANTOR with a STICK if he saw the mess
> > that CANTOR made of his CALCULUS!
>
> Newton's calculus, without the benefits of any proper theory of limits,
> was based only on a sort of intuition, not the logic offered by a proper
> theory of limits.

Newton invented Calculus merely to solve the orbit calculations of
other planets.

Like they invented PROLOG to solve Modus Ponens.

formula1(X,Y,Z) -> formula2(A,B,X,Y)

/\
|| UNIFY( f1,f3 )
\/

formula1(1,2,3) -> ?

----------------


? = formula2(A,B,1,2)

>
> At least Cantor had a proper theory of limits with which to do the
> Fourier analysis which led to uncountability.
>

such as?


LIM( X-->oo) 2^X = ?



Herc