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Topic: § 433 The reason for calling matheology matheology
Replies: 24   Last Post: Feb 22, 2014 5:23 PM

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fom

Posts: 1,968
Registered: 12/4/12
Re: § 433 The reason for calling ma
theology matheology

Posted: Feb 21, 2014 2:01 PM
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On 2/21/2014 6:10 AM, mueckenh@rz.fh-augsburg.de wrote:
> On Thursday, 20 February 2014 21:46:29 UTC+1, John Gabriel wrote:
>> On Thursday, February 20, 2014 9:51:26 PM UTC+2, muec...@rz.fh-augsburg.de wrote:
>
>

>> I know. But where does it say "Cantor's diagonal argument works exclusively in the domain of terminating sequences which he himself as proven to be countable." ?
>>
>>
>>
>> Okay. I get it. *You* are saying that. Cantor *does not* say that, but that's *exactly* what Cantor does deceptively. :-)

>
> Yes, set theory is a big fraud.
>
> If you have a list like this:
>
> 1
> 11
> 111
> ...
>
> then no entry has infinitely many digits. However, the entries are potentially infinite because there is no upper threshold for the number of digits. That makes the diagonal 111... also poetntially infinite: Its number of digits surpasses every natural number. But it does never become larger than all natural numbers, because the diagonal is restricted to the entries which all are finite. Obviously the diagonal cannot become larger than all entries.
>
> This is confused by set theorists: The diagonal has more digits than *every* entry but they claim that it has more digits than *all* entries. That's the intended fraud. That's why they strictly refuse to distinguish between potential and actual infinity. It is a simple quantifier exchange:
>
> For every subset {1, 2, 3, ..., n} of |N with cardinaliyt n there is a larger subset with larger cardinality. But by sleight of hand they claim: There is a cardinal number, aleph_0, that is larger than all others. And the newcomers are not able to distinguish these different infinities, are not able to recognize the fraud, and after a while they will refuse to recognize it because they have invested so much time and power in this nonsense that is completely useless for all scientific purposes.
>
> Regards, WM
>


Since there is so little (available) translated original
source material for me to examine, I cannot vouch for
your account here. But, if it is true, then why do
you keep blaming Cantor?

As with anyone being credited with an innovation,
Cantor worked at a time when meanings of certain
symbols were not as well-structured as they are
today. One could now make an attempt to argue for
some sort of game-theoretic interpretation of the
19th century quantifiers. An epsilon-delta proof
could be seen as a claim for a "winning strategy"
against your "finitely-definable" numbers.

Cantor's logical analyses are in the tradition
of Bolzano, Weierstrass, and Dedekind. But, this
is not the tradition of Peano, Whitehead, Russell,
Wittgenstein, and Carnap. Peano's axiomatization
of number had been much more amenable to the British
tradition influencing Whitehead and Russell. And,
because they succeeded in formulating a system of
logic with such extent, it might be much more
appropriate to blame them for how the modern
interpretation of quantifiers in mathematics came
about.

Russell may have been reluctant to accept an
axiom of infinity, but he had an agenda to ground
truth in such a way as to discredit the religious
faith of others. The principle of "logical atomism"
grounded in his "theory of acquaintance" is a
metaphysical theory about Bertrand Russell's
sense data. Oh joy!

But, you blame Cantor for this metaphysical
interpretation of quantifiers.

Wittgenstein's "logical atomism" is just as
bad, if not worse. But, because these men had
been reluctant to accept an axiom of infinity,
you count them as authorities in support of
your agenda.

I do not know Carnap's opinions on infinity,
but his "meaning postulates" and "analytical
truth" are just as much of a misdirection as
the "logical atomism" of the prior two philosophers
mentioned. I cannot say what "truth" is. But,
I am fairly certain that there had been "truth"
before there had been "dictionaries" and I am
fairly certain that "truth" for living organisms
that experience the world through sense data
is not to be found in "dictionaries".

But, these philosophers and their metaphysics --
which they vehemently denied as being metaphysics --
is convenient for you because it permits you a
luxury. To be precise, "words mean what you say
they mean".

All that Cantor really observed is exactly
what you are talking about. Numbers are
uncompletable.

If you take into account that Cantor's notions
arose from what we now call topology, then
the rape of his set theory at the hands of
philosophers is evident. Ultimately, his
competitor, Frege, had more influence on the
eventual formation of the logical system in
"Principia Mathematica" than Cantor. But,
this is because Frege's definition of number
had been proposed as an attempt to show
that Kant had been wrong. And, if there is
anything a couple of 19th century British
empiricists like Whitehead and Russell would
jump upon, it would be an opportunity like
that.

This metaphysics of "not metaphysics" continues
to this day -- and it continues with its
program of ignoring Cantor. This is evidenced
with the generalized continuum hypothesis.

According to Cantor, to the extent that one
treats infinities mathematically, they should
be treated like finite sets. Well, in the
early years of set theory, it is perfectly
understandable that a proof of the generalized
continuum hypothesis might be sought. Today,
however, many issues have been worked out. It
is known that the generalized continuum
hypothesis corresponds with Cantor's statement
of how to mathematically address infinities.

You can find the statement in the link:

http://en.wikipedia.org/wiki/Generalized_continuum_hypothesis#The_generalized_continuum_hypothesis

But, instead of simple direct statements, the
discussion of these matters is relegated to the
voodoo created by the metaphysics of "not metaphysics"
promoted by the philosophers mentioned above.

If one abides by the principle that what exists
in mathematics is determined by the principle of
non-contradiction, then there is a simple duality
associated with the generalized continuum hypothesis
which cannot be resolved.

The problem is understood. But, the metaphysics
of "not metaphysics" completely distorts the
mathematics.

You blame Cantor for things which are not his
responsibility. And, you do not take into account
how modern mathematicians and philosophers are
unwinding Bertrand Russell's adoration of George
Boole and contempt for the beliefs of others.

If you do not ignore the topological intuitions
of Cantor, then you get a different view of what
should be associated with Cantor's intuition. On
my bookshelves is a book entitled "Topological
Dynamics". Wikipedia has the link,

http://en.wikipedia.org/wiki/Topological_dynamics

That link refers to "semigroups" and leads to
the page,

http://en.wikipedia.org/wiki/Transformation_semigroup

But group theory arose in mathematics from both
a geometric and an algebraic origin. Moreover,
it entered mathematics at the same time as algebraists
began dealing with "semantic indeterminacy". So
the skeptical argument "It is just words" began
entering mathematics at this time.

If you look at the examples in the page on semigroups,

http://en.wikipedia.org/wiki/Semigroup#Examples_of_semigroups

you will find the set of positive integers with
addition. So, there is a rather direct route from
"topological intuitions" to the natural numbers.

And, one can now directly relate the natural numbers
to the mathematics of 19th century infinitesimals
by the ultraproduct definition of hyperreals,

http://en.wikipedia.org/wiki/Hyperreals#An_intuitive_approach_to_the_ultrapower_construction

If you read the link, you will find that both this
construction and the Cantorian construction in terms
of Cauchy sequences relate to ring theory. A Boolean
ring satisfies the axioms for a ring from abstract
algebra. But, a Boolean algebra is a construct that
only makes sense in universal algebra. It does not
integrate with the mathematical structures important
to the fundamental theorem of algebra or the fundamental
theorem of arithmetic.

There is also a reference to "ergodic theory" in
the link on topological dynamics (repeated here
so you may verify),

http://en.wikipedia.org/wiki/Topological_dynamics

In the link for ergodic theory,

http://en.wikipedia.org/wiki/Ergodic_theory

one finds a link to "topological entropy"

http://en.wikipedia.org/wiki/Topological_entropy

As stated in that link, topological entropy
is usually discussed in terms of compact Hausdorff
spaces.

Let me emphasize that in real analysis, "compact"
corresponds with CLOSED AND BOUNDED. In case it
is beyond you, boundedness in this sense means
that it is within a sphere of finite radius centered
at the origin. Philosophically, it corresponds to
the fact that "paradigmatic material objects" do
not have unbounded extent.

Moreover, you might consider Cantor's admonition
about treating infinite pluralities as if they
were finite in this context. That is, the
generalized continuum hypothesis is much like
a single-point compactification. And, this is
to be expected since he compared his notions with
the geometric points at infinity discussed in
the literature of his day.

Of course, none of this wonderful topology had
been worked out at the time of Cantor. What he
had to deal with had been Dirichlet's generalization
of "function" as a system of ordered pairs -- not
necessarily abiding by continuity constraints --
and a notion of dimension that is purely algebraic
in the sense of metamathematical subscripts.

Today, there is a topological characterization of
dimension, and, it is known that dimension is
preserved when functions are restricted to being
continuous.

There is more.

The definition of a "topological dynamical system"
in the link on topological entropy (repeated here
so you may verify),

http://en.wikipedia.org/wiki/Topological_entropy

discusses "continuous self-maps". Now, Cantor's
work occurred in the context of real analysis in
analytic spaces having a fixed origin. In the
link,

http://en.wikipedia.org/wiki/Directed_complete_partial_order#Properties

You will find the statement,

"Every set S can be turned into a pointed dcpo
by adding a least element BOTTOM and introducing a
flat order with BOTTOM <= s and s <= s for every
s in S and no other order relations."

If you notice that this is a reflexive partial
order. And, that its status as a reflexive
partial order is significant since this plays
a role in understanding the structure of
real numbers as "standard parts" with respect
to hyperreals,

http://en.wikipedia.org/wiki/Hyperreals#Properties_of_infinitesimal_and_infinite_numbers

I try to show a little respect for you. You
are a professional teaching at a university.
You are published in your field of study. You
have clearly made an effort at researches to
correct what you perceive to be a problem in
the teaching of mathematics.

When the character of usenet exchanges leads
to overstepping the bounds of respectful
discourse, I make apologies to you.

But, I sincerely feel that you are blaming
Cantor for mathematics that should not be
attributed to him. To the extent that it
originates with him, his views had been primarily
ignored. To the extent that it had been
altered by others, those alterations are
far more responsible for the distortions
which you find disturbing. You do not criticize
these individuals because these individuals had
been predisposed to a finitism, or to a
predicativism, which seems reasonable on its
surface. But no argument that leads, ultimately,
to "words have no meaning" or "it is just words"
can have any relation to the corpus of
facts derived through empirical studies
by the scientific community.

Cantor is not your culprit.




Date Subject Author
2/20/14
Read § 433 The reason for calling matheology matheology
mueckenh@rz.fh-augsburg.de
2/20/14
Read Re: § 433 The reason for calling matheology matheol
ogy
thenewcalculus@gmail.com
2/20/14
Read Re: § 433 The reason for calling ma
theology matheology
fom
2/20/14
Read Re: § 433 The reason for calling matheology matheol
ogy
mueckenh@rz.fh-augsburg.de
2/20/14
Read Re: § 433 The reason for calling matheology matheol
ogy
thenewcalculus@gmail.com
2/20/14
Read Re: § 433 The reason for calling matheology matheol
ogy
mueckenh@rz.fh-augsburg.de
2/20/14
Read Re: § 433 The reason for calling matheology matheol
ogy
thenewcalculus@gmail.com
2/21/14
Read Re: § 433 The reason for calling matheology matheol
ogy
mueckenh@rz.fh-augsburg.de
2/21/14
Read Re: § 433 The reason for calling matheology matheol
ogy
thenewcalculus@gmail.com
2/21/14
Read Re: § 433 The reason for calling ma
theology matheology
fom
2/21/14
Read Re: § 433 The reason for calling matheology matheol
ogy
mueckenh@rz.fh-augsburg.de
2/21/14
Read Re: � 433 The reason for calling WMytheology WMytheology
Virgil
2/21/14
Read Re: � 433 The reason for calling WMytheology WMytheology
Virgil
2/20/14
Read Re: � 433 The reason for calling WMytheology WMytheology
Virgil
2/20/14
Read Re: § 433 The reason for calling ma
theology matheology
fom
2/20/14
Read Re: § 433 The reason for calling matheology matheol
ogy
Tucsondrew@me.com
2/21/14
Read Re: § 433 The reason for calling matheology matheol
ogy
mueckenh@rz.fh-augsburg.de
2/21/14
Read Re: � 433 The reason for calling WMytheology WMytheology
Virgil
2/22/14
Read Re: § 433 The reason for calling WMytheology WMythe
ology
mueckenh@rz.fh-augsburg.de
2/22/14
Read Re: � 433 The reason for calling WMytheology WMytheology
Virgil
2/22/14
Read Re: § 433 The reason for calling WMytheology WMythe
ology
mueckenh@rz.fh-augsburg.de
2/22/14
Read Re: � 433 The reason for calling WMytheology WMytheology
Virgil
2/20/14
Read Re: � 433 The reason for calling matheology matheology
Virgil
2/20/14
Read Re: § 433 The reason for calling ma
theology matheology
fom
2/20/14
Read Re: � 433 The reason for calling WMytheology WMytheology
Virgil

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