fom
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Re: § 433 The reason for calling ma theology matheology
Posted:
Feb 21, 2014 2:01 PM


On 2/21/2014 6:10 AM, mueckenh@rz.fhaugsburg.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.fhaugsburg.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 wellstructured as they are today. One could now make an attempt to argue for some sort of gametheoretic interpretation of the 19th century quantifiers. An epsilondelta proof could be seen as a claim for a "winning strategy" against your "finitelydefinable" 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 noncontradiction, 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 singlepoint 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 selfmaps". 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.

