Re: Why does everyone do it?
- From: David Bernier <david250@xxxxxxxxxxxx>
- Date: Mon, 25 Aug 2008 05:24:33 -0400
mike3 wrote:
On Aug 24, 6:28 pm, david petry<david_lawrence_pe...@xxxxxxxxx>
wrote:
On Aug 5, 11:23 am, mike3<mike4...@xxxxxxxxx> wrote:
What is it about Cantor's theories that makes everyone want to try and "critique"Since you ask such a question, you evidently do not understand what
them, anyway? And why do they always seem to generate long discussions?
the debate is all about. The reason such debates go on forever is
simply that people find it hard to believe that other seemingly
intelligent people find it hard to understand very simple arguments,
so they get hooked on the idea that just one more insightful comment
will clue those people in.
In the sciences, and especially in physics, it is often said that a
theory is "not even wrong" if it is not falsifiable (i.e. if it makes
no testable predictions). This notion of falsifiability could be
applied to mathematics. That is, we can think of mathematics as a
science. As a conceptual aid, we can think of the computer as
analogous to the scientists' microscope, and then mathematics studies
the phenomena we observe when we look through that microscope. And
then a mathematical theory, being scientific, must be falsifiable,
i.e. we must be able to use the computer to search for counterexamples
to claims made by the theory. All of the mathematics that is used in
the sciences--that is, all of the mathematics that helps us model
phenomena in the real world, which is an enormous amount of
mathematics--must be scientific in that sense. The problem with
Cantor's theory is that it is not falsifiable; it cannot help us model
phenomena in the real world; it is not scientific; it is
pseudoscience; it has nothing to do with reality; it is not *even*
wrong; it is a fairy tale.
The "mathematicians" cling to the idea that the only criterion that
mathematics must meet is that it be consistent, and then they conclude
that critics of Cantor's theory evidently must fail to see the formal
consistency of the formal proofs of the theorems (which is simply
false), and so they think they are the ones who should be perplexed
that others can't understand simple arguments.
Anyway, there has been a somewhat interesting new development in the
debates about Cantor's theory. The "mathematicians" are now trying to
link criticism of Cantor's theory with antisemitism, which would seem
to be an act of desperation. Maybe the "mathematicians" are aware
that they are very close to losing the battle.
So what _exactly_ do you consider to be the _flaw_ in Cantor's theory?
What is it that makes Cantor theory _problematic_?
ZFC and non-constructivist logic is what most mathematicians use
almost all of the time; maybe it's worth pointing out that set theory
pretty much started in mathematical analysis, after Cauchy and
Dedekind had both given a definition of the field (R, +, *) from more
primitive concepts. And it was a great success for analysis.
The first proof that there are transcendental numbers was by
Liouville, unless I'm mistaken; it was done constructively.
At that time, it sure seems nobody was thinking about
countably infinite sets vs. uncountable sets. In fact, in
the very definition of countability, there is the word "function"
or "mapping". One way to prove the fundamental theorem of
algebra is by using polynomial algebra, standard inequalities, and
limits. Thinking of z |-> | P(z)| in the complex plane,
we can get a Cauchy sequence of "definite" complex numbers which,
if the Cauchy sequence converges, must converge on a z_0 such that
P(z_0) = 0. Alternatively, from Hungerford's algebra, I've read
that given enough algebra, one can get by knowing that every
polynomial of odd degree with real coefficients has a root.
But proving that, even for polynomials with integer coefficients,
seems to me to require the Intermediate Value Theorem. That
again brings us back to Cauchy sequences. Actually, it
might be an interesting exercise, given a list of
Cauchy sequences of rationals S_n = {q_{k, n} : 1<=k< oo}_{n = 1 ... oo} , q_{k,n} in
[0, 1], to find a new Cauchy sequence {p_n} where p_n has p_n in S_m for
some m>n, where the new Cauchy sequence can't possibly have the
same limit as any of the S_n. I guess this follows from [0, 1]
being sequentially compact ...
So Cantor had a very simple proof of the existence of
uncountably many transcendental numbers; it would be interesting
to know what mathematicians of his day thought
of a pure existence proof of infinitely many transcendentals.
There are all sorts in sci.math who argue against set theory.
Some, perhaps a minority, understand the logic of set theory
but consider, say, the power set of the Reals a bit in the
way Kronecker thought of mainstream 1880's mathematical analysis,
and even more so of Cantor's set theory:
<< Lindemann had proved that pi is transcendental in 1882,
and in a lecture given in 1886 Kronecker complimented
Lindemann on a beautiful proof but, he claimed, one that
proved nothing since transcendental numbers did not exist.
So Kronecker was consistent [...] >>
Cf.: The MacTutor biography:
http://www-history.mcs.st-andrews.ac.uk/Biographies/Kronecker.html
----
When it comes to uncountable ordinals, say those of cardinality aleph_1,
the total ordering each of them has is quite alien, in my view, to
our ordering intuitions coming from geometry, the naive view of time or
the real numbers. For example, any well-ordered subset of the reals
is at most countably infinite. And after the ordinals of cardinality aleph_1
come those of cardinality aleph_2, and so on. There's an important
basic concept in ordinals worth trying to understand:
the notion of the cofinality of an ordinal.
Cf.:
http://en.wikipedia.org/wiki/Cofinality
If alpha has cofinality alpha, the alpha is regular. The infinite regulars > aleph_0
are aleph_1, aleph_2, aleph_3 .... (not aleph_omega ...)
so then comes aleph_{omega +1} , aleph_{omega+2} ...
if cofinality(aleph_alpha) = aleph_alpha, and alpha is a limit ordinal
(infinite and having no successor), then (I think) we have what is
called a _weakly inaccessible_ cardinal [ cof(beta) is a cardinal
for any ordinal beta ]. AFAIK, it's possible ZFC can't prove
one exists.
Cf. Devlin, K. ``Joy of Sets":
http://www.amazon.com/Joy-Sets-Fundamentals-Contemporary-Undergraduate/dp/0387940944/
He writes that there are infinitely many cardinals kappa > omega
such that kappa is the kappa'th cardinal ... Say we call the smallest such
cardinal LambdaMu. LambdaMu exists. Does it exist because
of the ZFC axioms? Is there a cardinal Omicron
such that Omicron is the Omicron'th cardinal sharing
the "bigness" property LambdaMu has?
So from reading a bit of Devlin's book, I get the impression that
from ZFC, one can show that there are cardinals larger than one
might think at first. What intrigues me (to some degree) is the meaning
of sentences such as " There are quite large cardinals that exist with
properties (a) ... (b) ... (c) ... (d) .... " , if one sets aside proofs and axioms
and thinks instead of these properties.
One way of putting my view is:
(a) Integers, arithmetic properties, conjectures --> tangible
(b) Aleph_{aleph_{aleph_30}}, "advanced" set theory --> less tangible .
David Bernier
.
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