Re: Why does Cantor a target for cranks?

In article <1166185380.308915.39920@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Albrecht" <albstorz@xxxxxx> wrote:

Virgil schrieb:

In article <1166099791.719017.67260@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Albrecht" <albstorz@xxxxxx> wrote:

I don't know if Cantor is a target for cranks. In my opinion Cantor's
ideas are targets for questions because multiple causes.
1st: Cantor's ideas contradict not only the opinions of nearly all
great mathematicians before Cantor but also the opinions of nearly all
great philosophers before Cantor.

That's to be expected of most new ideas.

No. The most new ideas augment the former ideas.

Little new ideas may be merely minor extensions of old ideas, but big
new ideas are, ore tend to be viewed as, revolutionary. Consider, for
example, relativity and quantum theory.




2nd: Many mathematicians and nearly all mathematical laymen seems to
believe that the set of reals is greater than the set of naturals in an
absolute sense.

In the Cantor sense, perhaps. Cantor defines an order relation on set
"sizes", which he calls cardinalities, by saying:
Card(A) <= Card(B) means there is an injective function from A to B.
One can easily verify that these "sizes" are ordered by the stated
definition, so that one has a proper order relation on cardinalities.

If one wants a concrete model, on can define in ZFC or NBG, the cardinal
of a set to be the smallest, by inclusion, of all ordinals which biject
with the given set.

All this is well known. You need to affirm your own knowledge? But
that's not the subject of my 2nd thesis. You seems to belong to the
mathematical laymen I talked about. Without use of the axiom of
Infinity it is totally nonsensical to say that there are more objects
of any kind than natural numbers.

It is not nonsensical, it is merely unjustified without axioms to
justify it. But it is equally unjustified to claim that in the
non-physical world of mathematics there cannot be infinitely many
objects without having some sort of axiom to that effect.

My point is that, at least in in pure mathematics, one cannot conclude
anything at all without SOME assumptions. And what you get depends on
what you assume.



They can't see that the consequences of ZF only hold in
the framework of ZF.

And the conseqeunces of other axiom systems likewise only hold within
those systems. So what?

Axiom systems are not the world.

They are the world of pure mathematics, as you cannot draw any
conclusions from nothing.



They see in the nondenumerable sets platonic
objects and believe in their existance in an alogical manner.

That may be what you see, but I do not see anything mathematical being
valid outside of some axiom system. With NO assumptions at all, one
cannot deduce anything.

You miss totally the point. Surely we need always assumptions to
deduce something. But there are different kinds of assumptions. I
think, this is the centre of your problem. We may have the assumption
that the augmentation of a defined quantum with another defined quantum
leads to a defined quantum. The addition is based on this assumption.
We can prove this assumption in a finite number of cases and we can't
see any causes why this assumption should not hold in any of infinite
many cases. We call assumptions like this plausible assumptions. We may
use assumptions like this to build up axiomes.

It has been a plausible assumption for several millennia that there are
more than any finite number of points on a line.

Cantor uses an assumption which isn't plausible and which isn't
provable for any case:

"Infinite discret aggregations exist".

You use the assumption that it is false without being able to prove your
assumption. So you are doing the very thing you are criticizing.

3rd: Many people who discuss in sci.math have the opinion that ZF or
ZFC covers the whole mathematics. This opinion leads to constrictions
in the mathematical thinking. Maybe, all actually to be known math can
be expressed in ZF or ZFC. But if all mathematicians would further work
in ZF or ZFC only, they could overlook plenty of possible mathematical
worlds.

No mathematician claims that everything must be done in ZFC, or NBG, or
NF, or any single system of axioms, but many insist that anything one
wants to do should be done in SOME axiom system, i.e., there are no
universal truths other than logical tautologies that mathematicians are
bound by, and not all of them agree on which logical tautologies are
binding (e.g., the law of the excluded middle).

Futhermore, a great deal (though not all) of mathematics has been
successfully embedded in ZFC or NBG, so they serve as a unifying
principle or common basis for a good deal of mathematics.

Yes. And this may be a chance - or a danger. The danger lies in the
false assumption all math should be done in ZFC or any other axiomatic
system. That's what I've tried to say.

I am not aware that anyone has said that all mathematics MUST be done
all in one axiom system, though I can see that it might be a great
advantage if it were ever shown to be achievable.


4.: Nobody knows if ZF is inconsistent or not. Therefore deductions of
ZF must be proved very well and in some cases like e.g. the
Banach-Tarski-Paradoxon, math must be adapted to avoid antinomies.

No one can prove the consistency of any system sufficiently complex as
to allow the construction of standard arithmetic, so systems not known
to be inconsistent is as good as it gets.

Why are you unable to see that your argument isn't useful.

I think it supremely useful in showing that certain forms of logical
perfection are unachievable.



We live in a
real world. All we know is not sure. But there are things which are
more sure than other things.

The axiomatic method does not in any way dispute what you have just
said.

It is clear that the truth of what is derived from an axiomatic system
is ultimately dependent on the truth of its axioms, and that the truth
of its axioms can never be assured unequivocally.


Why do you try to suppress this truth to
defend the axiomatic method?

Why do you attack a method that has been a bastion of logic for
millennia?

Is Euclid your version of a mathematical anti-Christ?




There are much more causes. Maybe, I'd listed the most important one
why many deep thinking people criticise Cantor's work and the
consequences of Cantor's ideas.

Do they equally criticize Goedel's work?

Do you?

No.
.

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