fom
Posts:
1,097
Registered:
12/4/12
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Re: The Distinguishability argument of the Reals.
Posted:
Jan 4, 2013 12:41 PM
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On 1/4/2013 12:18 AM, Zuhair wrote:
> On Jan 4, 3:59 am, fom <fomJ...@nyms.net> wrote:
>> On 1/3/2013 10:30 AM, Zuhair wrote:
>>
>>> By the way I might be wrong of course, I'll be glad to have anyone
>>> spot my error, my analogies might simply be misleading.
>>
>> All right.
>>
>> Why did Dedekind make his investigations?
>>
>> Why did Bolzano feel compelled to prove the
>> intermediate value theorem?
>>
>> Why was Cauchy careful to not say that the
>> fundamental sequences converged into the
>> space from which their elements had been
>> given?
>>
>> I realize that you are not talking about
>> those subjects. But you are taking them
>> to the garbage heap -- along with every
>> plausible piece of mathematics that uses
>> the completeness axiom for the real numbers.
>>
>> You cannot prove the fundamental theorem
>> of algebra without results from analysis.
>> It requires the existence of irrational
>> roots for polynomials and the intermediate
>> value theorem. So, you are tossing
>> algebra onto the same heap with analysis.
>>
>> Now, there is a circularity in the topology
>> of real numbers. If you want to have
>>
>> x=y
>>
>> it must satisfy the axioms of a metric
>> space. But those axioms are too
>> strong.
>>
>> Go get yourself a copy of "General Topology"
>> by Kelley and read about uniformities and
>> the metrization lemma for systems of relations.
>>
>> What you will find is that the metric space
>> axioms (the important direction associated
>> with pseudometrics) depend on the least upper
>> bound principle.
>>
>> One can simply view it as fundamental sequences
>> being grounded by cuts. It is not circular
>> in that sense. It simply makes Dedekind prior
>> to Cantor.
>>
>> Before you continue with this mess, you should
>> take some time to learn what it means for two
>> real numbers to be equal to one another.
>>
>> It is not the Euclidean algorithm.
>
> Dear fom I'm not against Uncountability, I'm not against Cantor's
> argument. I'm saying that Cantor's argument is CORRECT. All what I'm
> saying is that it is COUNTER-INTUITIVE as it violates the
> Distinguishability argument which is an argument that comes from
> intuition excerised in the FINITE world. That's all.
Fair enough.
Much of what you have been posting seems
to be "confused" between the naive finitism
such as WM is asserting and the kind of
finitism that leads to constructive mathematics.
It seems, however, that you are merely
"investigating" matters.
At the heart of matters is the question of
how mathematics seems to have the explanatory
force that it has. Perhaps it is geometry. But
when we go to represent the geometric form
arithmetically, we are confronted with the
fact that a system of names is different from
the ostensive dubbing of a name.
Descartes introduced the problem. Every
piece of analytic geometry presupposes a
consistent global labeling of points.
Newton and Leibniz made it apparent with
the calculus.
The modern calculus gets around the foundational
problem posed by differentials with the
"little-oh" mechanism that ignores errors
by virtue of the fact that errors approach
zero at least as quickly as the differential
takes its value. It is somewhat humorous
since Berkeley's analysis of Newton's calculus
asserted that its effectiveness was analogous
to bad bookkeeping in which an error in one
set of books is countered by another error
in a second set of books. History seems to
have forgotten that even though calculus has
banished the infinitesimal.
In any case, have fun. I will leave you
to your "investigating".
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