Re: Why is Cantor a target for cranks?



On Dec 22, 8:32 am, "Newberry" <newbe...@xxxxxxxxxx> wrote:
R. Srinivasan wrote:
On Dec 15, 11:25 pm, "apoorv" <sudhir...@xxxxxxxxxxx> wrote:
david petry wrote:
Andrew Usher wrote:

Why are there so many on this groups whose mathematical goal is to
disprove uncountability? I can't imagine, really, why it would be a
crank target

There are several arguments against Cantor's theory.

1) It leads to inferences that do not fit in with the physical reality.
Zeno's paradoxes are a good example of this situation. Cantor's
theory allows us to imagine of an infinite seq. of discrete events
taking place in a finite time, with no last event. The principle of
time reversibility allows us to visualize these events taking place in
the reverse order. Thus, we have a sequence of events where we move
from zero events to an infinity of events without ever encountering the
first event.
[...]

This objection has been addressed in the following paper:

http://arxiv.org/abs/math.LO/0506475

Sec. 4 is readable independently of other sections and explains a new
method for doing real analysis in a new logic (NAFL) that would take
care of your concerns.

Your objection, essentially, can be re-stated as follows. If we had
infinitely many real intervals [-1,1], [-1/2,1/2], [-1/4,1/4], ..., the
intersection of these intervals is empty, yet *all* these intervals
have infinitely many (in fact, uncountably many) points. I.e.,none of
the end-points ever come *infinitely close" to zero, but yet the
intersection is empty-- how can that be? Why doesn't the intersection
contain an interval?

Similarly, a related paradox is as follows. If one takes the infinite
summation

1/2+1/4+1/8+.......=1,

each of the infinitely many terms in this summation is of non-zero and
non-infinitesimal magnitude, yet the sum is a finite magnitude (which
might represent the distance travelled by Achilles in order to catch up
with the tortoise). So the question is why isn't the sum inifnite.

Nobody ever said that the sum was equal to 1. (It is impossible to add
infinitely many operands.) The sequence CONVERGES to 1.

You are talking of the sequence of real numbers 1/2, 3/4, 7/8,...To go
from one term of this sequence to the next, you are adding a finite and
non-zero real number (there are no non-zero infinitesimals in standard
real analysis). And you repeat this endlessly. So why doesn't the
sequence increase without bound? Why does it converge to 1? That
precisely is the paradox. In classical real analysis, one simply
refuses to accept that there is anything paradoxical about this.

But in NAFL direct quantification over real numbers is banned, since
each real number is an inifinite object, say, as represented by a
Cauchy sequence of rationals. In NAFL, you must *construct* this
sequence without quantifying over reals. The NAFL method is basically
to represent (code) this sequence of reals by a sequence S of rationals
having each of the real numbers {1/2, 3/4, 7/8, ...} as limit points.
Note that rationals are just ordered paris of integers and there is no
problem with quantifying over rationals (quantification means that you
can use the universal and existential quantifier for rationals, or
equivalently in NAFL, an "arbitrary" rational number x exists despite
the fact that no construction is specified for x; but an "aribtrary"
real number in this sense does not exist in NAFL). But the sequence S
must necessarily have the real number 1 also as a limit point. So the
NAFL method of constructing the said sequence of reals must necessarily
construct all its limit points as well and so the limit points must be
included in the sequence.

The NAFL resolution of Zeno's paradoxes is as follows.

FIrstly you are not "endlessly" adding finite, non-zero and
non-infinitesimal quantities (starting with 1/2) and yet having a
sequence that is bounded by the real number 1. You could think of the
NAFL construction of {1/2, 3/4, 7/8, ..1} a "simultaneous" construction
of the terms of the sequence, including the limit point 1. You are not
able to explicitly identify a decreasing sequence of infinitely many
non-zero, non-inifnitesimal reals (in this case {1/4, 1/8, 1/16, ... }
which are successively added to the initial real number 1/2 because the
said decreasing sequence must necessarily include zero, when
constructed by the NAFL method.

Secondly it is not valid in NAFL to ask exactly "how many" terms are
present in any sequence of reals with a valid NAFL construction (say,
{1/4, 1/8, 1/16,...0}). Because quantification over reals is banned,
there is simply no cardinality for any super-class of reals (which are
used to denote "collections" of reals in NAFL, such as, the said
sequence). So you are not able to assert that the above super-class
contains "inifnitely many" reals.

The above two requirements of NAFL resolve Zeno's paradoxes. Note that
quantification over infinite entites is generally banned in NAFL and
that is what leads to the other conclusions that you agree with, like
Goedel's theorems not going through. NAFL is thus ruthlessly consistent
in its logical requirements.

Regards, RS

which would essentially mean that Achilles could never catch up with
the tortoise, as asserted by Zeno?

Classical real analysis does not really address these paradoxes, it
simply refuses to acknowledge that these are paradoxes.

In contrast, the NAFL version of real analysis proposed in my paper
does confront these questions and addresses them. The conclusions (e.g.
open intervals of real numbers do not exist, dy/dx is just 0/0, etc.)
may be startling and highly unpalatable to mathematicians/logicians,
but in my opinion they provide logically and philosophically
satisfactory resolutions to Zeno's paradoxes.

Regards, RS

.

Relevant Pages

  • Re: infinitely many nns = infinite nns?
    ... I have already explained in sci.logic threads as to how NAFL deals ... quantification over infinite classes is banned. ... sequence of closed real intervals: ... can only be possible if open intervals of reals ...
    (sci.logic)
  • Re: Multiple infinities - one more look
    ... continued for lager length of digit sequences without limit. ... infinite digit sequences... ... so the resulting reals have an order. ... (i.e. having a finite program to output their digits in sequence). ...
    (sci.math)
  • Re: Galileos Paradox and the Project of the Reals
    ... The way you get reals by a neverending process of generating digits ... and which thereby constructs a sequence of nested intervals whose ... It isn't *already* in the rationals you started with. ...
    (sci.math)
  • Re: Review of Mueckenheims book.
    ... If infinite, the limits of the ... that the reals are not countable. ... He starts with the sequence of rationals: ... in the building of the diagonal *each* digit has to be changed. ...
    (sci.math)
  • Re: Cantor Confusion
    ... least appears to totally disconnect the set of those remaining. ... sequence of reals with each term less than all terms of a strictly ... This is not the case for rationals. ...
    (sci.math)
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