Re: Why real numbers and points can't model continuum

On Nov 24, 6:38 pm, Venkat Reddy <vred...@xxxxxxxxx> wrote:
On Nov 25, 4:46 am, mike3 <mike4...@xxxxxxxxx> wrote:

On Nov 20, 2:56 am, Venkat Reddy <vred...@xxxxxxxxx> wrote:
<SNIP!!!>

The continuum is DEFINED as the set R of real numbers, and the
"points" in the continuum are DEFINED as the elements of that
set -- the real numbers.

Yes. No doubt about whether they are defined are not. It's there in
math books.


Good.

The real numbers can be defined in
a number of equivalent ways,
and none of them seem to cause a
lot of trouble like you claim.

Different but equivalent(?) definitions are not causing any new
trouble, because they are all equivalent :).

Again, good. At least we've got that resolved.

The question is about the
attempt to model the continuum using points and real numbers, which
have zero measure and trying to fill up space using them.

Attempt to "model" the continuum using real numbers? The
real numbers *ARE* the continuum, not just a "model" of it.
You need to remember that: The Real Numbers ARE the
Continuum, or are "a" continuum. They are not merely a
"model" of it. "The Continuum", in the singular, refers
to the real numbers (R), not a model of it but the reals
themselves.

http://mathworld.wolfram.com/Continuum.html
http://en.wikipedia.org/wiki/Continuum_(mathematics)

And also
about using infinite as an excuse to justify that zero measures can
really "fill" up space.

Interestingly, one also seem to agree that the sequence, 1/2, 1/4,
1/8, ... will never end up with zero, but then also agrees that when a
line segment is cut using infinite divisions, the piece ends up as a
point of zero length.


In the former case it will never reach zero after any _finite_
number of terms. If you are willing to admit an _infinite_
number of terms you could say in some sense that it "reaches
zero". The same goes for cutting up the line segment, or
splitting the interval into subsets.

You might ask, what makes me think that infinite number of cuts can't
make the piece length vanish.

They don't. You may want to examine the definition of measure,
which is the definition of the "length" you talk about:

http://en.wikipedia.org/wiki/Measure_(mathematics)

Note this part:

"***Countable*** additivity or σ-additivity: if E_1, E_2, E_3, ... is
a ***countable*** sequence of pairwise disjoint
sets in Sigma, the measure of the union of all the E_i's is
equal to the sum of the measures of each E_i."

Did you see that? COUNTABLE. The number of points on an
interval is countless. Therefore, this property need not
apply.

Also, see this:
http://en.wikipedia.org/wiki/Sigma-algebra

to see how the collection of sets the measure is defined
on works. See this as well to see how the topology of the
reals provides that:
http://en.wikipedia.org/wiki/Borel_algebra

I think so because, when you try
reasoning the opposite process - building of finite length using zero
length - if this is true, then all non-existent measures should spring
to a finite measure, by the magic of infinite, which I think is a
simple belief in super-natural powers of infinite in converting no-
existence into existence.


Furthermore, the Cantor set shows that an infinite amount
of points need not "magically" have positive measure. The
Cantor set has beth-one points in it, the same countlessly
infinite number of points as in a solid line segment.
There's more to length than just having an infinite &
countless amount of points. For one, it's not an interval,
or a union of countably many intervals. So not every
infinite collection of points has "length". It depends on
what points you put together. You have to quit thinking of
points as little BBs or something like that. Infinity
is _not_ an intuitively obvious concept that common sense
works well with. Quite probably because common sense is
just that -- COMMON sense, and infinity is far from our
common experience.

I hope what I'm saying is much clearer now.

- venkat

.

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