Re: Is continuum completely filled up?


"Dave Seaman" <dseaman@xxxxxxxxxxxx> wrote in message
news:eot851$mee$2@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx

My concept of interval is not actual one. I cannot discribe infinity
visually. It is
representation that we cannot descrive whole reals, and logical
conclusion.
If reals have any size and it represent distance, we cannot set up them
on
line
withot gap between them.

Reals don't have size. The "size" (Lebesgue measure) of a single point
on the line is 0.

Sorry what I meant is size of number. number 2 has size 2.

Where is the "gap" between the negative reals and the nonnegative reals?

If there is no gap between negative reals and nonnegative reals, it means
that
they lacks boundery which distinguishs them. Neighboring numbers which are
bounded by "0", or "0" itself are not distinguishable. The "gap" between
the negative reals and the nonnegative reals is the same as gaps imagined
among the other reals. The expression that there are another reals between
any two reals, is a explanation that keep away from logical conclusion that
leads to the limit condition around "0", just as the axiom of infinity may
leads to the existence of omega. This uses Zeno's arguments.
Zeno's paradox state that we cannot reach to infinity, but not that there is
something infinite already. I say that this cannot be decided without
difinition or assumption.
I propose two components of real line, that is, line and reals. "0" is mere
mark, as well as other numbers. Though I wonder whether this idea build
interesting,useful theory now.

Recursive explanation as for that there are no gaps among rationals, is
the
same
as the story of Achilles and tortoise. They never reach the goal.
Difference
from
the latter is that uncountable reals are assumed that fill line,and
distinguishable each other.

The reals are not "assumed". They are explicitly constructed, either as
Dedekind cuts or as equivalence classes of Cauchy sequences of rationals.

What are explecitly constructed by Cauchy sequenes, is the limits of
sequences,
which we can follow certainly, the others bare uncertain. Dedekind cut is
the same.
We can indicate only reals which know the existence from algebraic way.

The uncountability of the reals is not "assumed". Uncountablility is
proved from the definition.

With disputable proof. I wonder that the argument that diagonal number is
diffrent from any other numbers of the list in some digit decisive or not.
As for these circumstences, you may well know. Cannot we make another choice
as well as this like axiom of pararell lines?

And diffinition of uncountable. Points acquire volume, when they gather
uncountably many.

Yes, because Lebesgue measure is countably additive, but not uncountably
additive.

I don't opose this thepry, because I think that we cannot deal with infinity
without some vagueness. And mathematics deals with reals well as you done so
far. But sudden change that infinity and uncountable bring about, may not be
accepted by everybody, because its base is disputable as I have shown above
one of them.

To sum up my understanding of your objection, you say that in the
limit
(there isn't one?) because there are an infinite set of reals, you
cannot identify adjacent reals generated by the construction, so it
falls apart.

Whether or not there is a limit has nothing to do with the behavior of
infinite sets. You can't find out how infinite sets behave by taking
limits over finite sets.

Answer to this question is found in first assumption. Cardinality is the
one
to one crrespondence
, and not the amount of objects. but their total constructs a real line.

I don't know what you mean by "this question". I didn't ask a question
there, and the answer to Andy Smith's question is just what I said. He
was inappropriately asking about limits in a situation where they are not
applicable.

Yes, what I said above may be another question.

Reals are derived from rationals, and how much we might add point on
line,
it cannot be filled.
This observation may leads us to concude that reals are more than
countable.
But numbers except countables are not unspesifiable,too. Their existence
remain in assumption.

There you go again. The existence of the reals is a matter of proof, not
assumption. Unless you are referring to the assumptions contained in the
axioms of ZFC.

I don't know well about axiomatic proof. Probably axiom of infinity,axiom of
power set, and axiom of choice have problem, when they are appried to
infinite set. This is only my guess.

.First intuition about infinity is that line are constructed from
points. I
imagined this way for long time.
Perharps most of people entertain this idea at first.
Secondary there comes Zeno's reflection. This might have older origine.
I
have rejected this, because this differs from reality.
I think that we cannot deal with infinity without any assumption or
inconvenience. Standard theory is useful and interesting, but bring
about
some strange results

The only assumptions we need are the axioms of ZFC. As for your
assertion about points filling the line, I think you have it backwards.
First we define the points (the reals) and then we define the line R = {
x : x is a real number }. By definition, the real numbers fill the line.

I know. I only don't adopt this diffinition. I think this diffinition deals
with line well, and conveniently, but matters come to infinity, behaviers of
numbers derived from ZFC don't match to my intuition.

The reason there are no intervals is *not* that the reals are an
infinite
set. After all, the integers are likewise an infinite set, but there
is
still an interval between each integer and the next one.

Naturals are not finite and there are no largist member. But when their
size
is represented
with distance, can they produce infinte size of line?

I don't know what you mean by that. The measure of the integers is 0 and
the measure of the line is infinite. Does that answer your question?

Sorry, lack of my English ability delay communication. I meant number size
or number itself. But, probably the answer for my question may infinite too,
because there are no largist number in N?

We can assume infinite line at the beginning, but what is the size of
it?
Why finite size of reals
fill a line without gap and without contact( in such case, they are
identified), or they have undefinable size?

We don't "assume" the line; we *define* it. I am not going to address
any more questions in which you use the word "assume" inappropriately.
Find another way to ask your question.

Yes, then I use "define" instead of "assume"as long as possible, after
having concidered applicability.

The reason there are no intervals when considering the rationals or the
reals is that these sets are dense in the line. Given a member of Q or
of R, there is no "next" member and therefore it makes no sense to
speak
of intervals between adjacent elements. This is not something that you
can find out by considering only finite sets, which is what a limit
process would do.

We can define rational line though it is hard to deal with, and cannot
in
plane. I am not certain, but if we
can construct extended reals, and extended Cauchy sequence, then reals
might
not become complete.

I don't understand any of that. Why are the rationals hard to deal with,
and why can't we consider QxQ, the subset of the plane consisting of
points with rational coordinates?

I wonder whether we can deal with geometrical objects except one
dimensional.

I would say that at each iteration we know where our existing set of
numbers are, and can therefore move to the next iteration. And, as
you
have taught me re Zeno "there is no last term" , so there is no last
iteration?

There is no last iteration, and there is no iteration that results in a
dense set.

I don't know the meaning of iteration well. Is the correspondence
between
naturals and rationals not
iteration?

Iteration in the sense mentioned above means a sequence of objects, in
this case a sequence of subsets of the reals.

Thanks.

It isn't the sum that bothers me. It is the fact that one can argue
that
the sum is one thing, and then argue that it is another.

I think you have misunderstood my question. Given a series
sum_{k=1}^oo
a_k, we *define* the sum to be the limit of the sequence of partial
sums,
given by

s_n = sum_{k=1}^n a_k,

*provided* the sequence { s_n } converges. Notice that this definition
does not assign a meaning to the sum of the series in the event that
the
sequence of partial sums does not converge. Hence, my question, which
you have evaded: what meaning do you have in mind for the sum of an
non-convergent series?

I think that it depends on procesure to take sum like an example of
"balls
on a vase"
, if we want to define the resonable sum.

You have evaded the question. I asked for a definition of "sum of an
infinite series" that would apply to the case where the sequence of
partial sums does not converge. You have not provided such a definition.
..

If there might be uncountable collection of strings, we can definitely
put
uncountable reals on line.

There is a surjection from the set of all infinite bit strings to the set
of real numbers.

I don't know clearly the connection of proofs which cover from ZFC to
diagonal argument. If the axiom of power set implys uncountable reals,
problems originate ZF.
Typically (as an engineer) I view a set of bits as defining an
addressable range, so when I say a set of bits I meant the set of
possible different strings that can be represented by those bits.

I think so, too.

Andy Smith's statement was ambiguous. I was attempting to make the
distinction clear.

I thougt that above statement means that every bits of infinte strings are
not clearly defined.
Besides the Dedekind-cut definition of the reals, there is also the
Cauchy-sequence construction. You can find both constructions at

<http://en.wikipedia.org/wiki/Construction_of_real_numbers>

in sections 2.1 and 2.2. The point is, a countable sequence of bits
defines a Cauchy sequence of rationals and therefore corresponds to a
unique real number.

It correspons to definable real number. It is assumption that it
constructs
uncountable reals.

It is *proved* (not assumed) that the reals are uncountable. The proof
may or may not appear in that particular article, but it exists.

Set theory is interesting. What I want is that people don't confuse
assumption and
what we can say with certainty, and have wrong image about reality.
My inference may be wrong, but share with question of someone else.

That's exactly what you have been doing. Over and over you use the word
"assumption" where it does not apply.


Regards
Ozaki Toshiaki

.

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