Re: Is continuum completely filled up?

Dear Toshiaki,

"Saurav" <saurav1b@xxxxxxxxx> wrote in message
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toshiaki wrote:
"Saurav" <saurav1b@xxxxxxxxx> wrote in message
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toshiaki wrote:
"Saurav" <saurav1b@xxxxxxxxx> wrote in message
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I assumed 2. for convienience ( I also make assumption for infinity
,and
this assumption is based on my idea that we cannot difine infinity
consistently ).

It is difficult to answer; because, as soon as you accept the
existence
of infinity, the existence of a dense ordering comes true.

If possible , I don't want to admit unmeasurable set .

Unmeasurable sets? A dense ordering hardly needs some measure to be
defined; I don't understand why you are taking measure into the stage.

Sorry, this sentence is out of contexst here .
A point on a line is intuitionstic , but it's complement set is
unintuitionistic .

People are there who would reckon both as unintuitionistic.

May be .
I want to regard a point as mark not substanse of a line , its measure
is
0 ).

Explain.
My idea is to consider a line not a collection of points but that of
intervals .

The collection of intervals is something else than what we in topology
consider to be the space. You probably know that if we deal with
topological aspects, the collection of interval becomes something we
call the "basis" of the topology.
Well, if you are not satisfied to view a line as the collection of
points, I must allow you to have your own opinion. But in that case,
you have got to construct a new kind of topology.
A line is, in as much as we comprehend presently, regarded as a
linearly ordered set of points; and the collection of intervals
generate a unique topology on it, called the "order topology". To
construct in your own way, I think, you shall have to review the whole
conception of order topology and their related theorems.
I recommend a paper, by van Douwen, entitled "The Horrors of topology :
a nonnormal orderable space". This will make you understand how
critical it is to deal with even our present conception of order
topology; if you pioneer a new conception, remember, you'd have to
construct a parallel, vivid, more appealing theory of this.
Marks on it is only to indicate position , not to construct it .
When we subdivide a stick of cheese infinitely many ( much ) ,I wonder
this
cheese tranceform into cut points .

To divide something into infinitely many parts, you need to formalize
your thought by using transfinite induction. Put your dividing machine
in mathematics, ie, in transfinite induction. Is there any so?

I want to ristrict our argument in the range of ordinary language .
But ordinary language does in no way impede you from using transfinite
induction, and if you take "ordinary language" for a literal language,
I am afraid you'd fall through to put even a smidgen of your intuition
into considerably unambiguous form.
If you show it impossible by mathematical proof ,or by ordinary language , I
will try to
do so . If it is needed for yourself ,I agree ,and try to do also .
I may restate my sayings above that cheese is composed of points . hence
when it was cut at every its points, nothing is
remained .

See, what you mean by cutting is indeed a finite recursive process. You
cut nth time, then (n+1)th time. There is hardly any limit ordinal
there. But in cutting a continuous object, of whose the cardinality
surpasses d, there is a need to cut "omega'th" time; how could you
manage to do that?
About other my argument also , I want to know why some theory of mathematics
is different from our intuition .

What actually our intuition tells us is not that much clear.
If mathematical statement stand only in the range of its logic system ,its
no use for me .
What?
Everyone has a concept of sphere .Is the sphere that is difined
mathematically is only one?
Recently ,Poincare conjecture is solved . I t is about equevalence of two
difinition about three dimensional sphere .
So what?
I want that mathematics can expresse things naturally as far as possible .
I talk about what I took doubt about mathematics , because I want to know
why the matter is that .
Right, you should think how they are as they are, this is actual
philosophy. Thank you.
My misunderstanding may be much . I expect your answer will help readers to
know mathematics better .
My idea is that how much a cheese should be subdivided ,it remain as the
same cheese and points of subdivision .
And my aim is to build useful ,consistent real analysis and
measure theory as far as possible .

Good! Carry on.

This is possible only with your comments . If my idea is radically
wrong
,it
will not make form . I have not any perspective clue at present .
It seems to me odd to remove a point . Though we remove the boundery
,
there
remain boundery of points which belong to the set ,but cannot be
specified .
But my understanding may be insuffisient .

I don't understand what you actually mean by "boundary". First
you explain this.
This is the same as what Hero explained ."boundary" is a "point" in
topology

Sorry, I couldn't have found Hero's definition. Possibly he has defined
it to be the end popint or supremum or something like that.

Yes, the end point,if my interplitation is correct .
We can think points in interval as a distance from a base point .

What is distance? Can you define? More than just a humour, it has a
reason to raise this question.

The explanation of distance is the same as above . "length " .It's lack
of
my ability of expression .

The reason that I asked you this question is to remind you that length
is a real number, and you are trying to characterise points, which are
real numbers, by length. This is a cyclic reasoning, isn't it?

I want to think a line not as collection of points , but that of intervals .
I don't know wheather this idea goes well or not .
Read what I've written above. Your ideas ->might<- go well, but in
order to do that, you've got to toil.
If we think the boundery of compact set as a mark , cannot we think
that
the
diffrence between
open set and closed set is the difference of wheather to take into
acount
the boundery or not ?
I am reading your paper though it may take time read through . This
is
short
,but
foundamental issues of ordered field is nicely arranged .
Good for me .
This is only visualised idealistic explanation.
These are pictures that come from my idea that we can only deal
with
finite
objects.
Infinity is shown by following way.
countable infinity every number have its next. and
assumption
that
there
exist set including all of them.

Not merely! According to an assumption, which you have every
reason to
disagree with, every set, however large, can be well ordered; and
in
such an order, almost all elements in the set has a next element.

I accept your opinion as far as you admit that these are based on
assumption
.
We can actualy deal with only finite numbers of things .
I think, this is compatible with Lowenheim-Skolem theorem , is'nt it
?

No, no; certainly not. Lowenheim-Skolem's theorem deals with a deeper
concern. It tells that a first order theory based upon a countable
language, has a countable model. To comprehend the meaning of what it
means, shall need almost an entire devoted life of a mathematician.
For
it leads to a complexity, called Skolem's paradox, which has made
mathematicians all over the globe doubtful about the foundation of
mathematics. T Bays has devoted his life to the investigation of
whether it is indeed a paradox.

I don't think it paradox . We reason about infinity with countable
language
,and with difinition based on assumption except countable case .

One of the most appealing problems that arise in this area is that, the
relation "belongs to" behaves differently within and without a
particular model.
I shall study about this more ,and answer more correctly . As for "belongs
to " ,I know its role in the theorm .
Provably there are many argument about this problem , but what I meant is

NOT "provably", it has a different meaning, may be you meant
"probably".
that there don't exist uncountable
number of objects (or we cannot deal with them ).
There is much research on this. I myself am not sure what indeed has
created the problem. But as far I can tell you, "a is countable" may
have two meaning:
1) { x | x in a } is countable;
2) { x | M |= ( x in a) } is countable, where M is a particular model.
May be you already know this.
So that I thougt that
countable model only is sufficient to descrive a theory .
This is very controversial. You may read the thread
http://groups.google.co.in/group/sci.math/browse_frm/thread/122db6710267970b/c23b753fb8a4ecb3?lnk=gst&q=Saurav&rnum=2#c23b753fb8a4ecb3
Though this doesn't directly deal with the problem, I hope there is
something that relates to this problem.

I admit that there are things more than we can complehend . I had
thougt
that we can
complehend the world as far as time permits .
But now I wonder wheather that is possible. I dout about our ability
to
reason .

Right; about few years back, I also thought in this way; but as wisdom
surged up, I found that the whole realm recedes into the mystic region
of spirituality


First of all, you tell me whether there is at all an infinite set.
And
if such one exists, are we elligible to talk about its properties?
Remember, in our thought we seem to be rather finite beings than
infinite ones. Is it not a paradox? If we can talk about those, I
believe we can also talk about separating points.
You said with humor, just what I have said repeatedly so far .
You may know that it is not that is what I want to say .

But it is indeed what I believe I think; for I really doubt the
existence of infinity within our intuition.

I wonder what is the end of the universe , and where time go to .

Is there at all something like what you in your mind have sketched? I
mean, is there a "universe" as you think? I mean, again, is what you
consider to be the "universe" really the Universe? What is Universe?
Can any living being comprehend it?
I don't know what sort of thigs you imagined from what I said .
But for this question, my answer is that because I think , then I exist .
Who told you this answer? It is not you that has said this first, as I
know; but the implication is not clear. Who said, again, that you
think; and who said, that if one thinks then one exists?
Provabley we all are finite and mortal , so that , I know that I can not
complehend the Universe .
You _know_ that? How?
But I am in the Universe as well as you ,and communicating with you .
Probably you have read Russell's Paradox. You can not say " I have the
property p". For example, if you say " I am lying", are you lying
indeed? In view of this, you can heardly say, no matter how much urge
may you feel to utter it, "I am in the universe".
If you deny it , no communication is realised .
Don't take your decision so readily!
Possibly you have heard about Saints. They use something else than mere
intellect.
Don't ask me why I use the "mere" before intellect, for I think you
yourself should realize that in various aspects, intellect falls
through to satisfy us.

Do you say that this world doesn't exist or it is finite ? I don't dare
to
say decidedly that it is infinite .

Is the world within your comprehension? I think you've heard about
Russell's Paradox. As while thinking, you are a part of the universe,
you can't judge the universe, can you?

In higher portions of mathematics, you shall see, gradually, that
infinite things are not at all as simple as finite ones. For example,
you might already know, that d^d = d, but it is impossible for all
nonunit finite cardinalities.

Note: the term is "cardinality", not "cardinarity".


Cardinality is not a same concept with the number . But I wonder
that we
can
think it as expantion of
the number .
I'm afraid that I am perfectry wrong .

I am afraid, again, that everyone inside intellect is wrong. However,
it is difficult to settle what should be meant by true and what, by
wrong.

Regards and Best wishes,

Saurav


Regards OT

Concentrate on what I told you previously: I am afraid, again, that
everyone inside intellect is wrong. However, it is difficult to settle
what should be meant by true and what, by wrong.
Regards,
Saurav

Regards
Ozaki Toshiaki

.

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