Re: Is continuum completely filled up?


"Saurav" <saurav1b@xxxxxxxxx> wrote in message
news:1168425821.329154.54730@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

toshiaki wrote:
"Saurav" <saurav1b@xxxxxxxxx> wrote in message
news:1168186845.689228.110930@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

toshiaki wrote:
"Saurav" <saurav1b@xxxxxxxxx> wrote in message
news:1168059207.362458.27380@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx


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 .
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 .
If you show it impossible by mathematical proof ,or by ordinry 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 .
About other my argument also , I want to know why some theory of mathematics
is different from our intuition .
If mathematical statement stand only in the range of its logic system ,its
no use for me .
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 .
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 .
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 .
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
that there don't exist uncountable
number of objects (or we cannot deal with them ). So that I thougt that
countable model only is sufficient to descrive a theory .

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 .
Provabley we all are finite and mortal , so that , I know that I can not
complehend the Universe .
But I am in the Universe as well as you ,and communicating with you .
If you deny it , no communication is realised .
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

.

Relevant Pages

  • Re: Is continuum completely filled up?
    ... The collection of intervals is something else than what we in topology ... I talk about what I took doubt about mathematics, ... Infinity is shown by following way. ... I wonder what is the end of the universe, ...
    (sci.math)
  • Re: Zenkins paper on Cantor (reply of Dr. Zenkin)
    ... Every process within the universe is finite ... Mathematicians invented infinity without ... You will find Pi used in Physics. ... > Mathematics has nothing to do with theology. ...
    (sci.math)
  • Re: "Set Theory: Constructive and Intuitionistic ZF" on SEP
    ... classical and the constructive view of the universe of sets. ... extrapolated from the mathematics of finite sets, ... rejection of actual infinity ... Intuitionism stressed the dependency of mathematical objects on the ...
    (sci.logic)
  • "Set Theory: Constructive and Intuitionistic ZF" on SEP
    ... classical and the constructive view of the universe of sets. ... extrapolated from the mathematics of finite sets, ... rejection of actual infinity ... Intuitionism stressed the dependency of mathematical objects on the ...
    (sci.logic)
  • Re: Zenkins paper on Cantor (reply of Dr. Zenkin)
    ... Every process within the universe is finite ... Mathematicians invented infinity without ... You will find Pi used in Physics. ... > Mathematics has nothing to do with theology. ...
    (comp.theory)