toshiaki wrote: > "Saurav" <saurav1b@gmail.com> wrote in message > news:1168059207.362458.27380@51g2000cwl.googlegroups.com... > > > > toshiaki wrote: > > > "Bob Kolker" <nowhere@nowhere.com> wrote in message > > > news:4vvevuF1dt4kfU1@mid.individual.net... > > > > ooo wrote: > > > > > I am biginer in English and mathrmatics. > > > > > If real line is filled with points and each point is > > > > > distinguished,then each point has difference from every other > points. > > > > > Therfore real line has void. > > > > > > > > > > Thanks for advance. > > > > > > > > There are no isolate points on the real line. And the real line is > > > > dense. In addition every cauchy sequence of points on the real line > > > > converges to a point on the real line. The real line is locally > compact. > > > > > > > > So to answer your question: no holes. > > > > > > > > Bob Kolker > > > > > > > > > > > > Thanks. > > > I can't imagine the condition that there are no isolated points , and > they > > > have no contacts each other. > > > My imagination is that there exist condition whether > > > 1. all points are isolated , > > > 2. or each points are undistinguishable each other. > > > > Explain, first, what you mean by "there exist condition whether..". In > > some kind of ordering on any set, the order topology assumes condition > > 1, it's true. For the second condition, I would tell you that, even if > > there is a density in the set, the points are, as you said - > > idealistically - distinguishable. This seems to contradict in some way > > to our intuition. But remember, we do not always trust our intuition in > > mathematics; and indeed, our natural intuition sometimes turns out to > > be false, especially in the transfinite cardinality related areas. > > > 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.
> 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.
> 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.
> 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 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
> ( This assumption cause someone to imagine something completed total ). > > > uncountable infinity > > > reals > naturals assumption that new diagonal number is different > from > > > all listed number because we can choose different numbers at every > digits . > > > P(S) > S assumption that there exist one to one correspondence among > sets > > > of infinte obects . > > > These assumptions couldn't be refuted logically . I restrict my argument > on > > > reals. > > > At present objects that may be useful for our mathematical operation are > > > computable numbers. But ather more proper may be found. > > > And the rest are undistinguishable , but exist as closure.They cannot be > > > picked up in explicit form. > > > I hopefully think that this picture can avoid paradoxes which are based > on > > > the assumption that each points are separable . > > > And we can use only countable axiom of choice. > > > > 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.
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.
