It is a well known fact explainable via the notion of linear scale (used already in some answer above), which - being not sufficiently precise - can be (for didactical reasons) replaced by ruler, known from elementary geometry (see e.g.

https://www.mathsisfun.com/definitions/ruler.html

for getting my understanding of the notion).

If one assumes that between every two different points on the line there is another third point, then the infinity becomes clear (without referring to much deeper axioms of the set theory). However, there are two axioms involved about the (mathemtical) line:

- any point B in the closest neighbourhood of a point say A is not equal A

- between every two different points on the line there is another third point

It is not sufficiently strict definition of the line, only an example of possible formulation of some possible postulates. For strict definition fulfilling standard requirements accepted by the mathematical community is much more complicated; getting into the details - indeed - one needs studying math for couple of years.

Another problem is, whether the mathematical line corresponds to the line or the real world; by some atomistic philosophers - it is not the same, if the atoms need to have some size greater than some minimal, be it 10^{-123} meter. And what is the unit meter? Which (non-visible) points should be taken as the base points. Again we are coming to postulates - this time about the real world.

I think that the main problem of this thread is the definition of infinity, and separation of mathematical from physical models, and many other meta-questions like:

How can be accepted someones (non-professional) ideas if they are not written in commonly accepted terms (by the professional readers)?

How to explain briefly rules of some domain to an outsider who is falsely convinced that he/she understands the domain sufficiently well?

Hi, the continuum hypothesis formulated by Cantor says that there is no set whose cardinality is strictly between that of the integers and the real numbers. Your question is not exactly related to this hypothesis. In fact, each interval [a,b] has the same dimension as R itself and contains an infinite number of points.

Examples of various types of point sets

📷

1. The closed interval [a, b] on the x-axis corresponding to a 📷 x 📷 b is a point set in one-dimensional space that consists of points a, b and all points in between.

2. The open interval (a, b) on the x-axis corresponding to a < x < b is a point set in one-dimensional space that consists of all points between a and b (but not a and b themselves).

📷

3. Figure 1 shows a set A of points in one-dimensional space — an open interval, half open / closed interval, closed interval and two isolated points.

4. The points satisfying the equation

x2 + y2 < 25

Dear Peter Brauer,

Thank you so much for kind, extremely useful and encouraging answer. I am grateful for the same.

PB: "But haven't you contradicted yourself by saying that there is another number (indeed, an infinity!) between each number and its "immediate neighbour"? Wouldn't that new number be the immediate neighbour then,not the first?"

I know .. given any number a and its immediate neighbour b, we'll designate (a+b)/2 as "your" immediate neighbour for a, while I'll stick with b. How about that?

NG: If we represent the points by real number system, then `yes', you are right. I think representing the points in 1D space by real numbers are equivalent to attaching a linear scale to the 1D space. If this linear scale is removed, and the 1D space is looked as mere collection of points; each point being equivalent to any other point, then it becomes possible to define something like immediate neighbor, but still there will be infinite number points between them and this particular property seems to be a defining property for a continuum space. I am writing my findings under the title `A new $\epsilon$ - $\eta$ representation theory for a generic number system'....Later this representation can be mapped into any number system with a well-defined, natural and mathematically consistent operation.

It may not be so nice to discuss these things with words since they are mathematical. I just started writing these results and within a month or two, I will upload them. Before doing that, I will send the first draft to you seeking your valuable comments, remarks and advice.

Thanking you so much for your positive and very encouraging response.

Dear Jack Son,

Thank u so much for the information.

When someone is going into a public space like ResearchGate, it would be nice to have his/her questions correctly and precisely formulated. As of a question of N. Gurappa, I would recommend taking a basic course in fundamentals of mathematics where one can find immediate answers to these questions.

Respected Roman Sznajder,

Thank you for your kind answer. I really appreciate for its universality b'cos it can always be given as an answer, of course with appropriate modifications, to any question in RG.

RS: "As of a question of N. Gurappa, I would recommend taking a basic course in fundamentals of mathematics where one can find immediate answers to these questions".

NG: Before saying that, why don't you first provide me an answer or related relevant references and then say what you want to say? I am asking for that only b'cos within my limited knowledge and available literature for me, I am unable to locate it.

I think my question is more than clear for any properly trained mathematician.

`Take any arbitrary point and its immediate neighbor'; here, I mean that, there is actually no `distance' between them but still there exist an infinite number of points in between them. If this didn't sound crazy for you and you already know the result from the basic course you have taken in your childhood, please provide me the answer. I want that only....then I can stop writing that part of the result b'cos I would have came to an awareness that it's already known. In this regard, I request you to read my question once again.

I really appreciate your culture for recommending me to learn math fundamentals without really knowing whether I have undergone such a basic training or not.

Thanking you and best regards.

It is a well known fact explainable via the notion of linear scale (used already in some answer above), which - being not sufficiently precise - can be (for didactical reasons) replaced by ruler, known from elementary geometry (see e.g.

https://www.mathsisfun.com/definitions/ruler.html

for getting my understanding of the notion).

If one assumes that between every two different points on the line there is another third point, then the infinity becomes clear (without referring to much deeper axioms of the set theory). However, there are two axioms involved about the (mathemtical) line:

- any point B in the closest neighbourhood of a point say A is not equal A

- between every two different points on the line there is another third point

It is not sufficiently strict definition of the line, only an example of possible formulation of some possible postulates. For strict definition fulfilling standard requirements accepted by the mathematical community is much more complicated; getting into the details - indeed - one needs studying math for couple of years.

Another problem is, whether the mathematical line corresponds to the line or the real world; by some atomistic philosophers - it is not the same, if the atoms need to have some size greater than some minimal, be it 10^{-123} meter. And what is the unit meter? Which (non-visible) points should be taken as the base points. Again we are coming to postulates - this time about the real world.

I think that the main problem of this thread is the definition of infinity, and separation of mathematical from physical models, and many other meta-questions like:

How can be accepted someones (non-professional) ideas if they are not written in commonly accepted terms (by the professional readers)?

How to explain briefly rules of some domain to an outsider who is falsely convinced that he/she understands the domain sufficiently well?

Dear Peter,

The ruler is a good picture for understanding that there are infinitely many points between two different points, since everyone can imagine that the scale can be get in 0.1mm, then refined into 0.01mm between the closests points etc. By the way I have gone a step further with respect to (wasn't it your ?) concept of the rationals as explanation of the infinity. Is the ruler with possible refinements not a much simpler proof? The algebraic, topological, and any other structures are not needed for this thread, since the question is if it is known in math that

"In a one-dimensional real space, the number of points between any arbitrary point and its immediate neighbor is indeed infinite. "

My answer shows that yes. That's all:)

Regards, Joachim

Actually, the question I asked is not at all a question. I just stated a result which is merely a corollary for some general result and wanted to know whether such a thing is already known in math literature. There were so, many questions raised regarding the precession and correctness of the question itself. I don’t think I can write a detailed question with all necessary proofs etc, b’cos then it will not be a question but writing the whole paper.

Please don’t take the following as a criticism but my opinion.

(1) There are many correct and mathematically precise questions but answers are unknown. So it doesn’t mean that when correct and precise questions were asked, then answers will be given immediately.

(2) If answer already exists, then the question need not be posed with 100% precession b’cos experts can easily understand for what the question actually stands for; they will first correct the question itself to the exact shape and then will provide a suitable answer along with appropriate references.

(3) If answer didn’t exist, then further questions will be raised and questioned, complaining the validity of the very question itself.

In this regard, the very first reflex answer given by Peter Bruser is sufficient for me. That’s what actually I was looking for and I really thank him for that

.

I also thank Joachim Domsta for his very useful answer, though I am aware of it and it’s not what actually I am trying to do.

Here, I don’t mean the immediate neighborhood as an $\epsilon$ neighborhood. When I say `distance’, it’s not really with respect to some metric b’cos a zero distance must be independent of any arbitrary metric. If some metric gives a non-zero distance, then I am not taking about such a structure. Consider some set, S, with the following properties:

S (direcproduct) S = S ; and also `S is an element of S’.

Actually I am analyzing this set S. Within my limited awareness of math literature, I don’t know whether such a thing was considered earlier or not. If so I would like have the references. If not, and the above set looks crazy, strange and weird, then I am not bothered b’cos it comes in connection with the foundational issues in quantum mechanics which I am presently investigating. And precisely here, I came across the initial question I posed in RG.

I will certainly keep the remarks raised by Peter Brauer while writing the paper and try to maintain the needed precession b’cos I am actually not aiming at a full fledged pure math paper but a mathematical physics kind of one w.r.t some quantum systems I considered.

The answer given by Roman Sznajder is not all pleasing. I don’t know what he mean by the public place. If I can’t raise some questions in RG, then I don’t know why I should be its member. I do think raising a question, however improper and imprecise it may be, is a not crime. Somebody will surely feel for it and give some answer. If somebody thinks it’s a meaningless question, then no need to answer. Just scroll down.

I thank RE, PB, JS and JD for their valuable answers and suggestions. With best regards, NG.

@ N. Gurappa

Dear sir, thank you for thanks. I am, however surprised that you are stating now a different meaning of you question. I have tried to answer the question as it was formulated, i.e. whether the result is was known. My elementary answer - I think - is fully consistent with this aim. Moreover, purpousely the question of the nearest point has been omitted as it is not possible to understand what is meant under this strange notion. As Roman suggested, anyone reading this formulation has to ask himself/herself if the author is prepared sufficiently in mathematics to follow further explanation. I am still not sure how far you are understnding your statements/claims in the way commonly accepted in mathematics. Let me be more concrete: in your last comment you are talking about 'S is an element of S' and about S \times S = S' . This is completely outside the usually admitted theories of sets (see the Bertrand Russel pradox in any encyclopedia, please). In such new circumstnced I cannot take part in this dispute since your theory is not formulated as a complete system of postulates, which could be a basis for further discussion. Contrarily, some of your statements are inconsistent like the following text:

"it’s not really with respect to some metric b’cos a zero distance must be independent of any arbitrary metric."

Nobody can read from this what is a distance, what is your metric and what is the independence of distance zero of the metric. Moreover, your goal

"I am actually not aiming at a full fledged pure math paper but a mathematical physics kind of one w.r.t some quantum systems I considered."

cannot be any excuse for resigning of the stricness. To the contrary, mathematics in physics must be well constructed since it is the basic tool for predicting properties of not observed yet phenomena. Please, do not spread any suggestion that the math for phys (as well as for any other application) can be less consistent. If you are in hurry to write some ideas, call them ideas with these and these notions undefined yet. In particular, let me suggest you to avoid asking if this is a known result. Simply, you would be the first who is introducing the completely new notions and structure(s). Rather ask if the structure is consistent. The material shown till now does not fulfil any rigorous requirements of neither mathematical, nor physical sciences.

Good luck in getting closer to the correct formulation of your ideas!

Joachim

Dear Gurappa,

The answer to your question is available in an introductory course in real analysis. It is called the density of rational and irrational numbers. In fact, using Archimedean property, one can prove that for any two distinct real numbers a and b, there is a rational number r such that a < r < b. Similarly, for any two distinct real numbers a and b, there is an irrational number q. In conclusion, for any two distinct real numbers a and b there are infinite numbers of rational and irrational numbers between a and b. Concerning the connectedness of such points, we can prove that any real interval of numbers (a,b) is connected. Now, Can we consider R (or any Euclidean space) as a continuum space? The answer is fascinating and full of conjectures and contradictions. To follow the answer, I suggest reading about Zeno's paradoxes and other related topics. See the attached file.

Best wishes

Dear Peter Brauer,

Thank you so much for your answer.

PB: "There are many ... mathematically precise questions". Those are the only kind there are. Mathematics is about precise questions. Anything else is philosophy.

NG: Certainly, I completely agree. But, please respond or correct me if I am wrong in the following: Even precise math is based on certain definitions or staring concepts/assumptions, which can’t be precisely proven but has to accepted. Once accepted, then the remaining can be treated with precession or rigor. For example: the concept of a mathematical point….How can I precisely and mathematically prove that something is a point? I have to show that that given thing does have all properties of a point, which was defined and accepted at the beginning and itself can’t be proved precisely; but, acceptable intuitively. Precession appears once one accepts the stated starting properties of the mathematical elements.

PB: That seems to define "everything". But of course it is. This is just Whitehead and Russell's construction of set theory starting from nothing (the empty set) and "forming pairs" and "enclosing in set braces" and "applying predicates".

That's me interpreting your "S

@Joachim Domsta

Dear Joachim Domsta,

Thank you so much for your answer.

JD: I am, however surprised that you are stating now a different meaning of you question.

NG: Certainly not. I just mentioned my staring consideration from which a corollary was derived and that’s the one posed as a question in RG. Most probably I will upload the paper within a couple of months. Anyhow, whatever it is, I got the answer I wanted. Thank you so much for your kind answers and response.

With best regards, NG.

@Issam Kaddoura,

Dear Issam Kaddoura,

Thank you for the kind information.

With regards, NG.

Dear @ NG,

1. You are writing (some lines repeating, already):

“Space is in general an infinite-dimensional vector space of continuous (and also discrete) dimensions. “ (I am sorry, if it sounds crazy; but, it's the starting assumption I make)

Let S be the set of elements denoting that space. Then the most crucial properties S actually satisfies is

S (dir.product) S = S; and S is an element of S itself.

Any quantum mechanical Hilbert space is a sub-set of S.

Please take into acount in your next comments/answers/questions, that

this is not creasy, this is highly inconsistent.

2. You are still not responding to the fact raised also by Peter, that (in my words):

in your last comment you are talking about 'S is an element of S' and about 'S \times S = S' . This is completely outside the usually admitted theories of sets (see the Bertrand Russel paradox in any encyclopedia, please).

Not answering to this objection means that you are running away from the key problems. Let me repeat: if your theory of sets differs from those with minimal set of axioms - then the reader have right to expect a presentation of the new system of axioms. Promissing a paper in few months means that you don't have the system. Therefore also you are not in right to supply statements like the quantum mechanical Hilbert space is a subset of S. Is in your theory the Hilbert space an element of itself, too (in the same way as S \in S)? But OUR Hilbert space does not fulfill this! Thus, we cannot discuss your problem at all!

3. Accordingly, can you consider removing your question from RG?

Joachim

@ peter Breur

Dear PB,

Can you please prove for me, with all rigor and mathematical precession, "What is a number? "

Thank you and best regards, NG.

PB: There is nothing else but points in the space of points you will consider. Simple, no? Here: let X be space of points, with a vector addition and field multiplication defined. Done!

NG: Yes! You are telling me to accept the notion of a point. I didn't say anything different in my reply. Since I am not a well trained in basic math, let me ask you something. With what kind of a points X is spanned? (or) Given X has a given unique type of points? Is there any necessity to prove this? If so, how to prove this mathematically with precession and rigor?

The vector addition and field multiplication comes later right? That also you are defining; you yourself wrote. Did I say anything different from this?

PB: According to the traditional first baby model used, 0 is the empty set {}, 1 is the set containing the empty set {{}}, 2 is the set containing 1 and 0, i.e. {{},{{}}}}, and one keeps on going from there......

NG: Thank you for your answer. I do agree. Your construction/representation for numbers based on first baby model using the empty set. As a baby, let me ask you now, prove for me mathematically rigorously and precisely, "What is an empty set?". In the proof, please don't use any notions from numbers b'cos, we have to use the empty set to construct numbers.

PB: When I asked you to prove for me what is a number, is the construction/representation a proof? If so, it's fine for me.......

Thanking you and regards, NG. .

Dear @ JD,

JD: You are writing (some lines repeating, already):

“Space is in general an infinite-dimensional vector space of continuous (and also discrete) dimensions. “ (I am sorry, if it sounds crazy; but, it's the starting assumption I make)

Let S be the set of elements denoting that space. Then the most crucial properties S actually satisfies is

S (dir.product) S = S; and S is an element of S itself.

Any quantum mechanical Hilbert space is a sub-set of S.

Please take into acount in your next comments/answers/questions, that

this is not creasy, this is highly inconsistent.

NG: Inconsistency in what sense? There is no finite set as a solution or there is not even an infinite set as a solution?

“Space is in general an infinite-dimensional vector space of continuous (and also discrete) dimensions. “

(actually here I wrote the reason. But after reading towards the end of your reply, I felt meaningless to keep it)

PB: You are still not responding to the fact raised also by Peter, that (in my words):

in your last comment you are talking about 'S is an element of S' and about 'S \times S = S' . This is completely outside the usually admitted theories of sets (see the Bertrand Russel paradox in any encyclopedia, please).

NG: That's fine, b'cos eventually I want to do something physical, not to set theory.

PB: Not answering to this objection means that you are running away from the key problems. Let me repeat: if your theory of sets differs from those with minimal set of axioms - then the reader have right to expect a presentation of the new system of axioms.

NG: That's fine right. I do research and find somethings means to presnt it only. Any suspicion?

PB: Promissing a paper in few months means that you don't have the system.

NG: Is it that way? I won't simultaneously calculate and type. I starting typing only after completing my calculations. During the calculations, I got some result and wanted to know whether it's already known or not and pose a question in RG means I don't have anything? I can't work the same way you work. I already started a project on this topic in RG. I feel more responsibility to RG than to the researchers like you.

PB: Therefore also you are not in right to supply statements like the quantum mechanical Hilbert space is a subset of S. Is in your theory the Hilbert space an element of itself, too (in the same way as S \in S)? But OUR Hilbert space does not fulfill this!

NG: Actually, I already wrote a paper (arXiv:1710.09270{physics.gen-ph], also available in RG) in which I made that statement about S, and based on it, I have shown some results in the foundations of quantum mechanics. The present paper is just its continuation. I am not responsible for your doubting mind.

PB: Thus, we cannot discuss your problem at all!

NG: Thank you. I am also not interested to discuss my problem, especially with researchers like you. Initially I was seeking some information and that was my actual initial question in RG. Later discussions came b'cos of the kind of questions raised.

PB: Accordingly, can you consider removing your question from RG?

NG: If my question is silly, why don't you ignore it? What's really happening?? What's bothering you????

Give me one rigorous and precise reason .... I will remove it.

Just b'cos my considerations did not fit into you box? Only you are given licence to think and do research or what?

What are my other contributions I have to remove from RG? Please list them out.

Thanking you,

best regards, NG.