Dear Sarvesh Singh ,

Many of the engineer's face the same problem about the exact meaning of the concept (Limit of sequence, limit f(x), etc, ). This because the short methods for computations they learned in calculus, without focusing on the analysis of the method. So, you need to read more about REAL ANALYSIS, in particular, the Archimedes Property.

In fact, the two concepts in your question are the same.

Using Lim(1/n)=0 ,we obtain the infinite ∩(b - 1/n, b+1/n ] ={ b}.

To see this,

Let x ∈ ∩(b - 1/n, b+1/n ] for any n, then

b - 1/n < x ≤ b + 1/n for any n.

As n goes to infinity (1/n) goes to zero, we obtain

b - 0 < x ≤ b + 0 hence, x = b the only possible value.

we deduce that ∩(b - 1/n, b+1/n ] ={ b}.

You may ask, are there another point a which is very close to b

and belongs to the intersection?

Assume that b - (1/n) < a ≤ b + (1/n) for any n,

- (1/n) < a - b ≤ (1/n) ( by subtracting b )

as n goes to infinity, we obtain 0 < a - b ≤ 0,

provides a = b.

Therefore, ∩(b - 1/n, b+1/n ] = { b}.

Best regards

Let us start with your words

----------------

{b} = infinite intersection over 'n' of (b-1/n , b+1/n]

but according to my understanding(as above), it still should represent a range of real number and not single point.

---------------

... and put b=0. Can you find any number different from 0 that belongs to infinite intersection over 'n' of (-1/n , 1/n] ?

Vladimír Janiš :

I think i can find some numbers.

e.g. -1/n2 , 1/n2, -1/n3 , 1/n3 ... etc. ?

1/n^2 is a different number for different n. Please include ONE such number, something like 0.00000000003.

Dear Sarvesh,

Your first error is in the statement:

"lim (n-->infinity) (1/n) = 0+", where you claim that "0+" is "not exact 0".

However, according to the definition of limes ("lim" is for "limes" (latin word which means "limit")) lim (n-->infinity) (1/n) = 0 (this is "exact 0").

Dear Sarvesh Singh ,

Many of the engineer's face the same problem about the exact meaning of the concept (Limit of sequence, limit f(x), etc, ). This because the short methods for computations they learned in calculus, without focusing on the analysis of the method. So, you need to read more about REAL ANALYSIS, in particular, the Archimedes Property.

In fact, the two concepts in your question are the same.

Using Lim(1/n)=0 ,we obtain the infinite ∩(b - 1/n, b+1/n ] ={ b}.

To see this,

Let x ∈ ∩(b - 1/n, b+1/n ] for any n, then

b - 1/n < x ≤ b + 1/n for any n.

As n goes to infinity (1/n) goes to zero, we obtain

b - 0 < x ≤ b + 0 hence, x = b the only possible value.

we deduce that ∩(b - 1/n, b+1/n ] ={ b}.

You may ask, are there another point a which is very close to b

and belongs to the intersection?

Assume that b - (1/n) < a ≤ b + (1/n) for any n,

- (1/n) < a - b ≤ (1/n) ( by subtracting b )

as n goes to infinity, we obtain 0 < a - b ≤ 0,

provides a = b.

Therefore, ∩(b - 1/n, b+1/n ] = { b}.

Best regards

Issam Kaddoura I see you answer often this kind of question and I just want to tell you that you are very kind to take the time to do it. Have a nice day!

Bruno

I agree with Prof. Issam Kaddoura .

Hi Vadim Khodorovskiy / Issam Kaddoura ,

If lim(n-> infinity) 1/n = exact 0, then i am convinced. Here is an example which makes me wonder.

E.g.

lim(n->infinity) (1/n + 1/n + 1/n + .........n times) =

These two ways give different results.

Thanks

For any fixed k you have lim (1/n+1/n+...+1/n) (k times) = lim k/n = 0 = k lim 1/n = k times 0. No problem. You can replace the limit of a sum with a sum of limits only for a finite number of summands.

Sarvesh Singh Don't take what is following the wrong way. The example you gave:

lim(n->infinity) (1/n + 1/n + 1/n + .........n times) =

just shows that you don't know anything about real analysis, except the notation used in calculus, and that if these questions interest you or are required for your work/studies, you absolutely need to learn about basic real analysis.

There is an almost uncountable number of books about the subject but you could look at "Understanding Analysis" by Stephen Abbott.

https://www.amazon.ca/Understanding-Analysis-Stephen-Abbott/dp/1493927116/ref=sr_1_1?qid=1553881111&refinements=p_27%3AStephen+Abbott&rnid=927726&s=books&sr=1-1

Welcome to the joy of real analysis (that's where I started really liking maths)!

Thanks Vladimír Janiš .

Thanks Bruno Martin for the pointer.

I guess i really need to look deeper for better understanding and asking better question.

I guess that inside the real numbers 1/n tends to 0 for n -> infinity. But this is not true inside the surreal numbers (see wikipedia), where the limit becomes 1/omega, an infinitesimal number.

Dear Thomas Krantz,

I have to inform you (and all other gentlemen) that number "omega" does not exist and there is also no infinitesimal number. The number "omega" is a fiction by Georg Cantor.

Vadim Khodorovskiy , alright, but do for you numbers like square root of 2, like square root of -1, like pi and the Euler number, and so on "exist"?

omega is defined to be the smallest equivalence class of infinite well ordered sets. Provided one goes beyond finitary mathematics, surreal numbers are no less real than "real" numbers. (But I admit that one should start with studying real numbers...)

Dear Thomas Krantz,

I have appreciated your sense of humour but our dialog regards so great mistakes in mathematics that I advise you rather think deeper about the things you describe than write jokes on unusual opinion about them.

I see that the “definition” you wrote above - “the smallest equivalence class of infinite well ordered sets” – does not define any number (and something else).

There are infinite sets (and any one of them is equivalent to any other one), but there is no infinite numbers.

But it is too serious problem to discuss it in details here.

Dear Vadim Khodorovskiy,

I am not writing any jokes. I am trying to understand which principles you allow and which not. I am very used to "standard mathematics".

Do you agree with the following argument?

For any set E, there is no bijection f from E to the set P(E) of subsets of E.

For, otherwise, one could consider the subset A of P(E) of the elements x not belonging to f(x). Suppose a in E such that A=f(a). If a belongs to A then a does not belong to f(a)=A. If a does not belong to A=f(a) then a belongs to A.

I begin to understand your point: You don't admit the excluded middle-principle.

But then, what is your answer about the following classical question?

Are there irrational numbers a and b such that a to the power b is rational?

Classical mathematics solves this problem as follows:

Let r be the square root of 2. If s := (r to the power r) is not rational then the pair (s,r) gives an answer as s to the power r is 2.

If one does not admit excluded middle-principle, there could two models of the reals: one for which r to the power r is rational and one for which it is not.

Can you explain your point of view, the principles you admit?

Dear Thomas Krantz,

Well written exposition. I think you are pointing towards a semi rigorous derivation of the fundamentals of real analysis and the philosophical foundations of Mathematics.

Dear Ganitarthi Sp,

I am not familiar with the domain of intuitionistic (constructive) real analysis, though I know some people have done investigations in this domain (Troelstra, Schwichtenberg,...) Very interesting: From a proof of existence of a number one can deduce an algorithm calculating the number.This is also true to some degree in "classical" number theory, say for existential statements for instance.

But for general statements it is not true by an argument of Yuri Matiyasevich.

For instance for a general predicate P the proposition "there exists x, forall y, P(x) => P(y)" cannot be proved without using the law of the excluded middle, otherwise one could construct this particular x. Classically it is not difficult to deduce by an easy general argument.

My point is the following: If one proves some particular statement one should clearly say which axioms one has used to prove the result like law of the excluded middle, axiom of choice, etc.

Sorry for that excursion into constructive real analysis, I am not sure everyone shares the same axioms.

Dear Thomas Krantz,

Of course, I know such type of consideration which you stated above. And I agree that "there is no bijection f from E to" the power-set P(E). But there are two different reasons why it is so for finite set and for infinite set. Namely:

1) For finite set E the statement is obvious since both sets (E and P(E)) are finite and P(E) has more members than E. So we don't need the consideration you gave in your post.

2) For infinite set E the statement is right since in this case the power-set P(E) does not exist. So the consideration is not available. I know that it sounds very unusual but one need to think (not just read) about it in order to understand it.

Thus, the consideration is useless. This is the main error in your post. The second error is your "understanding" that I "don't admit the excluded middle-principle". I do admit this principle, although I know that it may not be applied to the propositions of special type (your examples are not of such type).

Dear Vadim Khodorovskiy ,

I admit that I am not familiar with the sort of theory you describe. It is not a set theory because in set theory to every object E one can associate the object of its parts P(E). I will not figure out the axioms underlying your considerations unless you give them.

Dear Thomas Krantz,

I try to tell you that the widely accepted set theory (more exact: theories, beginning from the Cantor’s naïve one to the famous axiomatic theories) contains a number of fundamental errors.

One of them (and main) is the statement that for an infinite set E there exists the power-set P(E). Many other errors stem from this main one: e.g., the existence of uncountable sets.

Some errors stem from the acceptance the self-contradictory idea of actual infinity: e.g., the existence of transfinite numbers (like omega).

But, as I write before, it is too serious problem to discuss it in details here.

Dear Vadim Khodorovskiy,

Can you write a formal proof of the inconsistency of classical set theory? You seem to state there would be an inconsistency. Or do you state simply that you consider another system better adapted to truth?

The notion of set theory has been chosen to allow most rich mathematical developments. I am not sure for instance that a valuable analysis could be developed if one does only admit finite sets and an infinite set without proper parts.

I would like to add a remark concerning the archimedian property.

Clearly the surreal numbers do not have it. One could then say: as far as we are concerned with real space or with "reality" it is a necessary requirement. Could someone comment on this?

Dear Thomas Krantz,

You are right: I state that there is an inconsistency of set theory. And the main reason of it is the acceptance the self-contradictory idea of actual infinity. Shortly it is described in Conference Paper On the absurdity of the idea of so-called actual infinity (К...

Now I cannot give you a text in more details (and in English). Maybe later.

I think it's totally useless to discuss the existence of actual infinity. It works properly (e.g. sum of series, integrals)...

Dear Gabriel Thomas,

You are "totally" wrong since in your examples potential infinity "works properly" (sum of series and integrals are limits, i.e. they are the results of the use of potential infinity).

Actual infinity cannot "work properly" because it does not exist. And if you think about it, you will have understood that you cannot define what is actual infinity.

Dear Vadim Khodorovskiy,

Nobody can ever contest finitary mathematics. But if you admit a set isomorphic to a proper part, you open Cantors paradise from which Hilbert said that no one could ever chase us...

By the way Aristotle, if I remember well, accepted also only potential infinity. Zenons paradox is solved then because Achilles can never join the tortoise.

I have a general question which could depend on the existence of actual infinity:

the status of non computable real numbers. Do they exist?

Another fact is the following:

The proof of Goodstein's theorem depends on the existence of actual infinite:

https://en.wikipedia.org/wiki/Goodstein%27s_theorem

The point is, if you don't accept actual infinite, you cannot disproof the existence of an infinite Goodstein sequence. (Peano's axioms for the natural numbers are not enough.)

Dear Thomas Krantz,

What did you write about? Why did you mentioned “finitary mathematics”?

You are wrong again: I admit a set isomorphic to a proper part (this is correct for any infinite set), but I see that “Cantor’s paradise” is hell in fact, because it is founded on the gross error of actual infinity.

What did you mean by mention of Zenon’s paradox?

Regarding non computable real numbers: as far as I know this concept, these numbers do not exist.

Dear Vadim Khodorovskiy,

I admit that the axiom of the existence of the"set of parts" is very familiar to me. I just found out that there are set theories in which it does not hold: For instance the set theory of Kripke and Platek.

All right, if I understood well, you admit the existence of countable sets, but all together they do not form a set.

Let me ask you another question: Can you present the axioms of your system? May be you are speaking of truth concerning the integers? This "theory" cannot be made explicit, though it is clear what is meant. But it is not decidable either and Gödel's argument does not apply.

PS: I still don't see your point about actual infinity. You pretend it is wrong, but can you prove your conviction? What does it mean to you that a set exists or does not exist?

Definitely, mathematics cannot progress with discussions on closed topics, like the existence of actual infinity.

Dear Thomas Krantz,

I have not comprehended: in your phrase "but all together they do not form a set" what does "they" mean?

Regarding the axioms system I need to state that it is a great error to think that the formalism is the way to avoid the mistakes. On the contrary, mistakes are hidden in axioms. For ex., in the Zermelo-Fraenkel system the power-set axiom is unacceptable for infinite sets (this axiom (like almost all other ones) is abstracted from our perception of the finite sets). Nevertheless, the ZF system may be appropriate if some of axioms will be restricted. All restrictions needed are connected with the principal difference between finite and infinite objects.

About my "conviction" on actual infinity. I have not prove it since this is not just "conviction", but comprehending. The idea of so-called actual infinity is in contradiction with potential infinity which is accepted by all mathematicians. Therefore, the adherents of the idea of actual infinity have to prove its existence. But it is impossible.

Any set is defined by its members. Therefore, a set exists if his definition defines - all members (for finite sets) or the method to generate every member of the set (for infinite sets).

The definition of the power-set (the set of "all subsets") of infinite set does not define such method. Thus, this "set" does not exist.

Dear Gabriel Thomas,

It stems from your previous post that you do not understand what is "actual infinity" (I note again: limit is based on potential infinity). And for you "the existence of actual infinity" is "closed topic"? It is very funny.

Dear Vadim Khodorovskiy,

According to what you said before the notion of "set of proper parts of an infinite set" is meaningless for you. This is not in contradiction with the comprehension axiom. In classical set theory as well you could not speak of "set of sets with a certain property" only of "the subset of elements of a given set with a certain property". The powerset axiom is not deduced from the comprehension axiom.

Hello,

the actual infinity is a concept. Manipulating a Sigma whose higher index equals "+\infty" means useing an infinite sum, so an actual infinity.

But for me discussing these kinds of points is useless, I assume.

Funny for you if it makes you satisfied.

Dear Gabriel Thomas,

You need to learn what the sum of series is.