Sir, I cannot simply borrow the following "If there is a smallest positive number then half it is even smaller and still positive, so by contradiction there is no smallest positive number ". If the numbers less than every positive real are non reals, then from the above statement, the half of the smallest positive real number is non real or infinite simal.A mathematics student cannot accept a statement by just borrowing from others without valid proof.I showed method of contradiction fails in this proof.Do you have any other method with valid logical arguments?

Sir,Questions1. Can you give me the logical arguments used to prove -"smallest positive Real number doesn't exist".2. Can you give me real number system as an interval.R = ?If you answer above questions in your own reasoning that you have studied, I will explain my argument on your mistakes ."The details are all included, but as usual they are tedious and not too instructive."[14].

Questions

1. Can you give me the logical arguments used to prove -"smallest positive Real number doesn't exist".

2. Can you give me real number system as an interval.R = ?

If you answer above questions in your own reasoning that you have studied, I will explain my argument on your mistakes .

Sir, Mr.Peter escaped without giving clear answer.In beginning of your explanation you wrote -every set has its own smallest element particular, there will be a smallest positive number you may call ε. But the statements in texts is, smallest positive real number doesn't exist.

Sir, for any non zero y, x/y exists.suppose y is an infinite simal other than 0, then what is x/y. ?

To Mr. Peter : Sir, what is meant by hyper reals.? To Mr.Wulf: Sir, Are you trying to prove that the smallest positive real number exists? Or trying a construction that deprives more time?

Mr. Wulf, my attempt was to find methods to modify proofs if there is mistake in already existing proofs.I am not interested to waste my time by doing fun in calculation that have no use to society. Mr.Peter ,I expected you may answer to R=?, as. R= (-infinite, infinite).Since you said it is grammatical error, I measured you, that you are not a mathematician as you claim.From your own answer without copying available explanation,,It is an axiom of the reals that x/y is a real number when y is nonzero. Whether y is "infinitesimal", "green", "charming" or not makes no difference. So the answer to your question is "a real number"., I concluded that what you are saying is wrong. But, when you give the available explanation s from other websites, it is 90% true.Thanks

Thanks for asking this question. Most of the mathematicians have doubts on it, but will not reveal that they don’t know its use.

Sir, Please do not under estimate .I will say to students you throw a ball up and look at distance traveled by it.Then think about the mathematical abstract of the distance traveled. Then I draw each bounce and measure its distance, That will represent a sequence converging to zero, The total distance can be represented as sum of series. (This is from a text book that teachers follow)

I feel now you can tell something on infinite sequence and series.

Philosophical view in Soul Analysis:

If you are provoked by a person at a particular time, then you may feel intense / extreme hatred to him. But as time goes, the hatred may reduce and gradually you may forgive and you may never feel hatred to him.

If I analyze the quality hatred based on equation of judgment and I simulate the points representing judgment as a graph, I can see your soul object hatred goes and turns to nothing, then there is a chance to create love ( Referece: Some screen plays of Indian Romantic Movies, Example: Thullade manavum Thullum, Vallavan.)

That graph represents a sequence converges /or not converges as per your feel. In films usually it will converge to other soul object love).

(In class room, I will give an advice to girl students in presence of boys, to be aware on tricks of boys after seeing the films)

In your terms I am enjoying fun with Soul Analysis. I have some satisfaction with that than playing with numbers as school students without knowing "for what" ?

Please do not feel disappointed. I am happy to read and answer.

Dear Peter, Now you are trying to answer my doubt. The standard axiomatization of the reals does not permit either infinitesimals or infinite numbers. Again I am asking you, if a is assumed as smallest positive, then a/2 is an infinitesimal. It will not follow standard axiomatization of the reals. Now come to my argument. What you trust or accepted- if there is a smallest positive number then half it is even smaller and still positive, so by contradiction there is no smallest positive number - wrong. To apply contradiction a/2 must be real. Now you look at my question.

Dear Peter Breuer and Wulf Rehder,

English People Will Never Have a Beautiful London Until They Refuse To Live In Ugly Houses.

Nishad T M

Hai, Peter and Wulf, Is the neighborhood of zero, (-a, a), for every a>0 included in Real number system?

Hai, Wulf

Can you give me the definition of infinitesimal?

Although you don’t seem to be a trained mathematician (there is no shame in that!), I want to reciprocate with three short videos, see below. But beware, the presenters are professional mathematicians. They make light of the somewhat unusual topic they are talking about, but what they are sketching has a grounding in mathematics proper.

Sir, Most of the trained mathematician are well known problem solvers. But we have a lot of problems in life and in society that need solutions .

Why the trained mathematicians fail to solve those types of problems?

Reason, Only mathematical view cant solve those problems.That is why I combined philosophical views and mathematical views in soul Analysis to solve such types of problems.

You try to learn all the articles from me in Soul Analysis.

Dear Peter : If "a" is a real positive number, then yes, (-a,a) is an open interval about zero, one of the base of neighborhoods of zero in the standard topology on R.

You are trying hard to give me details .Thanks.

In a number system that contains infinitesimals, a (positive) infinitesimal number is any positive number that is smaller than 1/n for each and every positive integer n (those are 1,2,3,4,....).

I agree with that

Now you look at my question that I asked 20 hrs ago,

Hai, Peter and Wulf, Is the neighborhood of zero, (-a, a), for every a>0 included in Real number system?

Your Answers

Wulf: If a is a real number and (-a,a) denotes the open interval defined by a, then, yes, (-a,a) is a “neighborhood of zero” and as such a subset of the real numbers, in other words, every number in (-a,a) is a real number – regardless how small a>0 is.

Peter: If "a" is a real positive number, then yes, (-a,a) is an open interval about zero, one of the base of neighbourhoods of zero in the standard topology on R.

2 days ago.

Peter: Dear Nishad, "infinite" is not a real number, as I pointed out. The standard axiomatization of the reals does not permit either infinitesimals or infinite numbers.

My argument :The statement " if there is a smallest positive number then half it is even smaller and still positive, so by contradiction there is no smallest positive number " is wrong

Reason: Half the number is smaller and positive.But there is chance to become it an infinitesimal. So it wont be a real number. To apply contradiction , we must show a positive REAL number. Or include the testing of x/2 to show it is not an infinitesimal

Post Script: If you both or other researchers accept my argument , then we shall together think on how can we modify the proof.Or I will find out a new proof. I know you are trying to protect the validity of the statement .But a goof proof is needed.

Nishad T M, Kerala, India

Suppose I am Modifying the proof as follows.

Statement: The smallest positive real number does not exist.

Proof:

Assume that x be the smallest positive real number.

Then there is an infinitesimal such that 1/n < x for some n in N

Now consider x/2.it is smaller than x and positive.

If x/2 is a real number , then by method of contradiction our assumption is wrong. Hence the statement holds.

So We have to check whether x/2 is a real or not.

It is enough to prove that x/2 is not an infinitesimal since the possibility is either x/2 a real or an infinitesimal.

To prove that x/2 is not an infinitesimal.

Since 1/n < x

1/2n < x/2

1/m < x/2, where 2n =m which is again a natural number in N Therefore x/2 is not an infinitesimal.

Hence it is a real number smaller than x.

Hence the proof.

Note : Infinitesimal number is any positive number that is smaller than 1/n for each and every positive integer n.

This proof is now acceptable to me. But I need more time to scrutinize and think in critics view if there is further modification needed.

Thanks Peter and Wulf for helping me to reach at this modified proof. In mathematics text books , this modification is to be included.

As per the view of above proof the Set of Real Numbers can be written as follows:

R = (-∞, -a ] U {0} U [a, ∞ )

where a is such that 1/n < a , for some natural number n.

Sirs, is the above True?

Dear Peter,

I know well the existing definitions from textbooks,

If I write a critics view , you must realize that I have been investing more time and trying to modify the proofs if there is any mistake in existing , to protect the positions of Maths Professors.

Now Wulf answered that

Your line about R is not true, unless a = 0. But when a = 0, your second line about 1/n

Sir, I deliberately committed that mistake as 1/n is infinitesimal. It is because, I know well, you will look at that small error to criticize my argument, and will neglect the huge mistake, that I derived from your explanation that 1/n < x.

Sir., if x is assumed as smallest, how can you write there exists some natural number n such that 1/n < x? I know well, using method of contradiction that statement cannot be proved correctly. One more question to you,

Is the set of infinitesimals other than zero an empty set?

If you are answering, Yes, then your claim --ifsomebody claimed that there are sabre-tooth tigers around (which there are not), then they would still be animals-- is right. If your answer is non empty. Then your above argument is wrong. Please consult with your friends. If you say it is empty, then you are accepting that zero is the only infinitesimal, which is also real and there are no non real infinitesimals.I don't think, other mathematicians will agree with that.

How can you write there exists some natural number n such that 1/n < x". That is the definition of x IS NOT AN INFINITESIMAL. So it's perfectly possible to write it.

Sir, it is written in a proof with an assumption that x is the smallest. If you write 1/n < x, that obviously means x is not the smallest. In layman's logic if one person say x is smallest, and if 1/n is real, then surely 1/n > x. If this is agreed by a student as in his own logic, how can he agree with its opposite only with the reason that it is said by a famous mathematician and engineer Mr.Peter ?

proof that there are NO INFINITESIMALS IN THE REALS: Sir, what about zero?.

Dear Peter and Wulf, You have now two options.

1. You shall agree with my arguments, my question and suggest your colleagues or other mathematicians to bring a modified proof.

2. You shall continue false arguments (that deprives your positions) to convince me that what I argued is wrong.

First Option is the Best one, the third option is an escapism.

Mr. Wulfs Position will never support him the third option.

Dear Wulf, We shall consider 3 devotees, one atheist and one mathematician. Devotee1 says-God is Jesus. Devotee 2 says-God is Allah, Devotee 3 says-God is Vishnu. Atheist says God doesn't exist. You are the mathematician.What will you say?

Dear Peter, If my objective is to teach sequence, then I focus on distance only. If my objective is polynomial equation, then I will use gravitational force etc, and will say about path traced by falling comet, orbit of planet, projectile, etc, similarly based on objectives,I use examples.I am guessing ,your that move is to lower my confidence to question you .

OK,

I am from India, so I studied from Sachin Tendulkar on how to react on sledging.

Again I am seeing mistakes in your above explanation. Any way try to answer my last question to Wulf

If you are a mathematician, you shall see mathematics in any phenomena. The reply shows you are not a good mathematician. I am seeing maths in screen play of films,marriage proposal, teaching methods, horoscope, etc. The example of ball is from one text book that teachers follow to teach application of sequence, if you are questioning that, you are actually questioning the author and you are arguing that -

I'm pointing out that incompetence or even deep inability at mathematics is not necessarily a barrier to teaching mathematics, as many primary school teachers of mathematics exemplify.

-is about the author of the texts. Your above argument again consist mistake, to point out, where the mistake, means, you are getting free review from me. You have to pay the fees for more reviews to point out your mistakes.

Can you answer my last question to Wulf, as a mathematician ?.

Sir, see the link

.http://www.classzone.com/eservices/home/pdf/student/LA211DBD.pdf

Do you feel that, the method used to teach in the problem is wrong?

In the problem other aspects like, speed, gravitation etc are avoided.

In my view the objective of that problem is just showing application of sum of series.

I asked the question on God and you both neglected, actual my target was to begin this explanation from your answer and show that you are just like an atheist who implements and experiences Real Part of God (in my view god has imaginary part that the devotees believe) in his life but he says God does not exist and he never needs a proof for it.

To under stand this concept I would like to suggest you to read.

https://www.ijser.org/onlineResearchPaperViewer.aspx?Origin-082017.pdf

Student can interpret from assumption,

If x is the smallest positive real number, that means (common logic), there is no real number smaller than x.

If x is not smallest then there is chance to find another real number smaller than x.

So the student will suspect on status of x/2 if x is the smallest.

i.e, What is the positive number which is smaller than the smallest positive real number?

Is it an infinitesimal?.

So in Proof, we must include a testing that x/2 is not an infinitesimal

I will be more glad if you give a proof for that

Statement: The smallest positive real number does not exist.Proof: Assume that x be the smallest positive real number. Now consider x/2.it is smaller than x and positive. If x/2 is a real number , then by method of contradiction our assumption is wrong. Hence the statement holds.

There are two cases either x/2 a real or x/2 an infinitesimal.

So We have to prove that x/2 is not an infinitesimal.

...........................................

I expect the proof of this part from you.

Through this attempt, I am trying to protect the positions of those Professors who teach this statement. Suppose after seeing our conversations if a student asks this doubt .The professor need to answer the question from point of view of students to convince him.

The Professor has to use the following steps in most of the proofs.

Assume epsilon be a small positive real number,

....................

As epsilon tends to zero (since epsilon is arbitrary).

....

Hence the proof.

If he cannot clear the basic , how can he use these in other proofs? This is the motive to ask this question.

So We have to show that x/2 is a not an infinitesimal.

..........................................

I expect the proof of this part from you.

You are saying

" Infinitesimals are reals too. Half an infinitesimal is a real. Half a non-infinitesimal is a real. It is an axiom. "

5 days back you answered

Dear Nishad, "infinite" is not a real number, as I pointed out. The standard axiomatization of the reals does not permit either infinitesimals or infinite numbers.

You are saying infinitesimals are reals but it will not satisfy standard axioms.

Please have a look on your explanations

Peter: DAY 1 The standard axiomatization of the reals does not permit either infinitesimals

Peter: DAY 2 After assuming x as the smallest, if we write 1/n < x for some natural number, It is true. Reason x is assumed as not an infinitesimal.

Peter: DAY 3 Infinitesimals are reals too.

My Conclusion

DAY 1 and DAY 3 arguments of Peter contradicts

DAY 2 (The main focus smallest can be given relaxation?)

Peter is considering smallest as a fuzzy quantifier smallest (x) can take values from 0 to 1, Smallest (x) =1 means strictly smallest, but after introducing the condition " x is not an infinitesimal" you are relaxing the condition smallest(x)

Nishad: " I am a Mathematician. - Thanks. Never expect any reply from me."

It's funny that the only real mathematics I have seen so far are from Peter and Wulf. I suggest you stop posting this kind of "questions" as you do not seem to master even the basics of undergraduate real analysis which should be a requirement to discuss these "questions" with a minimum of coherence.

Good luck

Hai Mr. Peter and Mr. Wulf.

I respect and value all of your criticisms on my arguments:

My Objective of asking this questions was

"Through this attempt, I am trying to protect the positions of those Professors who teach this statement. Suppose after seeing our conversations if a student asks this doubt .The professor need to answer the question from point of view of students to convince him.

To teach the existing proof is very easy if the students are not informed about definition of infinitesimal. Usually this trick is used in curriculum.

As in view of an ordinary student (who wish to know clear picture on real number system and has habit of searching internet ) I argued with Mr.Peter and Mr. Wulf .

Mr.Peter explains so many unwanted things to protect the in accurate proof.The explanations takes about a week to cover a simple proof .

Mr. Wulf tries a construction - that is an escapism or failure to admit the new derivation.

But both of them are trying to protect the existing in accurate proof by criticizing me. It may due to fear on my questions.

-If my argument is agreed and if no one can bring a better proof, then the Professors who teach the statement will fail to explain the proofs of several theorems-.

Mr. Peter's explanations to protect the existing proof brings more contradictions in his own arguments.

I tried one modified proof, but it had some difficulty to show x/2 is not an infinitesimal.

Last day I developed a new proof.

It is now available as an article in my RG profile.That proof is accurate than existing proof.

See the link

Article Smallest Positive Real Number Does not exist