taeto

Three questions: 1. Who is wrong? 2. What are they wrong about? 3. What would be right?

If your math book suggests to compute the derivative of \(y\) as the limit of the difference quotient of \(f\), without saying first that \(y=f(x)\), then it makes sense for you to throw it in the recycling bin. Maybe it can. But this is analysis, a particular branch of math that deals with functions of real and complex numbers. If you ask whether it makes sense to have a real number \(d\) with the property \(0 < d \leq x\) for all real numbers \(x\), then the answer is that such a \(d\) cannot exist, because, e.g., \(d/2\) has \(0 < d/2 < d\) in \(\mathbb{R}\) contradicts that \(d\) has the required property. If this is not the property that you would want an "infinitely small" number to have, then what is the property that you are thinking of? Will you evade to answer that question? Just writing up some string of symbols like \(1/\infty\) doesn't always point to something that makes any sense. I ask again, what does it mean to you? Apparently you can make sense of it, since you keep going on about it. Why don't you answer the question: "what does it mean?"? Now it looks like you use "finite" to mean "non-zero". In the context of analysis, distance is given by Euclidean metric, and then the only distance that is not "finite" is identically 0. You end up with \(f(x+dx)-f(x) = 0\) always, independently of \(f\) and \(x\). The most positive I can say would be something like just forget about the actual meaning of \(dx\). In ordinary usage, the functions \(x\) and \(dx\) belong to different species, they do not allow to be composed together by the binary operation of addition. It is something like trying to add a scalar to a 2-dimensional vector. The most you could do is to use your expression as a notational shorthand, which substitutes \(dx\) for \(h\) and removes the needs to write the \(\lim\) symbol. Maybe you can think of any advantages in doing so. Something like how the Leibniz notation allows to write the chain rule in the form \(\frac{dz}{dx} = \frac{dz}{dy}\frac{dy}{dx}\).

You were hiding the information about \(y(x)\) being the same as \(f(x)\)? I do not know the answer. What do you mean by \(\infty\)? And by \(1/\infty\)? Why do you think that \(1/\infty\) is infinitely small? What does it mean to you that something is infinitely small? You are saying things that have no place in usual mathematics. And do you know how to answer a question without introducing several more unknown quantities? What do you mean when you speak of "infinity"? What is that exactly to you? I have absolutely no idea what you are going on about. Why would we have to deal with objects that have contradictory properties, it would seem an extraordinary stupid thing to do, no? Again now, look: the adjective "finite" in mathematics applies to sets. A set X is finite if there is a natural number n such that the "size" of X is n, which means that there exists a bijection from {1,2....,n} to X, so that you can count the elements of X from 1 to n. So correspondingly, the adjective "infinite" applies to a set for which there is no such n. Which kind of set is your dx for which the adjective "infinite" applies to it, and how can you prove this? And if so, how does this fact apply to the rest of the things that you are saying?

There is a "y" on the left hand side, but an "f" on the right hand side. The equation relates the derivative of a function y of x to the derivative of a function f of x. That is not what the definition of a derivative is supposed to look like. What kind of thing is dx supposed to be? An integer, a rational number, a quaternion, what? What do you mean when you say "infinitely small"? Can you give any examples of mathematical objects that are "infinitely small"?

You mean \( \frac{df}{dx}(x)\). Maybe it is possible. But only after you explain the meaning of the expression on the right hand side. E.g. what exactly do you mean by saying that dx is "infinitely small, infinitesimal"? If there are several different dx that are "infinitely small, infinitesimal", then it would only make sense if you can prove that the value of the expression on the right hand side does not depend on which one of the possible values of dx that you apply. Obviously you have to explain what it means to do addition and division with objects that are "infinitesimal". You will have a lot of work to do before you can convince anyone that it makes sense. If you are willing to put in the work, then good luck.

I decide to take this seriously, because I just see it over and over, with lots of misunderstanding mixed in. Take another example than the concept of "real numbers". Let us say we talk instead about "cars". We look up the wikipedia definition "A car (or automobile) is a wheeled motor vehicle used for transportation. It runs primarily on roads, seats one to eight people, has four tires, and mainly transports people rather than goods." You may agree that by this description you now have a firm basis for recognizing when some object in your vicinity is a car or not. Now someone says that Porsche has provided a construction of a car, so thereby and henceforth, the definition of a car has to be something that is made by the Porsche factories. The big news, apparently, on this forum is that: no, that is not how definitions work. In the case of real numbers, or in indeed all mathematical objects, the first thing that comes about is a description of what are the supposed properties of such a thing. In the case of real numbers it happens to be not the number of wheels etc., but things like properties of addition and multiplication, ordering properties, and more. When you have those properties sorted out, you are ready to distinguish between objects that are real numbers and those that are not. If somebody says, wait, Cauchy has a construction of real numbers, so we should take the definition of real numbers to be equivalence classes of Cauchy sequences, then that is BS, it is still not how it works. Not with cars, not here either.

Do not worry. Nobody in this forum will troll you, the moderators are too strict. You seem confident when you use the word "infinity" that it means something specific to you. To me it means nothing in particular. For someone who works with elliptic curves, infinity is just an extra point that gets added to get the full group structure. It could be that infinity is something like the set with one element \(\{0\}\). Of course it is the name of the symbol \(\infty\) that gets used all the time in indefinite summations and integrals, and in limit calculations. It is just a letter in that context. So if you say "let infinity be a real number", then I am cool with that, except I would find "let x be a real number" easier to live with. But in contrast you mean something precise and absolute when you say "infinity"?

The wikipedia page says that \((\mathbb{R},+,\cdot)\) is a field, in particular \((\mathbb{R},+)\) is a group, which implies the three properties that I mentioned, if you add a simple notational convention. I do not know if you will consider the defining properties of a group as only "derived". No, I have to admit that I never came across such a definition. What is the name of your book? Maybe I will recollect. You mean the page on "Real Number"? I would like to know where on that page you can find this? I see mention of "Dedekind cuts", but I would be concerned to define real numbers in that way, it is only one of many possible constructions of reals, and the other ones are just as good. It is not a definition anyway, only a construction.

I do not recognize your definition of a real number. But it is not serious, because after all the precise definition does not matter, so long as we agree about the properties of real numbers. So at which step do I go wrong in the following: 1. if x is a real number, then -x is a real number, 2. the difference y-x is the same as y+(-x), 3. if x and y are real numbers, then x+y is a real number? Because if these are all correct, then the difference y-x between real numbers x and y will always be a real number.

In one of the first posts I already asked him to define a neighbourhood of infinity, which he did by the example of the complex plane, where it is given as the unbounded region outside a circle of radius R. The real line is special, since there are two regions defined by a circle \( \{-R,R\}\), a negative and a positive neighbourhood of infinity. In higher dimension the neighbourhood is always a connected set. But I suspect that he is thinking of a neighbourhood of infinity as something different yet.

I just pointed out that \(\widehat{\infty}\) is the difference between \(b\) and the real part of \(z_0\), and deduce that \(\widehat{\infty}\) is also real. Are you sure about your definition of a real number, because it sounds circular?

Isn't \( b - \widehat{\infty}\) the real part of your \(z_0\)? If \(b\) is also real then \(\widehat{\infty}\) is the difference of two real numbers, hence real. It is what you call "infinity"? But for any real number there are larger real numbers. And anyway, the question is why the real part \( b - \widehat{\infty}\) of your \(z_0\) is not \(1/2\). Is it because it is not real?

I believe the point is that it is already known that every nontrivial zero has real part strictly between 0 and 1 (the 'critical strip'). So where you say that there are nontrivial zeros with real part equal to \(-\widehat{\infty}+b\), we all see that as a consequence, \(b\) must lie strictly between \(\widehat{\infty}\) and \(\widehat{\infty}+1\). How do we make sure that \(b\) is not exactly equal to \(\widehat{\infty}+1/2\), in which case you would just have a point on the critical line, and not a counterexample?

I would argue that you cannot refer to "the neighbourhood" of infinity, when your definition of such a neighbourhood clearly depends on a positive real number R, unless you fixate the value of R. Grammatically correct would be "a neighbourhood" otherwise. But since every nonzero complex number belongs to some such neighbourhood, it would seem to add no further information.

Thank you Strange for the references! So we know only a little bit about \(\widehat{\infty}\) from that paper, in which it is a central quantity. Then for the OP: How can we learn enough about it to be able to read past the third line of the paper, where it first appears? Could you present an explicit example in which \(\infty\) and \(\widehat{\infty}\) both appear in the roles that they are supposed to occupy in this paper? The value of R goes into the definition of "neighborhood" in the beginning. Are you saying that this value does not exist, hence the definition is meaningless? If you think that the paper begins with meaningless definitions, then why would you present it here in the forum?

So in the article, the real numbers are divided into those x with |x| at most R and those with |x| > R. Does the numerical value of the radius R matter at all? It is not specified in the OP article? What is the meaning of \( \widehat{\infty} \)?

How is "neighborhood of infinity" defined?

Can you make this precise? Would you say that if we 'expand' each point of the uncountably infinite set R into a line of uncountably infinite many points, thereby naturally expanding R into R^2, then we have done anything impossible? Because R has measure zero in R^2? The cardinalities are still preserved.

If you want to throw in a notion which serves to invalidate your entire explanation, then this is a pretty good choice by the author of that piece. Surely you cannot expand anything other than an uncountably infinite collection of points into a universe of infinite volume (though the same would be true for any positive volume). Other than the trivial copout to expand by adding a sufficient amount of enough new points to do the trick, of course.

Excuse me. Please explain how the Axiom of Infinity has any bearing on the definition of a limit in Calculus?? And since you will not be able to do that, please explain why you thought that was the case.

I have not checked that particular wikipedia page, which I admit could possibly be infested with crackpot contributions. But finitism is basically a respectable theory, just to check how much one can get away with without the possibility of something like "infinite sets". Obviously it is not a lot, as it bans you from doing arithmetic, calculus, complexity etc. But in some contexts of fundamental maths it is a worthwhile study. Of course the cranks, like the OP, hate this theory. They want to argue that all of ZFC is "wrong", without admitting to any finitistic presumptions.

Then why do you want to discuss about it all the time? Noone else has an idea what it is about, and neither do you. Do you happen to know about the theory of "True Arithmetic"? It has only natural numbers 1,2,3... in it, and all kinds of true theorems on addition and multiplication of those numbers. It is also known as the standard model of the natural numbers. There are no infinite objects in this theory. It is a part of mathematics, but would you approve of it nonetheless?

It is Saturday evening here; everyone has a lot of time on their hands, sorry DannyTR already argued why there cannot be any number larger than all other numbers, which, assuming the total ordering of numbers, is the same as saying that there has to be infinitely many numbers. By inference he is not disputing that fact. A, ehm, mathematically inclined person might speak of the "cardinality" of the set of integers, the set of points on a line. Rarely of the "number". But you hint at the possibility that the OP is thinking about this "Actual Infinity" as a number? Q.E.D. as witnessed by this thread: OP's confusion and everyone else's time wasted.

That is pretty hilarious. I wasn't sure before that you are just trolling. Just for fun though this question: if the objects of set theory should not be called "sets", then what is your alternative suggestion?

The inductive set is familiar. How do we get from there and to a definition of "Actual Infinity"?