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Questions tagged [nonstandard-analysis]

Non-standard ordered fields are fields which have infinitesimals, that is, positive numbers which are smaller than any positive *real* number. Non-standard analysis is analysis done over such fields (e.g. hyperreal fields). Please specify the exact framework for non-standard analysis you are using in your question (e.g., what definition of "hyperreal number" you are using).

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Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?

In the book Thomas's Calculus (11th edition) it is mentioned (Section 3.8 pg 225) that the derivative $\frac{\textrm{d}y}{\textrm{d}x}$ is not a ratio. Couldn't it be interpreted as a ratio, because ...
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About 0.999... = 1

I've just happened to read this question on MO (that of course has been closed) and some of the answers to a similar question on MSE. I know almost nothing of nonstandard analysis and was asking ...
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23 votes
3 answers
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What is infinity divided by infinity?

This should be a simple question but I just want to make sure. I know $\infty/\infty$ is undefined. However, if we have 2 equal infinities divided by each other, would it be 1? And if we have an ...
Xitcod13's user avatar
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11 votes
4 answers
673 views

Why adjoining non-Archimedean element doesn't work as calculus foundation?

Consider the smallest ordered field that contains R and does not satisfy the Archimedean property. I assume this is a much simpler construction than ultrafilters and other big caliber artillery used ...
Tegiri Nenashi's user avatar
164 votes
7 answers
22k views

Is non-standard analysis worth learning?

As a former physics major, I did a lot of (seemingly sloppy) calculus using the notion of infinitesimals. Recently I heard that there is a branch of math called non-standard analysis that provides ...
29 votes
3 answers
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Are infinitesimals dangerous?

Amir Alexander is a historian of mathematics. His new book is entitled "Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World". See here. Two questions: (1) In what ...
Mikhail Katz's user avatar
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45 votes
9 answers
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Is the theory of dual numbers strong enough to develop real analysis, and does it resemble Newton's historical method for doing calculus?

I've been interested in non-standard analysis recently. I was reading up on it and noticed the following interesting comment on the Wikipedia page about hyperreal numbers, right after giving an ...
Mike Battaglia's user avatar
25 votes
6 answers
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What is the meaning of infinitesimal?

I have read that an infinitesimal is very small, it is unthinkably small but I am not quite comfortable with with its applications. My first question is that is an infinitesimal a stationary value? It ...
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3 answers
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A few questions on nonstandard analysis

I know that nonstandard analysis is analysis plus the existence of infinitesimal numbers. Does it mean that nonstandard analysis is the same theory as $ZF+\exists$infinitesimal numbers? From what I ...
Student's user avatar
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35 votes
5 answers
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Is mathematical history written by the victors?

The question is the title of a 2013 publication in the Notices of the American Mathematical Society, by twelve authors (of which I am one). The contention is that traditional history of mathematics ...
Mikhail Katz's user avatar
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28 votes
1 answer
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Has anybody ever considered "full derivative"?

When differentiating we usually take a limit and drop the infinitesimal terms. But what if not to drop anything? First, we extend the real numbers with an infinitesimal element $\varepsilon$ which ...
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Dedekind completion of ordered fields

Let $\mathbb S$ be an ordered field of cardinality larger than $\mathbb R$. Let $\mathbb S^*$ be the completion of $\mathbb S$ via Dedekind cuts. Now it is well-known that $\mathbb R$ is the unique ...
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3 answers
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Uniqueness of hyperreals contructed via ultrapowers

The construction I've seen of the field of hyperreal numbers considers a non-principal ultrafilter $\mathcal{U}$ on $\mathbb{N}$, then takes the quotient of $\mathbb{R}^{\mathbb{N}}$ by equivalence ...
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Why hyperreal numbers are built so complicatedly?

I have seen approaches at building hyperreal systems by using complicated notions like ultrafilters and the like. Why not just postulate the existence of infinitesimal element $\varepsilon$ and ...
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9 votes
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Why continuum hypothesis implies the unique hyperreal system, ${}^{\ast}{\Bbb R}$?

On page 33, Robert Goldblatt, Lectures on Hyperreals(1998): Now it has been shown under certain set-theoretic assumption called continuum hypothesis the choice of $\mathcal F$ is irrelevant: All ...
Metta World Peace's user avatar
52 votes
9 answers
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Is $dx\,dy$ really a multiplication of $dx$ and $dy$?

On the answers of the question Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio? it was told that $\frac{dy}{dx}$ cannot be seen as a quotient, even though it looks like a fraction. My question is: ...
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Hyperreal measure?

If AC be accepted, then there exists a Lebesgue unmeasurable set called Vitali Set. However, I'm curious about measure valued in hyperreal numbers. Argument in disproof of unmeasurability of Vitali ...
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What's the difference between hyperreal and surreal numbers?

The Wikipedia article on surreal numbers states that hyperreal numbers are a subfield of the surreals. If I understand correctly, both fields contain: real numbers a hierarchy of infinitesimal ...
Nathan Reed's user avatar
15 votes
3 answers
3k views

Who gave you the epsilon?

Who gave you the epsilon? is the title of an article by J. Grabiner on Cauchy from the 1980s, and the implied answer is "Cauchy". On the other hand, historian I. Grattan-Guinness points out in his ...
Mikhail Katz's user avatar
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10 votes
6 answers
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What does limit actually mean?

I have been in a deep confusion for about a month over the topic of limits! According to our book, the limit at $a$ is the value being approached by a function $f(x)$ as $x$ approaches $a$. I have a ...
austin's user avatar
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9 votes
2 answers
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Construction of the Hyperreal numbers

Several times I have seen questions/answers here about using the correct definition of derivatives. There are also questions about whether or not $1/0$ is defined. Sometimes there is a discussion ...
Thomas's user avatar
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9 votes
4 answers
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What are the disadvantages of non-standard analysis?

Most students are first taught standard analysis, and the might learn about NSA later on. However, what has kept NSA from becoming the standard? At least from what I've been told, it is much more ...
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8 votes
1 answer
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Infinitesimal calculus

I have been reading some non-standard analysis from Keisler's book and I think it is logically consistent till now but there are criticisms against it and why isn't non-standard analysis accepted more ...
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I would like to know an intuitive way to understand a Cauchy sequence and the Cauchy criterion.

My understanding from the definition in my book (Rudin) is this. A seq. $\{p_n\}$ in a metric space $X$ (I only really know $\mathbb R^k$) is said to be a Cauchy sequence if for any given $\epsilon ...
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11 answers
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What's an example of an infinitesimal?

If you want to use infinitesimals to teach calculus, what kind of example of an infinitesimal can you give to the students? What I am asking for are specific techniques for explaining infinitesimals ...
Mikhail Katz's user avatar
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2 votes
1 answer
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Hyperreals, other models and 1=0.999....

Please dont jump on me before reading it all. I am aware and l agree that Within the Standard Reals 1=0.999..... Now, I know only a bit about the Hyperreals and other non-standard models of the Reals....
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1 answer
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Multiplicative Derivative

I need a hint to recover the definition of the multiplicative derivative as $f^*(x)=e^{f'(x)/f(x)}$ starting from the definition with the limit $$f^*(x)=\lim_{h\to0}\Big(\frac{f(x+h)}{f(x)}\Big)^{\...
yngabl's user avatar
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47 votes
15 answers
5k views

Is it a new type of induction? (Infinitesimal induction) Is this even true?

Suppose we want to prove Euler's Formula with induction for all positive real numbers. At first this seems baffling, but an idea struck my mind today. Prove: $$e^{ix}=\cos x+i\sin x \ \ \ \forall ...
Aditya Agarwal's user avatar
44 votes
2 answers
9k views

What is the use of hyperreal numbers?

For sometime I have been trying to come to terms with the concept of hyperreal numbers. It appears that they were invented as an alternative to the $\epsilon-\delta$ definitions to put the processes ...
Paramanand Singh's user avatar
15 votes
1 answer
816 views

Does every Cauchy net of hyperreals converge?

This came up in a discussion with Pete L. Clark on this question on complete ordered fields. I argued that every Cauchy sequence in the hyperreal field is eventually constant, hence convergent; he ...
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10 votes
7 answers
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Products of Infinitesimals

In my physics class my professor was abusing the derivative, as per so many physics classes I've been in. This time, he took the quantity $(x+dx)(y+dy)$ and argued that the $dxdy$ term should ...
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8 votes
1 answer
4k views

Cardinality of the set of hyperreal numbers

What is the cardinality of the set of hyperreal numbers?
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Why do we need ultrafilter for construction of hyperreal numbers? [duplicate]

While trying to get some hold on the hyperreal numbers I found that their construction using the already existing real number system $\mathbb{R}$ requires the use of objects called free ultrafilters ...
Paramanand Singh's user avatar
7 votes
2 answers
1k views

Are the Hyperreals complete?

Since $^*\mathbb{R}$ does not form a metric space then it can not satisfy the Cauchy conditions for completeness. However, my intuition is telling me that it would satisfy conditions of completeness ...
Dustin McPhate's user avatar
6 votes
1 answer
528 views

Hyperreals - is there a "boundary" between convergent and divergent series?

Hypearreals are equivalence classes of sequences of real numbers. Is there a hypperreal number $ h $ such that for every convergent (real) series $ \sum a_n $ we have that the equivalence class of ...
Ormi's user avatar
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6 votes
3 answers
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Why Cauchy's definition of infinitesimal is not widely used?

Cauchy defined infinitesimal as a variable or a function tending to zero, or as a null sequence. While I found the definition is not so popular and nearly discarded in math according to the following ...
iMath's user avatar
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6 votes
2 answers
220 views

Can unlimited hypernaturals be represented by increasing sequences?

Consider the usual ultra-power construction of the hyperreals $^*\mathbb R$ with ultrafilter $\mathcal F$. Let $K = [k_n] \in ^*\mathbb N$ be an unlimited hypernatural. My question is does there ...
Hyperplane's user avatar
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5 votes
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What are hyperreal numbers?

My first encounter with hyperreal numbers was two months ago. I read a lot of articles about them, but I did not understand what and which they are, because of this: On the french wikipedia page I ...
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5 votes
2 answers
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how to create the set of hyperreal numbers using ultraproduct

As title says, can anyone explain how to create the set of hyperreal numbers using ultraproduct from the set of real umbers?
ttp's user avatar
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3 votes
2 answers
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$1/\infty$ is zero or infinitesimal? [closed]

Since $\infty>0$, so $1/\infty >0$, thus I think $1/\infty$ should be infinitesimal, but the calculus book says $$\lim_{x \to \infty} \frac{1}{x}= 0$$ So is $1/\infty$ zero or infinitesimal ? ...
iMath's user avatar
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2 votes
3 answers
617 views

Concept behind the limit to infinity?

I can across transfinite numbers and came up with a thought. What if$$\lim_{x\to\infty}f(x)=f(T)$$where $T$ was a transfinite number? Generally, in calculus, I have noted that it is two different ...
Simply Beautiful Art's user avatar
22 votes
6 answers
5k views

What are hyperreal numbers? (Clarifying an already answered question)

This question already has an answer here. That answer is abstract. Could you help me with some not-so-abstract examples of what the answerer is talking about? For example, give examples of hyperreal ...
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19 votes
2 answers
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Are there non-standard counterexamples to the Fermat Last Theorem?

This is another way to ask if Wiles's proof can be converted into a "purely number-theoretic" one. If there is no proof in Peano Arithmetic then there should be non-standard integers that satisfy the ...
Conifold's user avatar
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13 votes
2 answers
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What is the topology of the hyperreal line?

Denote by $\Bbb R$ the real line and by $\Bbb R^*$ the hyperreal line. For any real numbers $x < y < z$ and infinitesimal $\epsilon$ the following holds: \begin{equation} \forall a,b,c \in \Bbb ...
Andrea's user avatar
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12 votes
4 answers
902 views

Hyperreal field extension

In non-standard analysis, assuming the continuum hypothesis, the field of hyperreals $\mathbb{R}^*$ is a field extension of $\mathbb{R}$. What can you say about this field extension? Is it ...
FPP's user avatar
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11 votes
1 answer
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l'Hopital's questionable premise?

Historians widely report that l'Hopital's 1696 book Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes contains a questionable premise expressed by an equation of type $x+dx=x$ (...
Mikhail Katz's user avatar
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10 votes
1 answer
358 views

Is there a universal property for the ultraproduct?

Given an ultrafilter U on a set I and a collection of ser X_i ($I \in I$) one defines the ultraproduct as the quotient of $\prod X_i $ by the identification $x_i=y_i :\leftrightarrow \{i:x_i=y_i\} \in ...
Dominik's user avatar
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9 votes
2 answers
627 views

Trouble understanding how the Transfer Principle is applied for the Extreme Value theorem.

I am reading Keisler's Elementary Calculus (which can be downloaded here). I am having trouble understanding his proof sketch of Extreme Value Theorem and how he is applying the Transfer Principle. ...
Dair's user avatar
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9 votes
1 answer
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Completing the space of series so there is a slowest converging series

It is well known that there is no slowest converging infinite series (see e.g. here). But there is also no largest rational number whose square <=2. Once we complete the rationals to the reals, ...
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9 votes
6 answers
3k views

Basic Geometric intuition, context is undergraduate mathematics

At some point in your life you were explained how to understand the dimensions of a line, a point, a plane, and a n-dimensional object. For me the first instance that comes to memory was in 7th ...
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