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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).

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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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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 ...
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9answers
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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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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 ...
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1answer
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Does “cheap nonstandard analysis” take place in a topos?

Terence Tao's A cheap version of nonstandard analysis describes a way to do analysis halfway between ordinary analysis and nonstandard analysis which, if I'm not mistaken, cashes out to working in the ...
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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 ...
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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 ...
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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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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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2answers
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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 ...
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Can $\sum_{x \in [0,1]} e^x$ be represented as an integral?

In $$\sum_{x \in [0,1]} e^x,$$ $e^x$ is summed over all values in the interval $[0,1]$. Am I right to say that $$\sum_{x \in [0,1]} e^x = \int^{x=1}_{x=0} e^x \, \mathrm dx?$$
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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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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 sense are ...
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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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2answers
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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 ...
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2answers
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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 ...
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3answers
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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 ...
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Intuition for a physical real line vs. a physical “hyperreal line”

As a mathematical structure, I have no problem with the hyperreals. But I came across the following from Keisler's book "Elementary Calculus: An Infinitesimal Approach". "We have no way of knowing ...
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What exactly is nonstandard about Nonstandard Analysis?

I have only a vague understanding of nonstandard analysis from reading Reuben Hersh & Philip Davis, The Mathematical Experience. As a physics major I do have some education in standard analysis, ...
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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 ...
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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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Non-ZFC set theory and the hyperreals: problem solved?

The reals are the unique complete ordered field. The hyperreals $\mathbb{R}^*$ are not unique in ZFC, and many people seemed to think this was a serious objection to them. Abraham Robinson responded ...
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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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How can $0.999\dots$ not equal $1$?

First, by definition I assume that $0.999...$ actually is defined as: $$\text{lim}_{n\rightarrow\infty}\sum_{i=1}^n 9/10^i$$ Now by geometric series we already know that this equals one. But ...
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Can nonstandard analysis give a uniform probability distribution over the integers?

There exists no uniform probability distribution over the non-negative integers. This is because we would need to have $p(i)=q$ for all $i$, for some real number $0\le q\le 1$. But normalisation ...
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1answer
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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 $x+dx=x$ (...
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What is the Galois group of the Hyperreal Numbers?

The Galois Group of $\mathbb{R}$ as an extension of $\mathbb{Q}$ is trivial. That’s because any field automorphism of $\mathbb{R}$ is order preserving, so since $\mathbb{Q}$ is dense in $\mathbb{R}$, ...
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4answers
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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 ...
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6answers
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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 ...
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1answer
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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 ...
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4answers
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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 ...
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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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Which branch of mathematics rigorously defines infinitesimals?

I have some trouble doing standard computations in calculus because of the notion of a differential, otherwise known as an infinitesimal, being rather ill defined, in my experience. Are there any ...
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7answers
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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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3answers
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The nature of infinities

I have been thinking about the nature of infinity lately. I have no experience with higher mathematics or theorems regarding infinity, so please forgive me if my ideas on this topic are extremely ...
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2answers
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Relationship between nonstandard analysis and nonstandard models of arithmetic

So I've been reading through some introductory texts on nonstandard analysis, basically through the ultrafilter construction of hyperreals and the transfer principle. It seems that much of the time, ...
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1answer
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Juggling three non-Archimedean fields

I'm comparing the field of hyperreals, the Levi-Civita field and the Dehn's field for the first time. I'm not very familiar with their properties, so I'm looking for ways to understand and distinguish ...
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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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Is ${}^{*}\mathbb{Q}={}^{*}\mathbb{R}$?

This is a follow-up question of a previous one. Is it true ${}^{*}\mathbb{Q}={}^{*}\mathbb{R}$ ? It seems to me it's true because for any $x \in{}^{*}\mathbb{R}$, there is a $y \in {}^{*}\mathbb{Q}$ ...
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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 ...
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1answer
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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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2answers
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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. ...
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1answer
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A layman's motivation for non-standard analysis and generalised limits

Disclaimer: My apologies for making such a long question. The question is possibly also rather specific, but I hope that (some parts of) it might be useful in general. Background: I have recently ...
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What is… A Parsimonious History?

Interpreting historical mathematicians involves a recognition of the fact that most of them viewed the continuum as not being made out of points. Rather they viewed points as marking locations on a ...
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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 ...
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1answer
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Is it possible to have numbers that are to Hyperreal numbers what Hyperreals are to Reals numbers?

There are Hyperreal numbers that are smaller than any real number , also those that are larger than any real, they have properties analogous to those of Real numbers thanks to the Transfer principle ...
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1answer
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Does non-standard analysis still make use of classical topology?

I understand that non-standard analysis is generally viewed as an alternative to the $\epsilon$-$\delta$ approach of classical analysis. So for example, I'm guessing that the definition of a ...
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4answers
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Confused about differentiation

I'm new to calculus and have been taught that $\displaystyle \frac{dy}{dx}$ is the rate of change of y with respect to x. Does $\displaystyle \frac{dy}{dx}$ show how much the variable y changes as x ...
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1answer
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Two different definitions of big O notation

I find there are two different definitions of big O notations for $f(n)=O(g(n))$ as $n\rightarrow\infty$: There exist $M>0$, and $N\in\mathbb{N}$, such that $|f(n)|\leq M|g(n)|$ for $n\geq N$. ...
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1answer
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What is the significance of the Increment Theorem in non-standard analysis?

A bit of background: I'm an engineer, not a mathematician, and I need to review and improve my calculus. In college, I never liked how they said $dy/dx$ was a single symbol, not a ratio; and then ...