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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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Non-standard analysis - infinitesimals and archimedean property

I got a question about infinitesimals in non-standard analysis. If I understand correctly, they are defined to be the number that is closest to zero. However, at the same time, they satisfy all the ...
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Is there in anyway possible to prove that 0.999 recurring does not equal to 1 [duplicate]

I know that the reason why 0.999 recurring equals to one because it's goes on forever, and the difference between 0.999 recurring and 1 is 0 since it's infinite. But is it possibly to prove otherwise? ...
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What happens if to introduce infinite and infinitesimel quantities this way?

What if to introduce $\varepsilon$ and $\omega=1/\varepsilon$ from the following equation as a definition? $$\left(1+\varepsilon\right)^{1/\varepsilon}=e$$ or such $\varepsilon$ that $e^\varepsilon=...
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Hyperreals — convergent sum sets

Consider the set $X=\{1,\frac12,\frac13,\dots\}$, and let $X^*$ be its hyperreal extension, so that $X^*=\{\frac1n:n\in\mathbb N^*\}$. Call a subset $A\subset X$ convergent if the sum of elements in $...
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Limits and nonstandard analysis - is my intuition correct?

Having nonstandard analysis under our belts, would it be wrong to say that $$\lim_{x\rightarrow a^{\pm ^{}}}f(x)$$ is the same thing as $$f(x\pm ^{}{\mathrm{d} x})$$ where ${\mathrm{d} x}$ is ...
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Can this sequence be a hyperreal number? What would be its real part?

Consider the sequence $\{a_n\} = \{\sin(n) \mid n\in \mathbb N \}$. Can this sequence be viewed as a hyperreal number? What could be its real part? Any intuition would be highly appreciated :) ...
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Natural extension of a discontinuous function

Let $u : \mathbb{R} \to \mathbb{R}$ be the right continuous version of the Heaviside step function. What does the natural extension $u^*$ of $u$ to the set $\mathbb{R}^*$ of the hyperreals look like? ...
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Intuition behind constrution of the Hyperreals

Just want to attempt to check if my understanding/intuition for the construction of the Hyperreal numbers via an ultraproduct is correct. Appreciate any corrections or help. So Hyperreals are ...
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Basic question about nonstandard derivative

I'm trying to understand how the nonstandard derivative works. For instance, consider the function $f(x) = \frac{1}{2} x^2$ The derivative is $f'(x) = st \left( \frac{\frac{1}{2}(x + \epsilon)^2 - ...
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Set-theoretic Properties of a Universe

I would like to show that if {$A_{i}$: i$\in$I} $\subseteq$ $A$ $\in$ $\mathbb{U}$, then $\bigcup_{i \in I}$$A_{i}$ $\in$ $\mathbb{U}$, where $\mathbb{U}$ is a universe and the capital $A's$ are all ...
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Are there concepts in nonstandard analysis that are useful for an introductory calculus student to know?

Studying calculus I became aware that nonstandard analysis had some methods that that made the concept of infinitesimal concrete, so that $dx$ actually made sense. Can someone elaborate on this ...
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Limits and Series in Smooth Infinitesimal Analysis

I just learned a tiny bit about SIA. While it is interesting, that it handles derivatives so easily, I wonder: Can we still recover the concepts of limits (of sequences) and especially series, to ...
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138 views

Metrizability, Models, of Non-Standard Reals

according to compactness theorem in logic, there are models for the Reals of all infinite cardinalities, and these are elementary-equivalent to those of the "Standard" Reals ( Reals with uncountably-...
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The Hyperreal number system

Currently reading Infinitesimal Calculus by Henle and Kleinberg. In this text, page 25, they note that they define a hyperreal number system, not the hyperreal number system. This is because "there ...
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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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explain why ${\left(\frac{{1}}{{2}}\right)}^{\infty}=0$

Mathematica shows ${\left(\frac{{1}}{{2}}\right)}^{\infty}=0$, anyone can explain why ? I know we can get $\lim\limits_{{{x}\to\infty}}{\left(\frac{{1}}{{2}}\right)}^{{x}}={0}$ by taking limit , ...
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How is an infinitesimal $dx$ different from $\Delta x\,$? [duplicate]

When I learned calc, I was always taught $$\frac{df}{dx}= f'(x) = \lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{(x+h)-x}$$ But I have heard $dx$ is called an infinitesimal and I don't know what this means....
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Can I embed $\mathbb R^{\mathbb N}$ with a partial order into $^\ast\mathbb{R}$ with the linear order?

Define a relation $\prec$ on $\mathbb R^{\mathbb N}$ as, For all $f, g \in \mathbb R^{\mathbb N} $, $f \prec g$, if for all $n \in \mathbb N$, $f(n) \leq g(n)$, and there exists a $m \in \mathbb ...
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Limit points in nonstandard analysis [solved]

Let $A\subseteq\mathbb{R}$, $p\in\mathbb{R}$. I proved that the following are equivalent: $\exists\left(x_{n}\right)_{n\in\mathbb{N}}\subseteq A\cap\left\{ p\right\} ^{c}$ such that $x_{n}\rightarrow ...
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existence of unlimited hypernaturals

How can we prove that the extension *$\mathbb{N}$ of $\mathbb{N}$ contains unlimited elements? I have read a proof that shows that the only limited elements of *$\mathbb{N}$ are the standard ...
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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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1answer
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Construction of homomorphism between $^\ast\mathbb{R}$ and $^*\mathbb{Q\cap L}$

Denote by $\mathbb{I}$ the ring of infinitesimals and by $\mathbb{L}$ the ring of finite hyper-reals. Prove that $$\mathbb{R}\cong{^\ast\mathbb{Q\cap L/^\ast Q\cap I}}.$$ I thought using the first ...
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non-commutative infinitesimal extension of $\mathbb R$

Background: The transfer principle in nonstandard analysis implies that any nonstandard model of the reals is a commutative (for additively and multiplicatively). It is also well-known that the set $\...
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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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Nonstandard analysis: transfering a simple sentence

If $A$ is an infinite subset of $\mathbb{N}$, show that *$A$ contains aritrarily large unlimited elements. From "Non-standard Analysis for the Working Mathematician," p. 22 : "there is a Skolem ...
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1answer
2k views

What are the advantages/disadvantages of non-standard analysis?

I'm not interested in an in-depth answer. Here are some specific questions for which I couldn't find an answer: With non-standard analysis, can we solve problems that can't be solved using standard ...
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115 views

Size of a geometric point

It is well known that the geometric points do not have any length, area, volume, or any other dimensional attribute, also geometric object (for example "line") is made up of a infinite number of ...
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1answer
106 views

hyperreals standard part inconsistency

$\def\st{\operatorname{st}}$ I'm studying non-standard calc from Keisler's book. Taking "standard part" rule doesn't make sense... its not commutative. e.g. $a$ is finite non infinitesimal $b,c$ ...
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1answer
151 views

When “magnifying infinitesimals” why dont they have curvature ? (non standard) Infinitesimal calculus

Im reading https://www.math.wisc.edu/~keisler/calc.html. If you open up the chapter $2$ pdf... The $2$ diagrams (1st on page $14$ of the pdf (not the text book), 2nd on page $15$) have me confused. ...
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How to compute $(\int f(x) \, dx)^p$ with fractional number $p$?

It is well-known that one can say $(\int f(x) \, dx)^p = \int \prod_{i=1}^p f(x_i) \, dx_i$ if $p$ isa natural number. But what is if $p$ is a fractional ore even a real number? Is it possible to set $...
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3answers
104 views

Clifford algebra over non-Archimedean field

Usually the Clifford algebra is defined over the Reals $\mathbb{R}$ or the Complex $\mathbb{C}$ numbers. Can the definition be extended over non-Archimedean fields, such as the hyperreal numbers $\...
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2answers
108 views

Numerical system that includes the limit targets such as $0^+$, $0^-$, $1^+$ etc

I wonder what is the name of a mathematical system extending the real numbers that includes signed zero along with unsigned zero as well as other "limit targets", such as $1^+=1+0^+$, $5^-$ etc, so ...
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Fourier transform and non-standard calculus

The Fourier transform is not as easily formalizable as the Fourier series. For example, one needs to introduce tempered distributions to define the Dirac delta-function. Also, it is impossible to view ...
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Ultraproduct of a metric space

I am currently trying to understand "Curvature bounded below: a definition a la Berg--Nikolaev" by Nina Lebedeva and Anton Petrunin. They start with a complete, intrinsic metric and space $X$ and say ...
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273 views

Could “$\infty$” be understood by taking the reciprocals of the Hyperreal numbers?

When learning mathematics we are told that infinity is undefined. (*) Recently I read about the infinitesimal version of Calculus and how we can in fact treat $dy/dx$ as a fraction under this ...
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Is the empty set internal?

Is the empty set internal or not? And is there a proof (either way), or is it just a convention? If it's just a convention, why was that particular convention chosen?
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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 ...
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1answer
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How much choice is needed for the transfer principle?

To construct the hyperreals via ultrapower the Boolean prime ideal theorem apparently suffices. However, to prove the transfer principle for the extension $\mathbb{R}\subset{}^\ast\mathbb{R}$ ...
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Why those division by zero are formalized?

Easy example first: $f(x) = nx$ $f'(x) = (f(x+0)-f(x))/0 = (nx+0n-nx)/0 = (0n)/0 = n$ Hard one: $f(x) = a^x$ $f'(x) = (f(x+0)-f(x))/0 = (a^{x+0}-a^x)/0 = (a^x(a^0-1))/0 = (a^x(e^{\ln(a^0)}-1))/0 =...
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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 ? ...
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1answer
203 views

Derivatives of $\sin x$ and $\exp x$ using differentials / dual numbers

I want to introduce a concept of a differential $dx$ to my students and derive all the basic derivatives using it. Now, I define the differential to satisfy $dx \neq 0$, but $(dx)^2 = 0$. Therefore, ...
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1answer
218 views

Topologies induced by non-standard metric

Let $R$ be a set of points and $\mathbb{D}$ be a totally ordered field. Further consider a function $\rho:R\times R \rightarrow \mathbb{D}$. $\langle R,\mathbb D,\rho\rangle$ is a metric space if $\...
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Dirac Delta definition in non-standard analysis?

What is the definition of Dirac Delta in non-standard analysis? I would define it either as a standard distribution with $\sigma=\epsilon$ or maximum equal to $\omega$. Which is the correct answer?
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847 views

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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1answer
85 views

Prove, by nonstandard reasoning, that the limit superior of a sequence is a cluster point.

I'm working through Goldblatt's Lectures on the Hyperreals, and I've found myself quite stuck on this exercise: Prove, by nonstandard reasoning, that both the limit superior and the limit inferior ...
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111 views

Compute joint density function of exponential fuction

Consider a set of continuous random variablces $Y_1 ... Y_n$, i.i.d, exponentially distributed . with rate parameter $\lambda$. I showed first that for one single variablce (ie the first) its ...
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1answer
275 views

Hyperreal probability density?

I'm fairly new to the subject of hyperreal numbers and I'm wondering if there exists an infinitesimal number $a$ such that (in some reasonable sense) $$\sum_{n=1}^\infty a=1$$ ? In other words: Is ...
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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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Computing the standard part of $(3-\sqrt{c+2})/(c-7)$ where the standard part of $c$ is $7$

I'm working through Keisler's calculus book based on infinitesimals. The following problem has me a little bit stumped. Compute the standard part of: $$\frac{3-\sqrt{c+2}}{c-7}$$ Given that $c\ne7$ ...
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239 views

Nonstandard complex numbers and categoricity

Let ${}^*\mathbb{C}$ be a nonstandard complex number field (given, for instance, as a countable ultrapower.) By the transfer principle ${}^*\mathbb{C}$ is algebraically closed of characteristic zero, ...