Non-standard 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 it to the students' advantage to learn the language of infinitesimals?

A colleague of mine asked an interesting question reproduced below with his permission. It is reasonable to ask whether it is to the students' advantage to learn the language of infinitesimals - ...
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Proving a Function Continuous with Non-Standard Analysis

I am reading a text on non-standard analysis. I need to prove the following: Suppose that $f$ is non-decreasing on the real interval $[a,b]$ and that $f$ satisfies the intermediate value property. ...
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Probability theory with the hyperreals?

Forgive the undoubted ignorance of this question. I am out of my element both in probability and in nonstandard analysis. A mathematically curious layperson friend recently had a conversation with me ...
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Survey of varieties of non-standard analysis?

Is there a reliable, reasonably up-to-date, survey article doing a "compare and contrast" on varieties of non-standard analysis?
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Definite Integral of a infinitesimal

I did not study math, but have some foundations in it. I have been looking through some books on nonstandard analysis, and have (what I consider to be) a pretty simple question which I haven't been ...
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Constructing infinite field in which all subrings are subfields

A classmate posed a question in class as to if there existed an infinite field $F$ for which every subring $R \subseteq F$ was a subfield. We'd already determined that if $F$ was a finite field, then ...
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Characterisation of convergence of bounded sequences via ultra-filters

Let $\{a_n\}_{n\in\mathbb N}$ be a bounded sequence of real or complex numbers and $\mathscr F\subset\mathscr P(\mathbb N)$ be a non-principal ultra-filter. Then $a=\lim_{\mathscr F}a_n$ is ...
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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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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 them? What I am asking for are specific techniques for explaining infinitesimals to students, ...
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Is arithmetic with infinite numbers fictitious?

In 1933 Skolem constructed models for arithmetic containing infinite numbers. In a 1977 article Stillwell emphasized the constructive nature of Skolem's approach; see here. Is this at odds with ...
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Does evaluating hyperreal $f(H)$ boil down to $f(±∞)$ in the standard theory of limits? [closed]

Does evaluating $f(H)$ at an infinite hyperreal $H$ when doing calculus in Robinson's (Keisler's) framework amount merely to assigning $f(\infty)$ in the standard theory of limits? This question has ...
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Can I have something larger than infinite? [duplicate]

My question is "Can I have something larger than infinite?" Sometimes, we add infinite numbers into our set of numbers by simply extending our set and adding infinite numbers to it. But can't you ...
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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 ...
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Justification for manipulations according to Leibniz-notation

Is there a way to justify the manipulations according to Leibniz-notation without nonstandard-analysis. E.g. $\frac{dy}{dx} = x \\ dy = x dx\\ \int dy = \int x dx\\ y = \frac{1}{2} x^2$
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Non-standard version of Frechet derivative

Non-standard analysis offers very convenient tools to prove facts about continuity or differentiability. I am looking for such tool in infinite-dimensional calculus. To be more precise, let $X$ and ...
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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 \ \ \ ...
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Nonstandard analysis, Lie groups and universal enveloping algebras

The idea of nonstandard analysis is to combine finite quantities with infinitesimals. And, back in the day, Lie algebras were roughly considered the "infinitesimal elements" of Lie groups. Say we want ...
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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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First order logic,Step in Derivation of Non standard real numbers

This is from first order logic, specifically a section detailing the construction of the non standard real numbers, after Los' theorem And we have that:\ \ Let $L = \{+, ×, <, 0, 1\}$ be the ...
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Hyperreal star mapping isomophism

I've been reading through Goldblatt's book on the Hyperreals. And the star mapping is defined to be: *r=[r]=[(r,r,r,...)]. Where r is a real number, and [r] denotes the equivalence class of the ...
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Star mapping in Non-standard analysis

I'm trying to understand the star mapping in non-standard analysis in particular for the Hyperreals. I know that $*: \mathbb R\to \mathbb{^* R}$ is a mapping such that $^*(x)=^*x$ where $^*x= ...
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Should we say $0.\bar9$ *can* equal $1$ also in the hyperreals?

Consider the sequence $R_n$ of repunits, defined as $\displaystyle\frac{10^n-1}{9}$. We have $$\frac{R_{n+1}}{R_n}=\frac{9}{9} \frac{10^{n+1}-1}{10^n -1}=\frac{10^{n+1}-1}{10^n ...
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Mathematical descriptions of physical space

Bear with me as I'm a philosophy (not math) student. First some philosophical background, and then the math question. One philosophical view is that physical space is composed of infinitely many ...
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Is there such a thing as “hypertopology” (analogous to the hyperreals)?

The hyperreal number system adds infinities and infinitesimals, allowing Calculus to be done using these things instead of limits (sort of like when calculus was originally invented, but with ...
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In NSA (ZFC+IST), what can we say about generators for $\mathbb{Z}/\nu\mathbb{Z}$ for unlimited $\nu$?

Recently I've been going through a short text on Nonstandard Analysis that uses the axiomatic approach of Nelson (Internal Set Theory - IST). Its study has led me to be curious about the properties of ...
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Nonstandard analysis and integral transforms

Can integral transforms be evaluated without limits(i.e Laplace transform) such as in non standard analysis? Can the improper integral be bounded by a hyperreal number? I am not very familliar with ...
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Ultrapower Construction of Hyperreal System

I think I understand the ultrapower construction of hyperreals. Given a free ultrafilter $\mathcal{U}$ (take $\mathcal{U}\subset\mathcal{P}(\mathbb{N})$) for simplicity, then the hyperreal system is ...
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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 ...
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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 ${\mathrm{d} x}$ ...
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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 is useful for a 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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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 ...
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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 ...
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Inferring Probabilities from relative frequencies

I have an question concerning the converse strong law of large numbers By the Converse Strong Law of large numbers, i mean the general principle (2) which is the converse of the standard strong law ...
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Decoding the sign expansion of surreal numbers

One way to represent surreal numbers is the sign expansion. Now Wikipedia describes how to compare them, how to convert them to the standard representation of left/right sets, how to negate them, and ...
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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 ...
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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 ...