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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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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50 views

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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136 views

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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67 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 ...
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109 views

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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119 views

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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3answers
100 views

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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26 views

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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1answer
55 views

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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67 views

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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25 views

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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2answers
530 views

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
64 views

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 ...
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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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4answers
184 views

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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53 views

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
79 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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57 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
40 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
48 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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1answer
45 views

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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45 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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26 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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50 views

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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3answers
131 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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51 views

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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96 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 ...
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1answer
42 views

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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150 views

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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84 views

1/∞ is 0 or infinitesimal?

Since ∞>0 , so 1/∞>0, thus I think 1/∞ should be infinitesimal, but the calculus book says $\displaystyle \lim_{x \to \infty} \frac{1}{x}= 0$ So is 1/∞ 0 or infinitesimal ? P.S.I mean 1/∞ and ...
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1answer
79 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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159 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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111 views

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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1answer
82 views

What are hyperreal numbers?

Well excuse me if I make some english mistakes :) So I descovered hyperreal numbers two mouth ago and I read a lot of articles about them but I don't understand which they are... Because of one ...
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1answer
53 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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1answer
36 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
65 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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1answer
47 views

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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134 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, ...
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1answer
60 views

Which non-standard analysis framework to study?

Just recently I became slightly interested in non-standard analysis. After a preliminary check at the subject there seem to be at least two relatively common ways of establishing the framework: ...
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75 views

Internal Set Theory: $n$ is standard $\implies\;n+1$ is standard

I'm reading a bit about Nelson's version of nonstandard analysis and in the notes it is said that [$n$ is standard]$\implies$[$n+1$ is standard]. Right after that it is mentioned that an inductive ...
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236 views

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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60 views

Is there an algebraic-geometric solution to the problem of the Leibnizian formalism?

The precise question appears at the end of this entry. With all the recent advances in understanding infinitesimals, we still don't fully understand why Leibniz's definition of $\frac{dy}{dx}$ as ...
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How can one replace a set with a base set in nonstandard analysis?

In the superstructure approach of nonstandard analysis, one builds the superstructure $V(X)$ as ($X$ being a set) : $V_0(X)=X$ For $n$ in $\mathbb{N}$, $V_{n+1}(X)=V_n(X)\cup\mathcal{P}(V_n(X))$ ...
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42 views

Modified division, hyperreal numbers and transfinite derivatives

Suppose we are shooting from a cannon and measuring the speed of the projectile. The shorter period of time it takes for the projectile to reach the target, the faster it is. If the projectile hits ...
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1answer
96 views

Best mathematical object to describe speed

Suppose we want to describe speed of a particle, moving between two points on a real line: ---------------$0$----------------$1$----------------- If the particle starts at $0$ at $t=0$ and moves ...
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1answer
25 views

Why is a *-finite standard set a finite set?

In internal set theory, why is it that if there is a bijection between $x$ standard and $n\in\mathbb{N}$, then $n$ is necessarily standard ?
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126 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 ...