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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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Internal Induction and the Overflow Principle

On page 129 of Goldblatt's Lectures on the Hyperreals, I'm trying to understand the discussion between Internal Induction (Thm 11.3.2) and the Overflow Principle (Thm 11.4.1). For context: Theorem 11....
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usage of Leibniz notation for things like $\frac{d^2y}{dt^2}$ and $\frac{dy'}{dy}$

I've read the other posts on this site about whether you can treat $\frac{dy}{dt}$ as a fraction. There are a lot of conflicting opinions, but many seem to be saying that treating it as a fraction ...
Ishaan Jain's user avatar
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Existence of standard part

My favorite proof of the existence of standard part of a limited $x$ in the context of an extension $\mathbb R \subset {}^\ast\hskip-.5pt\mathbb R$ is to say that $x$ defines a Dedekind cut on $\...
Mikhail Katz's user avatar
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Converse of Intermediate Value Theorem via Nonstandard Analysis

I am working through Goldblatt's Lectures on the Hyperreals, and I am stuck on Exercise 7.8.4 (p. 82). It is a converse to the Intermediate Value Theorem: "Let the real function $f$ be monotonic ...
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Nonstandard Analysis research project ideas [closed]

Before I ask my question, this is my first post on MSE, so I apologise if I have not met the standard etiquette of a post. I have finished my first year at a UK univeristy for a maths degree. In our ...
demian's user avatar
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Solve an integral, e.g, $I = \int_{p}^{p+\delta p} f(x) dx=\int_{p}^{p+\delta p}\frac{e^{ax}(1-e^{-ax})}{x(1-x)}dx$ [duplicate]

I am trying to solve an integral over an infinitesimal interval, such as: $\begin{align}I = \int_{p}^{p+\delta p} f(x) dx=\int_{p}^{p+\delta p}\frac{e^{ax}(1-e^{-ax})}{x(1-x)}dx&\tag{1}\end{align}...
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Automorphism on the hyperreals

A field isomorphism $\phi:F\rightarrow G$ is a bijection such that (i) $\phi(x+y)=\phi(x)+\phi(y)$ and (ii) $\phi(xy)=\phi(x)\phi(y)$, where $F$ and $G$ are ordered fields. If $F=G$, then $\phi$ is ...
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How to define infinite sums in nonstandard analysis for general vector spaces?

I'm studying nonstandard analysis and I came across the concept of infinite sums in this framework. I understand that in standard analysis, infinite sums are typically defined through the concept of ...
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Is there distance (metric) between two points on the hyperreal line in Nonstandard analysis?

As is known, there is distance between two points on the real line. It's obvious. But if we imagine hyperreal line (see pic.) then we'll have infinitesimals and their corresponding points. For ...
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Is this a non-standard extension?

I started reading Henson's "Foundations of Nonstandard Analysis. A Gentle Introduction to Nonstandard Extensions" a couple of days ago, and I am a bit confused about something. Let $F$ be an ...
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Approaches to nonstandard measure theory

So I don't really know anything about measure theory or nonstandard measure theory or nonstandard analysis or anything like that, but my friend who is taking measure theory next semester told me about ...
dasffbsrewgfdsgfd's user avatar
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Why do we use symbol ";" in extended decimal notation for hyperreal numbers?

There is extended decimal notation for hyperreal numbers which was developed by A.H. Lightstone: $d.d_1d_2d_3...;...d_{H-1}d_{H}d_{H+1}...$ Why do we use symbol ";" in this notation? Thanks.
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Question in Davis' Applied Nonstandard Analysis

I am struggling over a proof in Davis' Applied Nonstandard Analysis book: Chapter 1, Section 4, Lemma 5 (page 18). Background: $N=\lbrace 0,1,2,3,... \rbrace$ $S_0=S$ (set of individuals) $S_{i+1}=...
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Equivalence between definitions of limit

In non-standard analysis, it is possible to define a limit as follows: $\displaystyle \left[\lim_{x \to a} f(x)=L \right] :=x\approx a\implies f(x)\approx L$ ($a\approx b$ denotes that the difference $...
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Is this a valid basis for a transfinite number system?

I've been curious about transfinite number systems including infinite ordinals, hyperreals, and surreal numbers. The hyperreals in particular seem particularly appealing for introducing a hierarchy of ...
Aidan Simmons's user avatar
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How is it possible that graph of function $y=x^2$ doesn't coincide with its infinitesimally small segment of tangent line in Non-standard analysis?

As is known, graph of function $y=x^2$ touches the axis $X$ not only at point $(0,0)$. There is infinitesimally small segment of tangent line (at the point $(0,0)$) that coincides with this graph (it'...
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Help with using "infinitesimal Riemann sums" to arrive at the formula for arclength

I am trying to arrive at the formula for arclength using infinitesimals. So far, I have a definition which says: $\displaystyle \mathrm{Re}\sum_{k=0}^{\omega}f(x_k)\Delta x:=\int_{a}^{b}f(x)\mathrm{d}...
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Why don't infinitesimals in nonstandard analysis have concrete size but infinitesimals in surreal numbers have?

I read on Wiki (https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis#Overview) that infinitesimals in NSA don't have concrete size but infinitesimals in surreal numbers have. How is it possible?...
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How are we to interpret the differential in the integral?

I have been working a lot with infinitesimals lately and related concepts such as derivatives and integrals. The for a function $y=y(x)$, the differential $\mathrm{d}y$ can be defined to be the change ...
Alice's user avatar
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What is the motivation for building the hyperreals using an ultrapower construction? [duplicate]

The hypperreal numbers are an extension of the reals that allow for a rigorous treatment of infinitely small and infinitely large values. Specifically it includes the number $\varepsilon$ where $$ 0 &...
Sage Mitchell's user avatar
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Is it possible to extend the domain and range of a function that maps from R to R to other sets?

I am currently working on a project where I would like to define infintesimals that can be used in conjunction with the real numbers (similar to the hyperreals). Right now, I am working on an ...
Alice's user avatar
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How big do hyper-reals get?

Let's assume there is some non-standard model of the reals containing a number $N$ that is larger than any real number. Suppose $\exists N\in {^*}\mathbb{R} ( \forall r\in\mathbb{R}: r<N).$ Now I ...
Numeral's user avatar
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Why isn't $\mathcal{P}(^*A)\subseteq {^*}\mathcal{P}(A)$?

I'm reading Goldblatt's Lectures on the Hyperreals, and he provides the following proof that $^*\mathcal{P}(A)\subseteq \mathcal{P}(^*A)$: Given sets $A,\mathcal{P}(A)\in\mathbb{U}$, the statement $\...
Numeral's user avatar
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Expressing trigonometric functions of infinitesimal arguments as algebraic quantities/elementary functions

I have recently been working with, and reading a bit about, infinitesimals and hyperreals and am currently trying to figure out how the trigonometric functions for infinitesimal inputs should behave ...
Alice's user avatar
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Thoughts about allowing arithmetic with infinitesimals to (mostly) solve limits, can it be done without contradictions?

I am currently working on a project for school where I would like to create an arithmetic framework that would make it easier to solve limits. Suppose we have a function $f:\mathbb{R}\smallsetminus \{...
Alice's user avatar
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Finite field with infinitesimals / nonstandard analysis over finite fields

I have two questions, which are really the same question phrased in two ways: Has there been any research on adjoining infinitesimal elements to finite fields? Has anyone considered extensions to ...
Jim's user avatar
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If $a_i/b_i$ converges then $\Sigma_m^na_i$ is infinitesimal iff $\Sigma_m^nb_i$ is too

I am trying to solve question ($6$) in section $6.11$ of Goldblatt's Lectures on the Hyperreals. The question asks: Given two series of positive terms $\sum_1^\infty a_i$ and $\sum_1^\infty b_i$ such ...
Numeral's user avatar
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3 votes
1 answer
154 views

Is it possible to replace hyperreal numbers with "good enough" alternatives?

The hyperreal numbers are undoubtedly interesting, generalizable, and have many nice properties, but are they really needed to solve the problems they solve? Would other, smaller fields work too? ...
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Nonstandard analysis without transfer principle and mathematical logic

I noticed that a big part of nonstandard analysis aimed to work with ordinary real valued functions. It is a bit strange for me because when we expand rational numbers to reals, we do not formulate ...
TheWildPalms's user avatar
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Problem from Keisler infinitesmal calculus book.

I'm going to Keisler's "Elementary Calculus, an Infinitesimal approach" , and I'm stuck on a problem: Given that $H$ is a positive infinite term, determine whether the given expression is ...
David Davidson's user avatar
5 votes
4 answers
308 views

Cauchy sequence for $0$ in non-standard analysis

In real analysis Cauchy sequence for $0$ is $(1/2,1/4,1/8,...)$. But in non-standard analysis (hyperreal numbers) this sequence is infinitesimal $\varepsilon$. Since hyperreal numbers are extension of ...
Mike_bb's user avatar
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Interpretation of the transfer principle.

I am reading an article written by W.A.J Luxemburg about nonstandard analysis (https://www.jstor.org/stable/3038221). My question is: Why does the transfer principle transform sentences and predicates ...
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How can I prove the existence of delta-incomplete / countable incomplete ultrafilters?

I am reading a W.A.J Luxemburg paper about nonstandard analysis (https://www.jstor.org/stable/3038221). He presents the following definition I am stuck trying to prove the existence of delta-...
DAGO's user avatar
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Are Distributions just functions with infinitesimal coefficients?

It is relatively well known that the Dirac delta, the classical example of a distribution, can be written as $$ \frac{a}{π(a^2+x^2)} $$ where $a$ is an infinitesimal such as a hyperreal. This can be ...
Daniel Schwartz's user avatar
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In nonstandard probability theory, does an event hold nearly everywhere iff it holds on an event whose complement has infinitesimal probability?

In Nelson's "Internal Set Theory: A New Approach to Nonstandard Analysis", he defines a predicate holding nearly everywhere if for all standard $\epsilon > 0$, there exists an event $N$ ...
Kellen Brosnahan's user avatar
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1 answer
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In IST, can nonstandard functions take standard objects to nonstandard objects?

I'm asking specifically about a proof given in Edward Nelson's "Internal Set Theory: A New Approach to Nonstandard Analysis". Here's the theorem statement: If every finite subset of a graph ...
Kellen Brosnahan's user avatar
2 votes
1 answer
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What is difference in approaches between using standard part in NSA and limit?

What is difference in approaches between using standard part in NSA and limit? I don't mean technique of differentiation or integration. Can somebody explain it on example? Thanks.
Mike_bb's user avatar
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How to use Internal Set Theory to prove Nelson's axioms in "Radically Elementary Probability Theory"

I'm currently reading through Nelson's "Radically Elementary Probability Theory". I don't believe Nelson gives a construction in that book, but Herzberg makes the claim that the axioms form ...
Kellen Brosnahan's user avatar
12 votes
2 answers
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Nonstandard infinite / hyperfinite sum in IST

TLDR: If anyone could provide a detailed proof that a sum indexed by an unlimited hypernatural number is well-defined using the axioms of IST, I would greatly appreciate it. I am studying Nelson's &...
singularity425's user avatar
2 votes
2 answers
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Can an Ultrafilter be defined in terms of the Convergence of a Sequence?

We know that the set $\{\sin(0), \sin(1), \sin(2), ... \}$ is dense in the interval $(-1,1)$. So now consider the sequence $S$ = $\langle \sin(0), \sin(1), \sin(2), \ldots \rangle$. This sequence ...
Jonathan Hoyle's user avatar
2 votes
2 answers
282 views

Are the hyperreal numbers separable? Can we construct a computable dense of them?

I am familiar with the construction of the hyperreal numbers $^*\mathbb{R}$ as an extension of $\mathbb{R}$ built of equivalence classes of real sequences up to equivalence with respect to some fixed ...
beanstalk's user avatar
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3 votes
2 answers
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Equivalence relationship defined over membership of a in halo of b. (Non-Standard Analysis, Monads, Halos)

Let's say we define an equivalence relationship such that $a \sim b \iff b\in \mu(a), \: \mu(a)$ is the halo\monad of $a$. By definition this would include all points an infinitesimal distance away ...
Roman Schiffino's user avatar
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1 answer
279 views

Is there a lack of rigor in the standard analysis?

Does the difficulty of defining exactly what infinitesimals and differentials are denote a lack of rigor in standard real analysis? For example, in an introductory course one may solve the ...
Gustavo Gabriel's user avatar
1 vote
1 answer
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Confusion about the model in Robert’s Nonstandard Analysis

I’m working through Alain M. Robert’s “Nonstandard Analysis”. I’m intrigued by his suggestion that his axiomatic approach, rather than adding elements to ℕ to create a nonstandard model *ℕ, “discerns ...
Rivers McForge's user avatar
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1 answer
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Is it a good idea to learn about hyperreals and how they are related to limits before beginning to learn about calculus?

I have some idea of what the main concepts of calculus are about but I have never actually taken a calculus class or studied myself. My general understanding is that before the concept of limits were ...
jacob78's user avatar
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1 answer
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How to prove that infinitely small and large numbers can be used as measures of rates of convergence?

I read "Lectures on the Hyperreals: An Introduction to Nonstandard Analysis" Robert Goldblatt. He wrote that infinitely small and large numbers can be used as measures of rates of ...
Mike_bb's user avatar
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Definition of arithmetic operations for hyperreal numbers

I read paper about hyperreal numbers (https://sites.math.washington.edu/~morrow/336_15/papers/gianni.pdf) I have few questions about definition of arithmetic operations. What is idea of this ...
Mike_bb's user avatar
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5 votes
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Limits vs infinitesimals

Many years ago in college when I went from Calculus to Differential Equations, there was a distinct difference in the classes. All three semesters of Calculus had spent considerable time on justifying ...
David Gudeman's user avatar
1 vote
1 answer
98 views

Proving that every cauchy sequence is bounded: Why is the set $\cal N$ internal?

I'm having some trouble understanding the following proof that every real Cauchy sequence is bounded: Proof: Let $(a_n)_{n\in \mathbb N}$ be a Cauchy sequence and suppose, for the sake of ...
Eduardo Magalhães's user avatar
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1 answer
107 views

Sum of Two Infinitesimals being Infinitesimal

Any hyperreal number greater than 0 and smaller than all positive real numbers is infinitesimal. We know the sum of two infinitesimals is infinitesimal. Let $A$ be the smallest positive real number. ...
Jonathan Lee's user avatar

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