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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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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}=...
1 vote
2 answers
119 views

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 $...
8 votes
6 answers
2k views

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 ...
2 votes
1 answer
154 views

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 ...
1 vote
1 answer
77 views

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'...
1 vote
1 answer
87 views

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}...
3 votes
1 answer
137 views

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?...
1297 votes
27 answers
145k views

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 ...
2 votes
1 answer
105 views

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 ...
8 votes
1 answer
2k views

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 ...
4 votes
1 answer
185 views

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 &...
45 votes
9 answers
10k views

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 ...
1 vote
2 answers
147 views

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 ...
2 votes
1 answer
128 views

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 ...
11 votes
3 answers
1k views

Uniqueness of hyperreals contructed via ultrapowers

The construction I've seen of the field of hyperreal numbers considers a non-principal ultrafilter $\mathcal{U}$ on $\mathbb{N}$, then takes the quotient of $\mathbb{R}^{\mathbb{N}}$ by equivalence ...
2 votes
1 answer
103 views

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 $\...
3 votes
2 answers
168 views

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 ...
1 vote
1 answer
204 views

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 \{...
5 votes
3 answers
445 views

structure of the hypernaturals

I want to understand the structure of the hypernaturals a little better. Let me recall the ultraproduct construction of the hypernaturals. On the set of all sequences of $\mathbb{N}$, we define an ...
-1 votes
1 answer
57 views

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 ...
4 votes
1 answer
67 views

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 ...
6 votes
2 answers
164 views

What is the “maximal hyperreal field”?

In many SE posts and the Wikipedia article on the surreal numbers I’ve seen references to a “maximal” hyperreal field that’s isomorphic to the surreals. If they’re isomorphic, then why is it that ...
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? ...
1 vote
1 answer
81 views

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 ...
0 votes
1 answer
70 views

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 ...
5 votes
4 answers
303 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 ...
2 votes
1 answer
165 views

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 ...
0 votes
1 answer
63 views

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-...
3 votes
2 answers
149 views

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 ...
5 votes
1 answer
67 views

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 $N$ ...
4 votes
1 answer
66 views

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 ...
2 votes
1 answer
68 views

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.
4 votes
2 answers
155 views

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 ...
12 votes
2 answers
844 views

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 &...
2 votes
2 answers
80 views

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 ...
8 votes
2 answers
992 views

How do we interpret and perform integrals using infinitesimals?

If $\mathrm{d}x$ is treated as a hyperreal infinitesimal we can easily do derivations. How do we interpret and perform integrals using infinitesimals? What is the $\mathrm{d}x$ in $\int x^3\,\mathrm{...
44 votes
2 answers
9k views

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 ...
10 votes
6 answers
11k views

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 ...
2 votes
2 answers
274 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 ...
3 votes
1 answer
217 views

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 ...
4 votes
3 answers
525 views

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 rigor)....
3 votes
2 answers
82 views

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 ...
0 votes
1 answer
277 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 ...
4 votes
1 answer
260 views

Ultraproduct construction: are finite hyperreals just a thinly disguised version of Cauchy sequences?

Periodically I've tried to wrap my head around nonstandard calculus and hyperreals, but I always thought I needed a lot more of a background in formal logic and/or set theory to understand what's ...
1 vote
1 answer
140 views

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 ...
2 votes
1 answer
86 views

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 ...
0 votes
1 answer
73 views

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 ...
2 votes
2 answers
101 views

Halos in non-standard analysis

Please consider this question in terms of the hyperreals. As per usual, the halo of a point $P$ is the set of all points separated from $P$ by an infinitesimal distance. Let $P$ be a point in a ...
3 votes
2 answers
180 views

Prerequisites for non-standard analysis

Today my real analysis teacher mentioned the existence of "non-standard analysis" and explained on a very basic level what it studies and it immediately caught my attention. After that he ...
5 votes
1 answer
239 views

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 ...

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