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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Does nonstandard analysis allow for a more powerful second derivative test?

The second-derivative test states that if $x$ is a real number such that $f'(x)=0$, then: If $f''(x)>0$, then $f$ has a local minimum at $x$. If $f''(x)<0$, then $f$ has a local maximum at $x$...
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Is the union of non-standard analogs of a family of sets a proper subset of the non-standard analog of the union of those sets?

The book on formal logic I'm using for self-study builds the foundation for non-standard analysis (to illustrate the usage of non-standard models) using the following construction. Consider the real ...
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What are elementary conclusions using $^*$-polynomials?

Let $^*$-polynomials be defined as hyperfinite polynomials over the hyperreals, i.e. elements of the set $\{ p\in \mathbb{R^R}\mid \exists\big( a:\{0,..,n\}\to\mathbb{R}\space \big)\forall x\in \...
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1answer
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Juggling three non-Archimedean fields

I'm comparing the field of hyperreals, the Levi-Civita field and the Dehn's field for the first time. I'm not very familiar with their properties, so I'm looking for ways to understand and distinguish ...
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1answer
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How can $0.999\dots$ not equal $1$?

First, by definition I assume that $0.999...$ actually is defined as: $$\text{lim}_{n\rightarrow\infty}\sum_{i=1}^n 9/10^i$$ Now by geometric series we already know that this equals one. But ...
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Let $f:\mathbb{^*R}\to \mathbb{^*R}$ be an external function with $(x,y\in\mathbb{^*R},x\approx y , x\le y ) \implies f(x)\le f(y)$. Is $f$ monotonic?

My main problem is that there's two methods yielding two different results: 1.We can count through $\mathbb{^*R}$ using nothing but infinitesimal steps. For example, we can partition the interval $^*...
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54 views

Derivatives in positive and negative x directions

For a function $f(x)$ the definition of its derivative is $$f'(x) = \lim \limits_{\Delta x \to 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}.$$ The derivative $f'(x)$ is supposed to be the same for $\Delta ...
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Nonstandard-Analysis: Showing L'Hospital

Let there be two functions $f,g:(a,b) \to\mathbb R$ that are differentiable in $(a,b)$ with either $$\text{Case 1:}\qquad\lim_{x\to b} f(x) = \lim_{x\to b} g(x) = 0$$ or $$\,\,\,\,\,\,\,\text{Case 2:}\...
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1answer
69 views

Doubly-hyper-reals? Can we include another level of infinitesimals?

Is it possible (even if there is no reason to even want to do this) to expand the hyperreal number line at each infinitesimal to insert a "second layer of infinitesimals"? Let $\epsilon$ be an ...
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238 views

Can nonstandard analysis give a uniform probability distribution over the integers?

There exists no uniform probability distribution over the non-negative integers. This is because we would need to have $p(i)=q$ for all $i$, for some real number $0\le q\le 1$. But normalisation ...
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Can the hyperreals be used to describe the “gold nuggets” found with nonconverging series and Casimir forces? [closed]

In one of Numberphile's videos they describe the $-\frac{1}{12}$ as a "gold nugget" inside the sequence $1+2+3+4+\dots$ surrounded by a bunch of "rock" that is infinity. Can these numbers that we get ...
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Nonstandard-Analysis: What are traits of sets that are “strange”?

By the power of the transfer principle, the principle of internal definition and the overspill principle, most sets in the nonstandard-superstructure behave rather tame (or rather, standard). However, ...
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2answers
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What's the nonstandard way to argue $\lim_{n\to\infty}\sum_{k=0}^n\binom n k (\frac x n)^k = \sum_{k=0}^\infty \frac{x^k}{k!}$

First, the equality holds, as: $$\lim_{n\to\infty}\sum_{k=0}^n\binom n k \left(\frac x n\right)^k =\lim_{n\to\infty}\left(\left(1+\frac{x}{n}\right)^n\right) = e^x = \sum_{k=0}^\infty \frac{x^k}{k!} ...
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ultrafilter convergence versus non-standard topology

I have recently been reading about the non-standard characterisation of topological spaces, by saying which points of ${^*X}$ are infinitesimally close to which standard points. The theory looks a lot ...
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A question about a detail in Bell's “Primer of Infinitesimal Analysis”

On p.35,36 of J.L. Bell's A Primer of Infinitesimal Analysis (2nd ed.), Bell uses the book's basic methods to derive the formula for the area of a circle based on the circumference. Where $s(x)$ is a ...
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60 views

Nonstandard Natural Numbers

Let $n \in \mathbb{N}$ and consider the set of nonstandard natural numbers $^*\mathbb{N}$ in sense of nun standard analysis. I want to show that for each $m \in (^*\mathbb{N}) \backslash \mathbb{N}...
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Is this proof correct: A nonstandard-statement transfers exactly iff it's an internal set.

Let $\varphi\in \widehat {^*S}$ be a statement in a nonstandard-superstructure. The following proof is supposed to show that if and only if $\varphi$ is internal, there exists a corresponding ...
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2answers
175 views

Non-standard integers?

Edit: Realized almost immediately that this was a stupid question - see comment below. Was about to delete it when an answer appeared that seems like saving... Context: I just saw a presentation of ...
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What's wrong with the following (Robinson-) nonstandard proof?

Note: I do not know for sure that it's wrong, but have a strong suspicion, as the authors implicitly mentioned it but in the end chose another (more elaborate) proof. Let $\hat S$ be a superstructure....
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How do I prove that every hyperreal has a standard part after constructing the reals from the hyperrationals?

In texts on nonstandard analysis, I've come across references to the following construction of the real numbers: starting from the hyperrationals $^*\mathbb Q$, say that $\mathbb R$ is the quotient ...
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1answer
113 views

Completeness in Levi-Civita field

I've been wondering for quite a time about Levi-Civita field (you can read it simply in https://en.wikipedia.org/wiki/Levi-Civita_field). I remember that I've read somewhere that Levi-Civita field is ...
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4answers
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Is the Archimedean principle necessary to prove the density of $\mathbb Q$ in $\mathbb R$?

I've noticed that most proofs of the density of $\mathbb Q$ in $\mathbb R$ use the Archimedean principle. For example see @Arturo Magidin's checked answer here. Density of irrationals I'm puzzled by ...
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Can we ever reach $\mathbb{R}$ from $\mathbb{R}*$? [closed]

The construction of the hyperreals is quite the ways over my head, but, after watching a series of videos that try to conceptualise the hyperreal number line from sources such as mainly YouTube and ...
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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 ...
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How to prove set of hyperreals {1, 2, 3, …} are > every real number

I'm working through Infinitesimal Calculus by Henle and Kleinberg on my own and I'm having some trouble with one of the exercises. Here's what I have currently: Exercise: Let $j$ be the hyperreal $\{...
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What is the basis of the hyperreal numbers?

Let us consider the Hyperreal numbers as a vector space over the real numbers. This vector space is quite interesting. Here are some interesting subspaces: The finite numbers The infinitesimal ...
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1answer
71 views

Searching for an example of a theorem wich is “easy” to prove in a classical way but way more difficult in the setting of non-standard analysis.

I´m learning some non-standard analysis. Quite some basic properties and theorems are "easier" to prove in the setting of non-standard analysis. But I´m searching for the converse, does anyone know ...
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2answers
196 views

How are infinite sums in nonstandard analysis defined?

Since, in nonstandard one can have infinitely large numbers, I was wondering if I can assign divergent sums to them. However infinite sums are defined by taking the limit to infinity of a partial sum ...
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1answer
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Nonstandard Extension of the Characteristic Function

$1_{A}$, the characteristic function of the set $A \subset \mathbb{R}$, is defined by: $$ 1_{A}(x)=\begin{cases} 1,& \text{ for } x \in A\\ 0,& \text{ for } x \notin A ...
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Monads in [a,b), with rational a and b [closed]

This is a question in non-standard topology: I am given a set $X=\Bbb R$ and a topology generated by sets $[a,b)$ where $a,b\in \Bbb Q$ and $a<b$. Here is the definition of a monad in the context ...
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1answer
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Is $\approx$ actually an entourage?

I was looking at applying the ideas in the paper On Nonstandard Topology to Uniform spaces. Given a uniform space $(X,\Phi)$, we can define the relation $\approx$ on ${}^*X$ as follows $$\approx \, = \...
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1answer
112 views

Epsilon-Delta Continuity in Hyperreals

I am wondering what's exactly the reason why we do not use the $\epsilon-\delta$ definition of continuity in Hyperreals? I mean, I know that one of the purposes of building this set is that we replace ...
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2answers
107 views

Physical Calculator with Hyperreals and Multiple Dimensions?

I'm just curious if there are any physical calculators out there that deal in the hyperreal number line or with multidimensional (i.e., complex) numbers. It would be a fun project to do, but I'm ...
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2answers
85 views

Applying ultrapower construction to the field $\mathbb {Q} $ of rationals

Can the ultrapower construction (used for extending the field of real numbers to get the field of hyperreals) be applied to the field $\mathbb{Q} $ of rational numbers? In my view it should be ...
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1answer
256 views

Textbook recommendation for non-standard analysis

I would like to design a one-semester elective course on non-standard analysis for mathematics majors at my department. The target audience will be junior and senior students who, by the time they can ...
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Multiplication property of equality for infinitesimals

I am to prove this property of *$\Bbb R$: If $x \approx y$ and $u \approx v$ and $u,x$ are finite then $xu \approx yv$. My question is can I just use the transfer principle for the multiplication ...
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2answers
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Is this result related to the Taylor series?

We have, $$f(b)-f(a)=\lim_{n\rightarrow \infty}\sum_{k=0}^{n-1} hf'(a+kh)\:\:\:\:\:..(1)$$ where $h=\frac{b-a}{n}$ Now, $$f'(a)=f'(a)$$ $$f'(a+h)=f'(a)+hf^2(a)$$ $f^2(a)$ meaning the second ...
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3answers
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Are monotone limited hypersequences converging?

Consider the usual ultrapower construction of the hyperreals $^*\mathbb R$. Let $a:{}^*\mathbb N \to \mathbb L$ be an increasing limited hypersequence. Does $a$ have an upper bound in $\mathbb L$? ...
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2answers
149 views

Relationship between nonstandard analysis and nonstandard models of arithmetic

So I've been reading through some introductory texts on nonstandard analysis, basically through the ultrafilter construction of hyperreals and the transfer principle. It seems that much of the time, ...
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2answers
69 views

Do normal arithmetics apply for infinitesimal variables?

In physics, we tend to use non-standard analysis, meaning we use the same arithmetics (division, multiplication, even vector arithmetics) on infinitesimal variables ($df, dx, dy, ...$) as on normal ...
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1answer
45 views

Prove that $\mathrm{fin}(^*\mathbb Q)/\mathrm{inf}(^*\mathbb Q)$ is complete.

I am trying to prove the existence of a standard map for $\overline{\mathbb Q}=\mathrm{fin}(^*\mathbb Q)/\mathrm{inf}(^*\mathbb Q)$ with $\mathrm{fin}(^*\mathbb Q)$ the set of finite numbers of $^*\...
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If $f(q)=g(q)$ for all rationals, prove that $f=g$ by nonstandard methods.

I am trying to prove that if $f,g:\mathbb R\to\mathbb R$ are ordered ring homomorphisms, then, if $\forall q\in\mathbb Q,f(q)=g(q)$, then $f=g$. Is it true ? Can you give a nonstandard proof of this ...
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195 views

Unitary operator as an infinitesimal transformation

Somebody can give me a hand with this? Given the unitary operator $U=1+i\varepsilon F$ (where $\varepsilon$ is an infinitesimal escalar), in order to prove that $F$ is Hermitean: $$UU^{+}=1$$ $$(1+...
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In hyperreal, does EVP imply IVP? Other way?

So I define those two properties as ($\mathbb{R_H}$ denotes hyperreal numbers): EVP: If $I$ is an interval and $f:I\rightarrow\mathbb{R_H}$, we say that $f$ has the extreme value property iff $f$ has ...
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3answers
155 views

What is the relation between $\varepsilon$-$\delta$ and $dy$-$dx$ notations?

For what I know, $dy$-$dx$ aren't real numbers, exist as convenient notations to capture our intuitions about infinitesimal increments, while $\varepsilon$-$\delta$ are real distances, saying that $f$ ...
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2answers
85 views

Does the transfer principle really work in both directions?

Let $ \mathscr{S}$ be a statement in a superstructure $\hat{S}$. Let $^*$ denote the transfer of an element of $\hat{S}$ via the transfer principle. The transfer principle says that $\mathscr{S}$ is ...
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1answer
103 views

Can any Neutrix be written as a countable intersection or union of internal sets? (IST)

I need help understanding a counter-example in non-standard analysis. First a few preliminaries: Def.: A Neutrix is a convex additive subgroup of the hyperreals ${}^*\mathbb R$. The two simplest ...
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1answer
152 views

How can a universe include an infinite (sum) relation

Let the universe be $ \hat{V}$, which is constructed as: $ \hat{V} := V_0 \cup V_1 \cup V_2\cup ... $ where $V_0$ is the set of primary elements and $V_{v+1} := V_v \cup P(V_v)$ So, in other words,...
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Mathematical justification for scalar product of infinitesimals

Setup: Uniform circular motion under a central force. Claim: Work done by a central force (say gravity) in 'keeping a particle in uniform circular motion' is zero. Reasoning for the claim: At any ...
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1answer
281 views

Intuition of exterior derivative as infinitesimal quantity

Consider the following funny argument: Let $d$ be the well-known exterior derivative. We know for any $k$-Form $f \in \Omega^k(M)$ we have $ddf=0$ and $d$ is a linear map. Then $df$ must already have ...