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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).

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Ultraproduct of a Function versus Functions of Ultraproducts

Let $G$ be a group (or even a set for our purposes here) and consider functions from $G$ to $\mathbb{R}$. Now after choosing a non-principal ultrafilter of $\mathbb{N}$, we can construct ultrapowers $^...
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Is $e$ transcendental when working with hyperreal numbers?

When working with strictly real numbers, there are a number of proofs that $e$ is transcendental. However, when dealing with non-standard analysis, one can express $e$ as $(1 + \frac{1}{H})^H$ for any ...
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Intuitive Understanding of Order of Hyperreals

I'm trying to understand the ultrapower construction of the hyperreal numbers as described in Wikipedia. The motivation is given as the surprising ability to find a total order of sequences of real ...
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206 views

Does absolute infinity invoke Cantor's Paradox? [closed]

Let $\mathcal{P}(X)$ denote the powerset of $X$, $\mathcal{P}^2(X)=\mathcal{P}(\mathcal{P}(X))$, and $\mathcal{P}^n(X)=\mathcal{P}(\mathcal{P}^{n-1}(X))$; $\mathcal{P}^0(X)=X$. It is trivial to show ...
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Does every standard set have a standard cardinality?

Add a new primitive one place predicate symbol $``std"$ denoting "standard" to the language of $\text{ZF}$, now iff we add an omega rule to $``\text {ZF - Infinity + every set is finite}"$ formulated ...
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How to do this problem without using infinitesimal?

A rod of linear charge density a of length h, What will be the electric field at an axial point at a distance x from end of the rod (the end at which the origin is chosen for defining charge ...
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Vacuous truths in Superstructure approach to Nonstandard Analysis

Good evening everybody, at the moment I'm studying non-standard analysis, specifically the superstructure approach to it. This approximately works as described in chapter 3 of http://people.dm.unipi....
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Solving Indefinite integral without FTC

While I was watching some physics lectures, I saw a professor write down the $\int r*dr$. The writing multiplication sign (normally just implied) prompted me to attempt to solve this integral without ...
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Question related to the use of the axiom of choice in real analysis, nonstandard analysis, and constructive proofs.

So, as far as I'm concerned, real analysis depends quite largely on some weak variants of the axiom of choice (such as the axiom of countable choice), and there seems to be no controversy surrounding ...
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1answer
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Notation Question for “Generalization of L’hopital’s Rule” PDF by V. V. Ivlev and I. A. Shilin

Within this brief text starting on page 2, the author(s) have a habit of employing a sort of "mixed" notation of differentiation, mixing the "prime marks" of Lagrange's notation for differentiation of ...
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Resources for Learning Hyperreal Numbers

I've somewhat recently discovered hyperreal numbers, but I haven't gotten the chance to thoroughly research them. What resources do you all recommend for undergrad level study of the hyperreal number ...
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Where can I find a proof that nonstandard analysis is a conservative extension of standard analysis?

Willie Wong's answer in this post claims that nonstandard analysis (NSA) is a "conservative extension" of standard analysis (SA), meaning that every provable statement in NSA that can be interpreted ...
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Intuition behind the Idealization Axiom of Internal Set Theory

Wikipedia describes the idealisation axiom as follows: "The statement of this axiom comprises two implications. The right-to-left implication can be reformulated by the simple statement that elements ...
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Improper integral over the rationals

Question: Suppose that I wish to integrate a function over the natural numbers. How could I do this? Answer: Consider the definite integral $\int_a^bf(x)\ dx$. If we consider this as the 'area ...
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Intuition for nonstandard analysis from limits conception

I'm trying to gain an intuition for the use of nonstandard analysis over the limit approach. Traditionally the motivation for derivatives is that the derivative of a point $(x,f(x))$ for some ...
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Cardinality, Logarithms, and Hyperreals

Take some infinite hypernatural number, $M$, and consider the integers (finite and infinite) less than or equal to $M$. There are uncountably many. Then consider $\log_2 M$. Is there a straightforward ...
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1answer
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The value of the infinitesimal in integral doesn't matter?

I am studying calculus by the infinitesimal approach using "Elementary Calculus: An Infinitesimal Approach" textbook. in page 187, the author proved that the value of the infinitesimal we integrate ...
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Definite integrals using infinitesimals in Keisler's book

Encouraged by this question I ask a question concerning Keisler's treatment of definite integrals using infinitesimals. He starts with a continuous function $f$ defined on a closed interval $[a, b] $ ...
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Loeb Measure construction in axiomatic approach to NSA

Hello dear StackExchange, in my upcoming bachelor's thesis, I plan to present an overview of some topics in nonstandard analysis, including some higher applications, like Loeb Measures (among other ...
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Trouble understanding how the Transfer Principle is applied for the Extreme Value theorem.

I am reading Keisler's Elementary Calculus (which can be downloaded here). I am having trouble understanding his proof sketch of Extreme Value Theorem and how he is applying the Transfer Principle. ...
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1answer
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What is $t_k$ in this context?

I am reading this paper which provides an argument for factoring $\sinh$ and $\sin$ using infinitesimals as Euler did but in a more rigorous way. On the bottom of the page labeled 66 it states: ...
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1answer
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In non-standard analysis, should we necessarily consider derivative as slope between two infinitesimally apart points?

For standard analysis, an answer (in the concept of limit) is here. However, when dealing with infinitesimals in non-standard analysis, that limit technique will not work. So in non standard ...
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What does a hyperreal version of the Cantor Set look like?

I would like to construct a hyperreal version of the Cantor set. Let $X_0$ be the interval $[0,1]$ in the hyperreal line, and for any $n$, let and let $X_{n+1}$ be the set of hyperreal numbers ...
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Norm on hyperreals ${^*\mathbb R}$ using the ultrafilter construction

Suppose we construct the hyperreals by fixing a free ultrafilter $\mathcal F$ – formalizing the idea of "large subsets of $\mathbb N$" – and defining an equivalence relation between two real sequences ...
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Completing the space of series so there is a slowest converging series

It is well known that there is no slowest converging infinite series (see e.g. here). But there is also no largest rational number whose square <=2. Once we complete the rationals to the reals, ...
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Canonical hyperreal numbers

The hyperreal numbers are constructed by any free ultrafilter. We know that we can't exhibit a concrete example of a free ultrafilter on natural numbers (see here). Is it possible to give a canonical ...
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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
166 views

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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1answer
51 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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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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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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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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1answer
52 views

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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1answer
55 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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138 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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1answer
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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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2answers
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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
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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
233 views

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 ...