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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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Using hyperreals to express the interval of uniform convergence of $x^n$ in a closed form

The sequence of functions $f_n: E \rightarrow \mathbb{R}$ where $f(x) = x^n$ converges uniformly to $ g(x) = 0 $ for $E = [0,1-\varepsilon]$, $\forall \varepsilon > 0$. Yet, it doesn't converge ...
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Using hyperapproximations of not everywhere derivable functions to find local extrema

Let's say I have the function $f(x):=\max(f_1(x),..,f_n(x))$ and want to know its extreme values. Then I could approximate it using $$g(x):=(\sum_{i=1}^n f_i(x)^h)^{1/h}$$ where $h\in\mathbb{^*N\...
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What hyperreal functions satisfy this condition involving positive infinitesimals?

Let $g$ be a function from the hyperreal numbers to the hyperreal numbers which is an extension of a real function in the canonical way, satisfying the condition that for all real numbers $x$ and all ...
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Lacunary Sequence with positive upper density

A sequence $\{n_{k}\}$ is lacunary if $\forall$ k $\in\mathbb{N}$, $\frac{n_{k+1}}{n_{k}}$ $\geq\lambda\gneq$ 1. And natural upper density of a set S $\subset\mathbb{N}$ is defined as limsup$_{n\...
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It would be possible to define an uniform distribution on $\Bbb N$ using infinitesimals?

In standard analysis it is clear that it is impossible to define an uniform probability distribution on $\Bbb N$ because there is no constant $c\in\Bbb R$ such that $\sum_{k=1}^\infty c=1$. Using ...
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Can the real numbers be embedded into all non-Archimedean real closed fields?

Every Archimedean real closed field is isomorphic to a subfield of $\mathbb{R}$. But I’m wondering if something in the opposite direction is true. Suppose that $F$ is a non-Archimedean real closed ...
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Finding functions $f:\Bbb{R}\to\Bbb{R}$ for which $\sum_{x\in\Bbb{R}}f(x)$ converges

A while ago, a friend asked me "for which functions $f:\Bbb{R}\to\Bbb{R}$ does the sum $\sum_{x\in\Bbb{R}}f(x)$ converge?", and a few days ago I completed a sketch of a proof that the sum converges ...
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Hyperfinite intervals are uncountable but nonstandard models of Peano arithmetic can be countable?

My understanding is, by Lowenheim-Skolem I can find a countable nonstandard model of Peano Arithmetic. On the other hand, I have just encountered the following argument: For $\alpha \in {}^*\mathbb{N}...
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In Non-standard analysis, is the number of natural numbers a hyperreal number?

In Non-standard analysis, is the number of natural numbers a hyperreal number? In other words, if $H$ is the hyperreal infinite unit, does the sum $\sum\limits_{n=1}^H 1$ yield the number of natural ...
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Is this elementary, nilpotent-free approach to automatic differentiation strong enough for real analysis? How similar is it to Newton's system?

This is a sequel to this question: Is the theory of dual numbers strong enough to develop real analysis, and does it resemble Newton's historical method for doing calculus? The ring of "dual ...
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Non Standard Prime number

Problem : Prove that for any $m\in^* \mathbb{N}$ there exists $n \in ^* \mathbb{N}$ such that $n\geq m$ and $n$ is prime . My Attempt : If n is prime, we can write as : $( \forall m \in ^*N)(m|n \...
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Prove the nonstandard part is in a closed and bounded subset of $\mathbb{R^2}$

problem : Let K be a closed and bounded subset of $\mathbb{R^2}$ and $^*K \subset ^* \mathbb{R^2}$ it's * - extension. Prove that for any $x \in ^* K$ , it's standard part $st(x) \in K$ My Attempt ...
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Example of an internal function which is $ \epsilon - \delta - continuous$ but not $ s-continuous $

Problem: I was looking for a function which is $ \epsilon - \delta - continuous $ but is not $ s-continuous $ at some point. Here are the definitions : $ s-continuous $ : An internal function $f \...
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$s$-continuity in NonStandard Sense

Problem: Prove that if a standard function $f$ is continuous at a point $x \in \mathbb{R}$, then it's *-extension $^*f$ is $s$-continuous at the point $x$. The definition of the $s$-continuous: An ...
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How do non-principal ultrafilters 'know' the key elements of an infinite series when establishing the equivalence for hyperreals.

I'm currently reading about non-principal ultrafilters and their relationship with defining the hyperreals, I am struggling to really get my head around the notion that a non-principal ultra filter ...
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What is the Galois group of the Hyperreal Numbers?

The Galois Group of $\mathbb{R}$ as an extension of $\mathbb{Q}$ is trivial. That’s because any field automorphism of $\mathbb{R}$ is order preserving, so since $\mathbb{Q}$ is dense in $\mathbb{R}$, ...
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Bound for roots of a polynomial with coefficients in a non-Archimedean valued field

Is there any bound for the valuation of the roots of a given polynomial with coefficients in an algebraically closed non-Archimedean valued field? Any reference or insight would be appreciated.
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Is there any real difference between 'unbounded' and 'bounded by infinity'?

When treated as a limit, 'infinity' is essentially synonymous with 'never' - saying that 'a process terminates after an infinite amount of time' means exactly the same thing as 'the process does not ...
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Does it make sense to talk about rationality and countability when dealing with nonstandard quantities?

I have two closely related questions: Firstly, suppose that I can find two hyperintegers $P$ and $Q$ s.t. $\frac{P}{Q}=\sqrt{2}$. Obviously, both $P$ and $Q$ lie in an extension of the integers. ...
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Is there an element of the hyperreals minus the reals that isn't a hyperirrational?

I was looking to see if there exist elements in *R-R that aren't in *Q? The ideal answer would offer some different ways to understand and see this as I still lack an intuition for these sorts of ...
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Can we define the unordered sum of a set of hyperreal numbers?

If $\{a_i:i\in I\}$ is a set of nonnegative real numbers, then the unordered sum $\sum_{i\in I}a_i$ is defined as $\sup \Bigl\{ \sum_{i\in A}a_i\,\big| A \text{ finite, } A \subset I\Bigr\}$. And $\...
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i want reference about the method(below) of describing Infinity… [closed]

let think x is real number, then \begin{align} x&=x\\ x&=(x/2)+(x/2), \\ x&=(x/3)+(x/3)+(x/3), \\ x&=(x/4)+(x/4)+(x/4)+(x/4), \\ &\vdots \\ x&=(x/n)+(x/n)+(x/n)+ \cdots +(x/n), ...
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Discrepancy between calculus methodologies - Is it significant?

Two of the ways of doing calculus with algebra are non-standard analysis NSA and smooth infinitesimal analysis SIA. NSA has a technique called 'taking the standard part' which neglects incremental (or ...
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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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1answer
235 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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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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1answer
100 views

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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1answer
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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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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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1answer
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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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1answer
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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 $x$...