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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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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 to get an introduction to non-standard analysis?

I am a high school rising senior with an interest in mathematics, and I will be taking AP calculus AB next year. I have been doing research online, and recently came across hyperreal numbers, which I ...
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Difference between $\mathrm {d} x$ and $\delta x $

Are $\mathrm {d} x $ and $\delta x $ the same mathematical object from the point of view of the nonstandard analysis?
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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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Practical applications of non-standard probability

Recently I read a paper by Benci et al. describing an alternative to Kolmogorov's construction of probability where the probability measure $P$ takes values in a non-Archimedian field and we have $P(A)...
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Fourier transform and non-standard calculus

The Fourier transform is not as easily formalizable as the Fourier series. For example, one needs to introduce tempered distributions to define the Dirac delta-function. Also, it is impossible to view ...
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Is ordering done component wise in the hyperreals?

In the ultrafilter construction of the hypperreals, is ordering done component wise? I.e. if $a,b$ are hyperreal numbers then $a = [x_n]$ and $ b = [y_n]$ where $[x_n]$ and $[y_n]$ are equivalence ...
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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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Is this proof of the four color theorem for infinite graphs legit?

So you got an infinite planar graph $G$. I will prove that it is four colorable. So, construct an infinite number of statements about graphs: The first is "is four colorable" Next, for each vertex $...
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Nonstandard analysis, Lie groups and universal enveloping algebras

The idea of nonstandard analysis is to combine finite quantities with infinitesimals. And, back in the day, Lie algebras were roughly considered the "infinitesimal elements" of Lie groups. Say we want ...
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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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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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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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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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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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non-commutative infinitesimal extension of $\mathbb R$

Background: The transfer principle in nonstandard analysis implies that any nonstandard model of the reals is a commutative (for additively and multiplicatively). It is also well-known that the set $\...
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Size of a geometric point

It is well known that the geometric points do not have any length, area, volume, or any other dimensional attribute, also geometric object (for example "line") is made up of a infinite number of ...
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Transfert principle of a conservative extension of ZFC

In the following paper, there is a theory called $^*ZFC$ in the language $(^*,\in)$. The *-map is (more or less) defined on the Von Neuman hierarchy $S$ and verifies the following axiom schemata true ...
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Concurrent relation and enlargement

The superstructure $V({}^{\ast}X)$ [with respect to a monomophism $\ast : V(X) \to V({}^{\ast}X)$] is called an enlargement of $V(X) $ if for each set $A \in V(X)$ there is a set $B \in {}^{\ast} \...
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Law of iterated expectation in an algebraic axiomatization of probability theory

In the second chapter of Radically Elementary Probability Theory [PDF], Edward Nelson gives an axiomatization of probability theory based on algebras of random variables, briefly discusses a couple of ...
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Berkovich analytification of Robinson fields

Let $\rho$ be an infinitesimal and let $^\rho \mathbb{R}$ be a (non-archimedean) Robinson valued field. Is there anything known about the topological structure of $\mathbb{A}^{1,an}_{^\rho \mathbb{R}}$...
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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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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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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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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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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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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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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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Value groups of ultrapowers of $\mathbb{R}$.

$\DeclareMathOperator{\Noo}{No}$ $\DeclareMathOperator{\ee}{e}$ Let $U$ be a non principal ultrafilter on $\mathbb{N}$, let $^*\mathbb{R}$ denote the corresponding hyperreal field. Let $^*\Gamma$ ...
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digits of pi, the nature of its randomness, immunity to place selections (Von mises) reichenbach partial limits

The digits of $\pi$ uniform distribution non-standard analysis, and its partial limit I was wondering whether there has been any investigation into the distribution of the decimals of $\pi$ using non-...
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Hyperreal star mapping isomophism

I've been reading through Goldblatt's book on the Hyperreals. And the star mapping is defined to be: *r=[r]=[(r,r,r,...)]. Where r is a real number, and [r] denotes the equivalence class of the ...
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Should we say $0.\bar9$ *can* equal $1$ also in the hyperreals?

Consider the sequence $R_n$ of repunits, defined as $\displaystyle\frac{10^n-1}{9}$. We have $$\frac{R_{n+1}}{R_n}=\frac{9}{9} \frac{10^{n+1}-1}{10^n -1}=\frac{10^{n+1}-1}{10^n -1}=\frac{\overbrace{99\...
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Nonstandard analysis and integral transforms

Can integral transforms be evaluated without limits(i.e Laplace transform) such as in non standard analysis? Can the improper integral be bounded by a hyperreal number? I am not very familliar with ...
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Hyperreals — convergent sum sets

Consider the set $X=\{1,\frac12,\frac13,\dots\}$, and let $X^*$ be its hyperreal extension, so that $X^*=\{\frac1n:n\in\mathbb N^*\}$. Call a subset $A\subset X$ convergent if the sum of elements in $...
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Set-theoretic Properties of a Universe

I would like to show that if {$A_{i}$: i$\in$I} $\subseteq$ $A$ $\in$ $\mathbb{U}$, then $\bigcup_{i \in I}$$A_{i}$ $\in$ $\mathbb{U}$, where $\mathbb{U}$ is a universe and the capital $A's$ are all ...
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Limit points in nonstandard analysis [solved]

Let $A\subseteq\mathbb{R}$, $p\in\mathbb{R}$. I proved that the following are equivalent: $\exists\left(x_{n}\right)_{n\in\mathbb{N}}\subseteq A\cap\left\{ p\right\} ^{c}$ such that $x_{n}\rightarrow ...
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$\approx$ and $\ll$ for different-order infinitesimals

This seems like a pretty basic question, but I've been searching around and haven't come across the answer. Consider two infinitesimal numbers, $\epsilon$ and $\epsilon^2$. On the one hand, it would ...
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Is there a source linking Robinson's work in wing theory with his theory of infinitesimals?

Abraham Robinson worked in applied mathematics for several decades. MathSciNet lists 12 articles by Robinson in wing theory. His production included the book Robinson, A.; Laurmann, J. A. Wing theory....
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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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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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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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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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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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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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Generalizing mathematical induction by differential forms.

Suppose we want to prove a predicate in the $$HyperReal$$ Numbers,by (weak)induction,for example,for all: $$x\in\mathbb H,x^2>=0$$ Would we Proceed as follows?: $$P(dx)= dx^2>0,$$ $$P(x)=x^2>=...
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existence of unlimited hypernaturals

How can we prove that the extension *$\mathbb{N}$ of $\mathbb{N}$ contains unlimited elements? I have read a proof that shows that the only limited elements of *$\mathbb{N}$ are the standard ...
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Nonstandard analysis: transfering a simple sentence

If $A$ is an infinite subset of $\mathbb{N}$, show that *$A$ contains aritrarily large unlimited elements. From "Non-standard Analysis for the Working Mathematician," p. 22 : "there is a Skolem ...