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).

A non-zero element $\varepsilon$ of an ordered field is infinitesimal if $|\varepsilon| < \frac{1}{n}$ for all $n \in \mathbb{N}$. Nonstandard analysis is analysis done in fields with infinitesimals.

There are many ordered fields which contain infinitesimals, but the most common is the hyperreal field. Denoted by ${}^*\mathbb{R}$, the hyperreal field has a subfield isomorphic to $\mathbb{R}$ and is therefore the perfect setting for formalising the arguments of Leibniz and Newton (without the need for limits). This came to fruition in the 1960's thanks to the work of Abraham Robinson.

See also .

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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 the use of the fundamental theorem. So I wrote as…
Dude156
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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 a single-variable with the subscript notation…
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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 obtained if the middle third is removed from each…
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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 $x$. If $f''(x)=0$, then the text is inconclusive. I’m…
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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 function for the length of a certain portion of…
Malice Vidrine
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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 derivative at…
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Show that there is a hyperreal number system of the same size as the real numbers?

The hyperreal number system is defined as one that contains the real numbers, satisfies the first order properties of real numbers, and contain infinitesimals. It can't be as simple as stating the reals are a subset of the hyperreals. Do I need to…
J.S
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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 fact ? If not, can you give a standard proof ?
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Axiom Schema of Restricted Comprehension and Big O notation

In this blog post about ultrafilters and nonstandard analysis, it is stated that "if we attempt to formalise this by trying to create the set $A := \{ x \in {\Bbb R}: x = O(1) \}$ of all bounded numbers, and asserting that this set is then closed…
A.G.
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Star mapping in Non-standard analysis

I'm trying to understand the star mapping in non-standard analysis in particular for the Hyperreals. I know that $*: \mathbb R\to \mathbb{^* R}$ is a mapping such that $^*(x)=^*x$ where $^*x= (x,x,x,x,...)$ which is in $\mathbb{^* R}$. In other…
user253919
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How to compute $(\int f(x) \, dx)^p$ with fractional number $p$?

It is well-known that one can say $(\int f(x) \, dx)^p = \int \prod_{i=1}^p f(x_i) \, dx_i$ if $p$ isa natural number. But what is if $p$ is a fractional ore even a real number? Is it possible to set $\int f(x) \, dx = h \sum_k f(hk)$ for…
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How much choice is needed for the transfer principle?

To construct the hyperreals via ultrapower the Boolean prime ideal theorem apparently suffices. However, to prove the transfer principle for the extension $\mathbb{R}\subset{}^\ast\mathbb{R}$ apparently a stronger version of choice is needed. Does…
Mikhail Katz
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In what sense is $\Bbb R(x)$ an "instantiation" of the hyperreals?

I'm teaching myself about hyperreal numbers. My main motivation for doing so is that they include infinite numbers, whose existence I hear disputed & doubted often as "quantifying the unquantifiable". In this YouTube video, around twelve minutes…
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What is difference between derivative in standard and non standard analysis?

I am reading the book on complex analysis by Tristan Needham. In that book he explains derivative in an intuitive way as a quantity by which dx is expanded to get dy in both complex and real number plane. But In non standard analysis, derivative is…
Rahul
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What is the hyperreal multiplicative inverse of $1 + \epsilon$, and how do we show it exists?

What is the multiplicative inverse of $1 + \epsilon$, in the ordered field of hyperreals or surreals? Simple algebra shows it must be equal $1-\epsilon+\epsilon^2-\epsilon^3+\epsilon^4...$ But how do we prove that that number exists as a hyperreal…
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