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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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Euler and infinity

What do people mean when they say that Euler treated infinity differently? I read in various books that, today, mathematicians would not approve of Euler's methods and his proofs lacked rigor. Can ...
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1answer
631 views

A layman's motivation for non-standard analysis and generalised limits

Disclaimer: My apologies for making such a long question. The question is possibly also rather specific, but I hope that (some parts of) it might be useful in general. Background: I have recently ...
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2answers
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Nonstandard extension of a function with a limit

Question 1. Let $g : \mathbb{R} \to \mathbb{C}$ with $g(y) = \lim_{x \to \infty} f(x,y)$, where $f : \mathbb{R}^2 \to \mathbb{C}$. Is it correct that the nonstandard extension $^*g$ will have $x \in \...
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3answers
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What is infinity divided by infinity?

This should be a simple question but I just want to make sure. I know $\infty/\infty$ is undefined. However, if we have 2 equal infinities divided by each other, would it be 1? And if we have an ...
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1answer
251 views

A maximal system of hyperreal numbers

Let $( \mathbb{R}^\mathbb{N}/\mathcal{U} )_{\mathcal{U}\in\beta\mathbb{N}}$ be the set of all the hyperreal number systems, does there exist a set $\mathbb{X}%$ and embeddings $i_{\mathcal{U}}: \...
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2answers
255 views

Is there more than one infinitessimal among the hyperreal numbers

Take $\mathbb{H}=\mathbb{R}^\mathbb{N}/\mathcal{U}$, where $\mathcal{U}$ is some ultrafilter. Questions: Are there more than one independent infinitessimal in this field. This means $\epsilon_1 > ...
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3answers
381 views

Numbers between real numbers

I wonder if there can be numbers (in some extended theory) for which all reals are either smaller or larger than this number, but no real number is equal to that number?! Is there some extension of ...
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0answers
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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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5answers
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What exactly is nonstandard about Nonstandard Analysis?

I have only a vague understanding of nonstandard analysis from reading Reuben Hersh & Philip Davis, The Mathematical Experience. As a physics major I do have some education in standard analysis, ...
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3answers
258 views

Superstructure with sets as atomic/base entities

This question arose while learning nonstandard analysis. The superstructure $V(X)$ of a nonempty set $X$ is defined recursively: $$\begin{align*}V_0(X) &= X \\ V_{i+1}(X) &= V_i(X) \cup P(...
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1answer
727 views

Reference request: preparation for learning a little smooth infinitesimal analysis?

I'm interested in learning a little smooth infinitesimal analysis. There is a free book by Kock: Smooth Differential Geometry, http://home.imf.au.dk/kock/ . As I dive into it, I feel that I'm not ...
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4answers
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Complex numbers and Nonstandard Analysis

A finite hyperreal number $r$ is a number defined as a sum of a real number and an infinitesimal number $\omega$: $$r=a+\omega$$ Do you know if is it possible (and useful) to define a complex number ...
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1answer
955 views

Non-ZFC set theory and the hyperreals: problem solved?

The reals are the unique complete ordered field. The hyperreals $\mathbb{R}^*$ are not unique in ZFC, and many people seemed to think this was a serious objection to them. Abraham Robinson responded ...
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1answer
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Simplify _Elementary Calculus_ section 1.6 problem 33

Once again, I'm trying to simplify an expression from Elementary Calculus with hyperreals. Given that $H$ is infinite, compute the standard part of: $$\frac{\sqrt{H+1}}{\sqrt{2H}+\sqrt{H-1}}$$ The ...
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2answers
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Simplify _Elementary Calculus_ section 1.6 problem 25

Let's try this again. We're still on problem 25 in section 1.6 of Elementary Calculus. $$\frac{3-\sqrt{c+2}}{c-7}$$ My first thought is (again) to multiply by $3+\sqrt{c+2}$: $$=\frac{(3-\sqrt{c+2})...
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1answer
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Simplify _Elementary Calculus_ section 1.6 problem 25 (incorrect transcription)

Given $c\neq7$ and $st(c)=7$, simplify $$\frac{3-\sqrt{c+2}}{\sqrt{c-7}}$$ My inclination, based on one of the examples in the book, is to multiply by $3+\sqrt{c+2}$, yielding: $$\frac{(3-\sqrt{c+2})...
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3answers
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When are complex numbers insufficient?

Rationals can't solve $x^2=2$, and reals can't solve $x^2=-1$. Is there any problem that cannot be solved by complex numbers but can be solved by non-standard numbers? Every polynomial with ...
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2answers
248 views

Convergence of 1/2^(1+infinitesimal)

If $a$ is an infinitesimal or a non-standard number or a number strictly larger then 0 and smaller then all positive real numbers. What is $\sum_{n=1}^{\infty} \frac{1}{2^{1+a}} $ $\lim_{x\to \...
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4answers
482 views

Why adjoining non-Archimedean element doesn't work as calculus foundation?

Consider the smallest ordered field that contains R and does not satisfy the Archimedean property. I assume this is a much simpler construction than ultrafilters and other big caliber artillery used ...
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1answer
2k views

Cardinality of the set of hyperreal numbers

What is the cardinality of the set of hyperreal numbers?
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7answers
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Is non-standard analysis worth learning?

As a former physics major, I did a lot of (seemingly sloppy) calculus using the notion of infinitesimals. Recently I heard that there is a branch of math called non-standard analysis that provides ...
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1answer
685 views

Does every Cauchy net of hyperreals converge?

This came up in a discussion with Pete L. Clark on this question on complete ordered fields. I argued that every Cauchy sequence in the hyperreal field is eventually constant, hence convergent; he ...
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20answers
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Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?

In the book Thomas's Calculus (11th edition) it is mentioned (Section 3.8 pg 225) that the derivative $\frac{\textrm{d}y}{\textrm{d}x}$ is not a ratio. Couldn't it be interpreted as a ratio, because ...