# 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).

63 questions
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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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### 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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### 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 ...