# 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).

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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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### Noetherian Rings in Nonstandard Framework

I have been trying to go through some algebraic geometry using the nonstandard framework. Noetherian rings are of course fundamental in this subject and it is characterized by the attribute that every ...
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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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### Why isn't Riemann sum infinitesimal in nonstandard analysis?

I am trying to learn nonstandard analysis from Keisler's book. In the integration chapter it feels like the use of the Transfer Principle is some kind of magic that just requires us to believe results ...
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