# 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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### Internal Induction and the Overflow Principle

On page 129 of Goldblatt's Lectures on the Hyperreals, I'm trying to understand the discussion between Internal Induction (Thm 11.3.2) and the Overflow Principle (Thm 11.4.1). For context: Theorem 11....
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### usage of Leibniz notation for things like $\frac{d^2y}{dt^2}$ and $\frac{dy'}{dy}$

I've read the other posts on this site about whether you can treat $\frac{dy}{dt}$ as a fraction. There are a lot of conflicting opinions, but many seem to be saying that treating it as a fraction ...
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### Automorphism on the hyperreals

A field isomorphism $\phi:F\rightarrow G$ is a bijection such that (i) $\phi(x+y)=\phi(x)+\phi(y)$ and (ii) $\phi(xy)=\phi(x)\phi(y)$, where $F$ and $G$ are ordered fields. If $F=G$, then $\phi$ is ...
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### How to define infinite sums in nonstandard analysis for general vector spaces?

I'm studying nonstandard analysis and I came across the concept of infinite sums in this framework. I understand that in standard analysis, infinite sums are typically defined through the concept of ...
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### Is there distance (metric) between two points on the hyperreal line in Nonstandard analysis?

As is known, there is distance between two points on the real line. It's obvious. But if we imagine hyperreal line (see pic.) then we'll have infinitesimals and their corresponding points. For ...
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### Is this a non-standard extension?

I started reading Henson's "Foundations of Nonstandard Analysis. A Gentle Introduction to Nonstandard Extensions" a couple of days ago, and I am a bit confused about something. Let $F$ be an ...
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### Approaches to nonstandard measure theory

So I don't really know anything about measure theory or nonstandard measure theory or nonstandard analysis or anything like that, but my friend who is taking measure theory next semester told me about ...
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### Why do we use symbol ";" in extended decimal notation for hyperreal numbers?

There is extended decimal notation for hyperreal numbers which was developed by A.H. Lightstone: $d.d_1d_2d_3...;...d_{H-1}d_{H}d_{H+1}...$ Why do we use symbol ";" in this notation? Thanks.
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### Is this a valid basis for a transfinite number system?

I've been curious about transfinite number systems including infinite ordinals, hyperreals, and surreal numbers. The hyperreals in particular seem particularly appealing for introducing a hierarchy of ...
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### How is it possible that graph of function $y=x^2$ doesn't coincide with its infinitesimally small segment of tangent line in Non-standard analysis?
As is known, graph of function $y=x^2$ touches the axis $X$ not only at point $(0,0)$. There is infinitesimally small segment of tangent line (at the point $(0,0)$) that coincides with this graph (it'...