# 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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### Which branch of mathematics rigorously defines infinitesimals?

I have some trouble doing standard computations in calculus because of the notion of a differential, otherwise known as an infinitesimal, being rather ill defined, in my experience. Are there any ...
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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'...
1 vote
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### What is the use of hyperreal numbers?

For sometime I have been trying to come to terms with the concept of hyperreal numbers. It appears that they were invented as an alternative to the $\epsilon-\delta$ definitions to put the processes ...
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### What does limit actually mean?

I have been in a deep confusion for about a month over the topic of limits! According to our book, the limit at $a$ is the value being approached by a function $f(x)$ as $x$ approaches $a$. I have a ...
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### Are the hyperreal numbers separable? Can we construct a computable dense of them?

I am familiar with the construction of the hyperreal numbers $^*\mathbb{R}$ as an extension of $\mathbb{R}$ built of equivalence classes of real sequences up to equivalence with respect to some fixed ...
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### Is it a good idea to learn about hyperreals and how they are related to limits before beginning to learn about calculus?

I have some idea of what the main concepts of calculus are about but I have never actually taken a calculus class or studied myself. My general understanding is that before the concept of limits were ...
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### Is there such a thing as "hypertopology" (analogous to the hyperreals)?

The hyperreal number system adds infinities and infinitesimals, allowing Calculus to be done using these things instead of limits (sort of like when calculus was originally invented, but with rigor)....
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### Equivalence relationship defined over membership of a in halo of b. (Non-Standard Analysis, Monads, Halos)

Let's say we define an equivalence relationship such that $a \sim b \iff b\in \mu(a), \: \mu(a)$ is the halo\monad of $a$. By definition this would include all points an infinitesimal distance away ...
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### Is there a lack of rigor in the standard analysis?

Does the difficulty of defining exactly what infinitesimals and differentials are denote a lack of rigor in standard real analysis? For example, in an introductory course one may solve the ...
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### Ultraproduct construction: are finite hyperreals just a thinly disguised version of Cauchy sequences?

Periodically I've tried to wrap my head around nonstandard calculus and hyperreals, but I always thought I needed a lot more of a background in formal logic and/or set theory to understand what's ...
1 vote
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### Confusion about the model in Robert’s Nonstandard Analysis

I’m working through Alain M. Robert’s “Nonstandard Analysis”. I’m intrigued by his suggestion that his axiomatic approach, rather than adding elements to ℕ to create a nonstandard model *ℕ, “discerns ...
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### How to prove that infinitely small and large numbers can be used as measures of rates of convergence?

I read "Lectures on the Hyperreals: An Introduction to Nonstandard Analysis" Robert Goldblatt. He wrote that infinitely small and large numbers can be used as measures of rates of ...
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### Definition of arithmetic operations for hyperreal numbers

I read paper about hyperreal numbers (https://sites.math.washington.edu/~morrow/336_15/papers/gianni.pdf) I have few questions about definition of arithmetic operations. What is idea of this ...
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### Halos in non-standard analysis

Please consider this question in terms of the hyperreals. As per usual, the halo of a point $P$ is the set of all points separated from $P$ by an infinitesimal distance. Let $P$ be a point in a ...