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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). 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).

A non-zero element $\varepsilon$ of an ordered field is infinitesimal if $|\varepsilon| < \frac{1}{n}$ for all $n \in \mathbb{N}$. Nonstandard analysis is analysis done in fields with infinitesimals.

There are many ordered fields which contain infinitesimals, but the most common is the hyperreal field. Denoted by ${}^*\mathbb{R}$, the hyperreal field has a subfield isomorphic to $\mathbb{R}$ and is therefore the perfect setting for formalising the arguments of Leibniz and Newton (without the need for limits). This came to fruition in the 1960's thanks to the work of Abraham Robinson.

See also .

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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 asked whether the same is true for arbitrary Cauchy…
Akhil Mathew
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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, but wonder what the properties are that the…
Jens
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How can infinite sums be defined for a non-complete hyperreal field?

The definition of integration in non-standard analysis is $$\int_a^b f(x)dx:= st\left(\sum_a^b f(x)dx \right),$$ as given by Keisler (you could also use the transfer principle for internal functions). This sum involves adding up infinitely many…
Lave Cave
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Is ${}^{*}\mathbb{Q}={}^{*}\mathbb{R}$?

This is a follow-up question of a previous one. Is it true ${}^{*}\mathbb{Q}={}^{*}\mathbb{R}$ ? It seems to me it's true because for any $x \in{}^{*}\mathbb{R}$, there is a $y \in {}^{*}\mathbb{Q}$ such that $y < x$ and vice versa, and because for…
Metta World Peace
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Ultrapower construction of ${}^{\ast \ast}\Bbb R$

Suppose, we want to add positive elements that are smaller than all positive elements in ${}^{\ast}{\Bbb R}$. One way to do this, as shown in this master’s thesis, is to construct sequences of elements in ${}^{\ast}{\Bbb R}$ that range over…
Metta World Peace
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What is $\sin{ω}$?

I am reading about hyperreal numbers defined as (to my understanding) certain equivalence classes on all sequences of real numbers. $ω$ is defined as $(1, 2, 3, ...)$, and all functions are applied element-wise. This makes sense for sequences that…
IS4
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What are hyperreal numbers?

My first encounter with hyperreal numbers was two months ago. I read a lot of articles about them, but I did not understand what and which they are, because of this: On the french wikipedia page I can read: Un nombre hyperréel x est dit …
ParaH2
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Intuition Behind The Hyperreals

I know that there are an infinite number of hyperreals. But is it true that there are only two hyperreals with standard part equal to $0$ (the "finite" infinitesimal one and the "infinite" hyperreal)? Put differently, is it wrong to view the…
user1770201
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Is Dedekind completion of ${}^{\ast}\Bbb R$ a Archimedean field?

Here's Theorem 1.2 on page 6, Martin Andreas Väth's Nonstandard Analysis(See here on googlebooks) The Dedekind completion $\overline{X}$ of a totally ordered field $X$ is a complete Archimedean field with $\Bbb{Q}_{\overline{X}}$ as the…
Metta World Peace
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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. Let $^*$ denote the transfer of an element of…
Sudix
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How are infinite sums in nonstandard analysis defined?

Since, in nonstandard one can have infinitely large numbers, I was wondering if I can assign divergent sums to them. However infinite sums are defined by taking the limit to infinity of a partial sum of the sequence. But would that mean all…
ty.
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Which non-standard analysis framework to study?

Just recently I became slightly interested in non-standard analysis. After a preliminary check at the subject there seem to be at least two relatively common ways of establishing the framework: Hyperreals constructed via a non-principal ultrafilter…
J. J.
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In what way does the nonstandard definition of microcontinuity differ from that of epsilon-delta continuity, and related quibbles

During a very cursory glance over the Wikipedia articles on non-standard calculus, I spotted the following definitions of continuity, uniform continuity and "microcontinuity", and I wonder what the actual difference is between them and their…
FShrike
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Internal set theory: proof that limited integers are standard

I'm following this pdf from Edward Nelson about internal set theory: https://web.math.princeton.edu/~nelson/books/1.pdf I'm at page 6. Only two axiom schemes have been introduced so far. The transfer principle (writing here the dual version using…
John Smith Optional
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Resources for Learning Hyperreal Numbers

I've somewhat recently discovered hyperreal numbers, but I haven't gotten the chance to thoroughly research them. What resources do you all recommend for undergrad level study of the hyperreal number line and hyperreal numbers? The "undergrad"…
HarrisonO
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