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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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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 hyperreal numbers also if it is not possibile to…
asv
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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, there are sets that fall outside of the kingdom…
Sudix
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Nonstandard Natural Numbers

Let $n \in \mathbb{N}$ and consider the set of nonstandard natural numbers $^*\mathbb{N}$ in sense of nun standard analysis. I want to show that for each $m \in (^*\mathbb{N}) \backslash \mathbb{N} $ we have $n <^* m$ where $<^*$ is the extension…
user267839
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Does the transfer principle really work in both directions?

Let $ \mathscr{S}$ be a statement in a superstructure $\hat{S}$. Let $^*$ denote the transfer of an element of $\hat{S}$ via the transfer principle. The transfer principle says that $\mathscr{S}$ is true if and only if $ ^*\mathscr{S}$ is true. Let…
Sudix
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Ultrapower Construction of Hyperreal System

I think I understand the ultrapower construction of hyperreals. Given a free ultrafilter $\mathcal{U}$ (take $\mathcal{U}\subset\mathcal{P}(\mathbb{N})$) for simplicity, then the hyperreal system is the ultrapower of $\mathbb{R}$ associated with…
Shuchang
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Does the extension of a real function always agree on R?

I was reading a nonstandard analysis textbook, which says that any function $f: \mathbb R \rightarrow \mathbb R$ can be extended to a hyperreal function $f^*: \mathbb R^* \rightarrow \mathbb R^*$ such that $f$ and $f^*$ agree on $\mathbb R$. Does…
jmc442_coht
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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 convergence. It's intuitively clear for me. But how to prove this in…
Mike_bb
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Zero divisors in the hyperreal numbers

I am currently reading this introduction to hyperreal numbers. On the first page, to illustrate the problem with just taking hyperreal numbers to be sequences of reals, the following example is used: $$ (0,1,0,1,...) \cdot (1,0,1,0,...) =…
tolUene
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Internal subsets of nonstandard extensions.

I am studying the first chapter of L. O. Arkeryd et. al: Nonstandard Analysis. Theory and Applications. There it is shown that for the multiset $(\mathbb{X}, \mathcal{P}(\mathbb{X}))$ it is possible to find a nonstandard extension $(^*\mathbb{X},…
Christian
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Nonstandard extension of nonstandard hull

Let $(X_i, d_i, e_i)$ be a sequence of pointed metric spaces, let $\prod _\omega (X_i, d_i, e_i)$ be the ultraproduct of said spaces with respect to a nonprincipal ultrafilter $\omega$, and let $(\hat{X}, \hat{d})$ be the nonstandard hull of $\prod…
pseudocydonia
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Is it possible to construct a Loeb measure for $\{\epsilon: \epsilon\in[0,1], \epsilon \text{ infinitesimal}\}$?

Fix an infinite number $\omega$, and a finite number $n$. Let $$ \Omega = \left\{\frac{k}{\omega}: k=1,2,\ldots,n\right\}, $$ then $\Omega$ is a subset of the infinitesimal in $[0,1]$. Is it possible to construct the Loeb measure associated to $$ X…
user14108
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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 curved 2D surface $\sigma$. Every point in the halo…
Jimmy Rankerman
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Applying ultrapower construction to the field $\mathbb {Q} $ of rationals

Can the ultrapower construction (used for extending the field of real numbers to get the field of hyperreals) be applied to the field $\mathbb{Q} $ of rational numbers? In my view it should be possible to start with the set $\mathbb{Q} ^{\mathbb…
Paramanand Singh
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How can a universe include an infinite (sum) relation

Let the universe be $ \hat{V}$, which is constructed as: $ \hat{V} := V_0 \cup V_1 \cup V_2\cup ... $ where $V_0$ is the set of primary elements and $V_{v+1} := V_v \cup P(V_v)$ So, in other words, the universe is pretty much the ordinary…
Sudix
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non-commutative infinitesimal extension of $\mathbb R$

Background: The transfer principle in nonstandard analysis implies that any nonstandard model of the reals is a commutative (for additively and multiplicatively). It is also well-known that the set $\beta(\mathbb R)$ of all ultrafilters on $\mathbb…
Ittay Weiss
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