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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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Doubly-hyper-reals? Can we include another level of infinitesimals?

Is it possible (even if there is no reason to even want to do this) to expand the hyperreal number line at each infinitesimal to insert a "second layer of infinitesimals"? Let $\epsilon$ be an infinitesimal hyperreal in the halo of $0$. Can we…
jdods
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Multiplication property of equality for infinitesimals

I am to prove this property of *$\Bbb R$: If $x \approx y$ and $u \approx v$ and $u,x$ are finite then $xu \approx yv$. My question is can I just use the transfer principle for the multiplication property of equality? Also since, $x - y \approx 0$…
Halinka
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Why is u-equivalence useful?

If $X$ is a standard set, we define, for $\alpha\in\ ^*X$, $U_{\alpha}=\{A\subset X\mid\alpha\in\ ^*A\}$. We can see that $U_\alpha$ is an ultrafilter on $X$. We thus define an equivalence relation on $^*X$ which is $\alpha\sim\beta$ iff…
fyusuf-a
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Externality of a set

In nonstandard analysis (for example in $^*ZFC$ developed there), how can one easily see that $^\sigma X$ is external if $X$ is a standard infinite set ?
fyusuf-a
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Usage of the standard part function when calculating with infinities

We know that $\lim\limits_{H \to\infty}(\frac{H+1+d}{2H-1+3d})=\frac{1}{2}$. The concept of limits is something I understand perfectly fine. Though, in non-standard analysis, we have a function called the standard part function, st( ). For…
Andreas
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Is there a bijective map from $\mathbb{R}$ to ${}^{*}\mathbb{R}$?

Is there a one-to-one correspondance between the real numbers and the hyperreal numbers?
user2683246
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Hyperreal star mapping isomophism

I've been reading through Goldblatt's book on the Hyperreals. And the star mapping is defined to be: *r=[r]=[(r,r,r,...)]. Where r is a real number, and [r] denotes the equivalence class of the constant sequence r. It then goes on to say that the…
user253919
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Set-theoretic Properties of a Universe

I would like to show that if {$A_{i}$: i$\in$I} $\subseteq$ $A$ $\in$ $\mathbb{U}$, then $\bigcup_{i \in I}$$A_{i}$ $\in$ $\mathbb{U}$, where $\mathbb{U}$ is a universe and the capital $A's$ are all sets (p. 162-163, Lectures on the Hyperreals,…
theperennialstudent
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Why is a *-finite standard set a finite set?

In internal set theory, why is it that if there is a bijection between $x$ standard and $n\in\mathbb{N}$, then $n$ is necessarily standard ?
fyusuf-a
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Non-Standard Analysis Solution to Differential Equations

The non-standard analytical solution to the derivative of simple functions such as $x^2$ is well-known... Is there a similar solution for differential equations such as the heat equation or a simple ODE?
user14685
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Problem from Keisler infinitesmal calculus book.

I'm going to Keisler's "Elementary Calculus, an Infinitesimal approach" , and I'm stuck on a problem: Given that $H$ is a positive infinite term, determine whether the given expression is infinitesimal, finite (but not infinitesimal), or…
David Davidson
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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 definition? Why can we use componentwise operations? What…
Mike_bb
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Arguments illustrating advantage of hyperreal definition over sequential one

As is well known, fields of hyperreals $\mathbb R^*$ can be formed by an ultrapower construction, as quotients of the space of sequences of real numbers by a nonprincipal ultrafilter. In fact, some arguments using $\mathbb R^*$ can be reformulated…
Mikhail Katz
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Is it true $\operatorname{st}(^\ast A) = \overline{A}$, when $A$ is a bounded subset of $\mathbb{R}$?

Given a bounded subset of $\mathbb{R}$, let $\operatorname{st}(^\ast A)$ be the set of the standard parts of all elements in $^\ast$-transform of $A$, $\overline{A}$ is the closure of $A$.Is it true $\operatorname{st}(^\ast A) = \overline{A}$? It…
Metta World Peace
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Another question on the Hyperreals - regarding the monad at infinity...

I'm interested in exploring whether there is a monad at infinity. I guess we would define the infinitesimal space surrounding infinity as "A number that is greater than any Real number, but smaller than infinity". I can see some problems with it…
Spanki
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