Non-standard 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).

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How can one replace a set with a base set in nonstandard analysis?

In the superstructure approach of nonstandard analysis, one builds the superstructure $V(X)$ as ($X$ being a set) : $V_0(X)=X$ For $n$ in $\mathbb{N}$, $V_{n+1}(X)=V_n(X)\cup\mathcal{P}(V_n(X))$ ...
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Modified division, hyperreal numbers and transfinite derivatives

Suppose we are shooting from a cannon and measuring the speed of the projectile. The shorter period of time it takes for the projectile to reach the target, the faster it is. If the projectile hits ...
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Best mathematical object to describe speed

Suppose we want to describe speed of a particle, moving between two points on a real line: ---------------$0$----------------$1$----------------- If the particle starts at $0$ at $t=0$ and moves ...
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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 ?
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Hyperreals - is there a “boundary” between convergent and divergent series?

Hypearreals are equivalence classes of sequences of real numbers. Is there a hypperreal number $ h $ such that for every convergent (real) series $ \sum a_n $ we have that the equivalence class of ...
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Can hyperreal numbers be studied 'naively', like surreal numbers?

Surreal numbers are constructed 'explicitly', many of them have labels on which one can do arithmetic, an extension of the natural ordinal arithmetic on Cantor's normal forms, and their theory is ...
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$\approx$ and $\ll$ for different-order infinitesimals

This seems like a pretty basic question, but I've been searching around and haven't come across the answer. Consider two infinitesimal numbers, $\epsilon$ and $\epsilon^2$. On the one hand, it would ...
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A basic, basic question about the definition of a superstructure

A superstructure $V(X)$ over a set $X$ is defined as: $V_0(X) =X$ $V_{i+1}(X) = V_i(X) \cup P(V_i(X))$ $V(X) = ⋃_{i=0}^{\infty}V_i(X)$ My question is in regard the line item 2, where the set ...
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There is no smallest infinitely large prime

I'm reading Kunen's Foundations of Mathematics and trying to solve Exercise II.16.19, which constructs an elementary extension of the reals as an ordered field, and asks the reader to prove various ...
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34 views

Problem reconciling this proof with what I know about the Reals.

I'm currently reading a book on non-standard analysis. The proof states that there is an infinite integer greater than all reals. How is this possible? I thought that the reals contained the integers? ...
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275 views

l'Hopital's questionable premise?

Historians widely report that l'Hopital's 1696 book Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes contains a questionable premise expressed by an equation of type $x+dx=x$ ...
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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 ...
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Is the standard part function another devil's staircase?

The devil's staircase or Cantor function is an awesome function that increases value but has derivative zero everywhere (or "almost", whatever that means). I was incredibly amazed when I found out ...
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76 views

Difference between $\mathrm {d} x$ and $\delta x $

Are $\mathrm {d} x $ and $\delta x $ the same mathematical object from the point of view of the nonstandard analysis?
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Microcontinuous vs Continuous

I've been studying infinitesimals and came upon the idea of uniformly microcontinuous functions. My question is: if a function $f^*: \mathbb{R}^* \to \mathbb{R}^*$ the natural extension of $f: ...
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Ultrapower and hyperreals

The construction I've seen of the field of hyperreal numbers considers a non-principal ultrafilter $\mathcal{U}$ on $\mathbb{N}$, then takes the quotient of $\mathbb{R}^{\mathbb{N}}$ by equivalence ...
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Construction of the Hyperreal numbers

Several times I have seen questions/answers here about using the correct definition of derivatives. There are also questions about whether or not $1/0$ is defined. Sometimes there is a discussion ...
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Is $\epsilon^2/\epsilon^2=1$ or $0/0$?

Is it possible in the system of dual numbers ($a+\epsilon b$; $\epsilon^2=0$) to calculate $\epsilon/\epsilon =1$? How then does one deal with $\epsilon^2/\epsilon^2=1$ versus ...
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Is $dxdy$ really a multiplication of $dx$ and $dy$?

On the answers of the question Is $\frac{dy}{dx}$ not a ratio? it was told that $\frac{dy}{dx}$ cannot be seen as a quotient, even though it looks like a fraction. My question is: does $dxdy$ in the ...
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Construction of *ZFC

In the following paper, page 11 (Appendix), there is a construction of a model of a theory $^*ZFC$ (see the definitions in the paper included) from a model of $ZFC$. I have been trying really hard to ...
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Transfert principle of a conservative extension of ZFC

In the following paper, there is a theory called $^*ZFC$ in the language $(^*,\in)$. The *-map is (more or less) defined on the Von Neuman hierarchy $S$ and verifies the following axiom schemata true ...
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How do I represent such a transformation?

Let's say I have a 2d rectangle defined by $ [0,x_0] \times [0,y_0]$. Now lets say I cut out the middle rectangle $[\frac{1}{3} x_0, \frac{2}{3} x_0] \times [\frac{1}{3} y_0, \frac{2}{3} y_0]$. Now ...
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Axioms for the hyperrationals

I'm working on a comparison between a set theoretical and an axiomatic construction of the hyperrational numbers $^*\mathbb Q$. So far I have only found the construction of $^*\mathbb Q$ by using ...
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Is there a source linking Robinson's work in wing theory with his theory of infinitesimals?

Abraham Robinson worked in applied mathematics for several decades. MathSciNet lists 12 articles by Robinson in wing theory. His production included the book Robinson, A.; Laurmann, J. A. Wing ...
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Are infinitesimals dangerous?

Amir Alexander is a historian of mathematics. His new book is entitled "Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World". See here. Two questions: (1) In what sense are ...
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Is it possible to have numbers that are to Hyperreal numbers what Hyperreals are to Reals numbers?

There are Hyperreal numbers that are smaller than any real number , also those that are larger than any real, they have properties analogous to those of Real numbers thanks to the Transfer principle ...
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Saturation, (Complete) Ordered Fields and Model-Theoretic Methods in relation to Real & Non-Standard Analysis

I am trying to understand the following three questions: One and Two and Three. I'm under the impression that they're interrelated, though maybe not directly. What do I need to read to back-fill to ...
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An example of trans-series from Euler's equation

There seems to be a generalization of the notion of the "asymptotic series" to a notion of "trans-series". One "simple" example of this seems to arise in trying to solve the Euler's equation by ...
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infinitesimal calculus

I have been reading some non-standard analysis from Keisler's book and I think it is logically consistent till now but there are criticisms against it and why isn't non-standard analysis accepted more ...
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many infinities, how many zeroes? [closed]

although i am aware of the term non-standard analysis i have, as yet, no clear idea what it signifies. but i have often wondered about the pseudo-equation $$ \frac1{\infty} = 0 $$ which one may ...
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structure of the hypernaturals

I want to understand the structure of the hypernaturals a little better. Let me recall the ultraproduct construction of the hypernaturals. On the set of all sequences of $\mathbb{N}$, we define an ...
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Non-standard analysis way of proving that derivative of $e^x$ is $e^x$

What is the non-standard (infinitesimal) analysis way of proving that the derivative of $e^x$ is $e^x$? I tried to prove it myself, but I am having a hard time proving this without recourse to ...
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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 ...
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Intuition for a real line vs. a “hyperreal line”

I am a student of pure mathematics but I have no formal background in nonstandard analysis. I came across the concept of a hyperreal field recently, read just a little about them, and followed the ...
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define the reals in a non-archimedean elementary extension of the real field.

Can it be done? We have the real field $(\Bbb R,+,-,\times,0,1,<)$, of course $(0,1,-,<)$ are definable using the rest. We take an elementary non-archimedean extension. Can we define the ...
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Do Hyperreal numbers include infinitesimals?

According to definition of Hyperreal numbers The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form 1 + 1 + 1 ...
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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" ...
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Who gave you the epsilon?

Who gave you the epsilon? is the title of an article by J. Grabiner on Cauchy from the 1980s, and the implied answer is "Cauchy". On the other hand, historian I. Grattan-Guinness points out in his ...
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Is it true that infinite product of infinitesimals may not be an infinitesimal?

Is it true that infinite product of infinitesimals may not be an infinitesimal? Here's my attempt. Using ultraproduct construction, $$\epsilon_1=[\langle 1, \frac{1}{2}, ...
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Is there a nonstandard characterization of Lipschitz continuity?

Let $f: \mathbb R \to \mathbb R$ be Lipschitz continuous with finite constant $L$. Then $$ |f(x) - f(y) \le L |x-y|, \tag{1} $$ and, by direct transfer, this property holds for $^*\!f$. For ...
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The Domain of Hyper-real Functions

We know that the set of all real numbers is a subset of the hyperreal numbers, and the extension principle allows us to apply every real function to hyperreal numbers. For every real function ...
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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 ...
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Proof of Chain Rule using Nonstandard Analysis

I am trying to make an introduction and make myself comfortable with the Nonstandard Analysis in order to gain intuition for derivatives and integration. I am trying to prove myself the famous Chain ...
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$\sigma$-additivity of the standard part of an internal finitely additive measure on an internal algebra

Let $V({}^\ast{X})$ be an enlargement of ${}^{\ast}{V(X)}$ and $C$ be a fixed internal set in $V({}^\ast{X})$. $\{A_k\}_{k \in \Bbb N}$ is strictly increasing sequence of internal subsets of $C$. It ...
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non-archimedean in lay terms

I've dabbled with studying infinitesimals off and on for years ... Robinson, Keisler, Bell ("Smooth Worlds"), etc., even a bit of category theory. But I'm not a mathematician and tend to jump in way ...
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Show that, for a random walk $X$ on $\mathbb Z_2$, $X_\infty$ is independent of the past $\left\{ X_t \right\}_{0\le t<\infty}$

Define the *-finite time set $T=\{0,1,\ldots,n\}$, with $n \in {^*\mathbb N} \setminus \mathbb N$. Let $X=\left( X_t \right)_{t \in T}$ be a random walk on the cyclic group $\mathbb Z _ 2 = \mathbb Z ...
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Confused about differentiation

I'm new to calculus and have been taught that $\displaystyle \frac{dy}{dx}$ is the rate of change of y with respect to x. Does $\displaystyle \frac{dy}{dx}$ show how much the variable y changes as x ...
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Definition of the integral in non-standard calculus

In the sources I've seen, the integral is defined in non-standard calculus as the hyperreal extension of a function related to Riemann sums. E.g., Let $$ S(\Delta x) = \sum_{a}^{b}f(x)\Delta x$$ be a ...
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What is the meaning of infinitesimal?

I have read that an infinitesimal is very small, it is unthinkably small but I am not quite comfortable with with its applications. My first question is that is an infinitesimal a stationary value? It ...
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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 ...