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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Are surreal numbers isomorphic to formal power series? [on hold]

I wonder whether surreal numbers $No$ isomorphic with formal Laurent series? It seems that any surreal number can be represented in the form of Laurent series over $\omega$. For instance, ...
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20 views

Numerical system that includes the limit targets such as $0^+$, $0^-$, $1^+$ etc

I wonder what is the name of a mathematical system extending the real numbers that includes signed zero along with unsigned zero as well as other "limit targets", such as $1^+=1+0^+$, $5^-$ etc, so ...
2
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34 views

Fourier transform and non-standard calculus

The Fourier transform is not as easily formalizable as the Fourier series. For example, one needs to introduce tempered distributions to define the Dirac delta-function. Also, it is impossible to view ...
4
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3answers
107 views

Could “$\infty$” be understood by taking the reciprocals of the Hyperreal numbers?

When learning mathematics we are told that infinity is undefined. (*) Recently I read about the infinitesimal version of Calculus and how we can in fact treat $dy/dx$ as a fraction under this ...
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1answer
48 views

Is the empty set internal?

Is the empty set internal or not? And is there a proof (either way), or is it just a convention? If it's just a convention, why was that particular convention chosen?
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2answers
64 views

Are the Hyperreals complete?

Since $^*\mathbb{R}$ does not form a metric space then it can not satisfy the Cauchy conditions for completeness. However, my intuition is telling me that it would satisfy conditions of completeness ...
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1answer
34 views

How much choice is needed for the transfer principle?

To construct the hyperreals via ultrapower the Boolean prime ideal theorem apparently suffices. However, to prove the transfer principle for the extension $\mathbb{R}\subset{}^\ast\mathbb{R}$ ...
3
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3answers
131 views

Why those division by zero are formalized?

Easy example first: $f(x) = nx$ $f'(x) = (f(x+0)-f(x))/0 = (nx+0n-nx)/0 = (0n)/0 = n$ Hard one: $f(x) = a^x$ $f'(x) = (f(x+0)-f(x))/0 = (a^{x+0}-a^x)/0 = (a^x(a^0-1))/0 = (a^x(e^{\ln(a^0)}-1))/0 ...
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1answer
70 views

1/∞ is 0 or infinitesimal?

Since ∞>0 , so 1/∞>0, thus I think 1/∞ should be infinitesimal, but the calculus book says $\displaystyle \lim_{x \to \infty} \frac{1}{x}= 0$ So is 1/∞ 0 or infinitesimal ? P.S.I mean 1/∞ and ...
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40 views

Derivatives of $\sin x$ and $\exp x$ using differentials / dual numbers

I want to introduce a concept of a differential $dx$ to my students and derive all the basic derivatives using it. Now, I define the differential to satisfy $dx \neq 0$, but $(dx)^2 = 0$. Therefore, ...
5
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1answer
148 views

Topologies induced by non-standard metric

Let $R$ be a set of points and $\mathbb{D}$ be a totally ordered field. Further consider a function $\rho:R\times R \rightarrow \mathbb{D}$. $\langle R,\mathbb D,\rho\rangle$ is a metric space if ...
5
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2answers
80 views

Dirac Delta definition in non-standard analysis?

What is the definition of Dirac Delta in non-standard analysis? I would define it either as a standard distribution with $\sigma=\epsilon$ or maximum equal to $\omega$. Which is the correct answer?
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1answer
65 views

What are hyperreal numbers?

Well excuse me if I make some english mistakes :) So I descovered hyperreal numbers two mouth ago and I read a lot of articles about them but I don't understand which they are... Because of one ...
2
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1answer
45 views

Prove, by nonstandard reasoning, that the limit superior of a sequence is a cluster point.

I'm working through Goldblatt's Lectures on the Hyperreals, and I've found myself quite stuck on this exercise: Prove, by nonstandard reasoning, that both the limit superior and the limit inferior ...
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1answer
29 views

Compute joint density function of exponential fuction

Consider a set of continuous random variablces $Y_1 ... Y_n$, i.i.d, exponentially distributed . with rate parameter $\lambda$. I showed first that for one single variablce (ie the first) its ...
0
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1answer
58 views

Hyperreal probability density?

I'm fairly new to the subject of hyperreal numbers and I'm wondering if there exists an infinitesimal number $a$ such that (in some reasonable sense) $$\sum_{n=1}^\infty a=1$$ ? In other words: Is ...
2
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1answer
41 views

Computing the standard part of $(3-\sqrt{c+2})/(c-7)$ where the standard part of $c$ is $7$

I'm working through Keisler's calculus book based on infinitesimals. The following problem has me a little bit stumped. Compute the standard part of: $$\frac{3-\sqrt{c+2}}{c-7}$$ Given that $c\ne7$ ...
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1answer
112 views

Nonstandard complex numbers and categoricity

Let ${}^*\mathbb{C}$ be a nonstandard complex number field (given, for instance, as a countable ultrapower.) By the transfer principle ${}^*\mathbb{C}$ is algebraically closed of characteristic zero, ...
2
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1answer
48 views

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: ...
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2answers
62 views

Internal Set Theory: $n$ is standard $\implies\;n+1$ is standard

I'm reading a bit about Nelson's version of nonstandard analysis and in the notes it is said that [$n$ is standard]$\implies$[$n+1$ is standard]. Right after that it is mentioned that an inductive ...
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4answers
212 views

The nature of infinities

I have been thinking about the nature of infinity lately. I have no experience with higher mathematics or theorems regarding infinity, so please forgive me if my ideas on this topic are extremely ...
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53 views

Is there an algebraic-geometric solution to the problem of the Leibnizian formalism?

The precise question appears at the end of this entry. With all the recent advances in understanding infinitesimals, we still don't fully understand why Leibniz's definition of $\frac{dy}{dx}$ as ...
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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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1answer
33 views

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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1answer
92 views

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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1answer
23 views

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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1answer
102 views

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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1answer
124 views

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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23 views

$\approx$ 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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1answer
29 views

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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488 views

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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1answer
40 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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1answer
299 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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1answer
49 views

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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1answer
142 views

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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5answers
194 views

Why can't the reals be constructed from the infinitesimal?

If the infinitesimal gives an unlimited precision as 1/∞ --> 0 Which can be thought of as the decimal 0.000000.....00000... then Why can't the reals, which demands, simply, unlimited precision (this ...
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1answer
84 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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49 views

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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3answers
128 views

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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2answers
198 views

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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195 views

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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9answers
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Is $dx\,dy$ 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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1answer
236 views

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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44 views

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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1answer
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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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1answer
89 views

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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1answer
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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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1answer
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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 ...