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 is the derivative exact if it's the standard part?

I understand that no such infinitesimal value exists in our universe, so we round to the nearest real value when taking the derivative. Such as how $2x+\Delta x$ equals $2x$ when we take the standard ...
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Why do we need ultrafilter for construction of hyperreal numbers?

While trying to get some hold on the hyperreal numbers I found that their construction using the already existing real number system $\mathbb{R}$ requires the use of objects called free ultrafilters ...
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What is the use of hyperreal numbers?

For sometime I have been trying to come to terms with the concept of hyperreal numbers. It appears that they were invented as an alternative to the $\epsilon-\delta$ definitions to put the processes ...
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What's so different about limits compared to infinitesimals?

If you find the limit is 2 for a given function, wouldn't this be the same as $2 + \epsilon$ with $\epsilon$ being a negligible value? This different way of defining limit-like behavior seems rigorous ...
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43 views

What properties do hyperreal extensions of real functions have?

If I have a function $f : \mathbb R \to \mathbb R$ and extend it to the hyperreal function $f^* : \mathbb R^* \to \mathbb R^*$, what are some of the properties that I know $f^*$ must have? ...
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Why do the infinitely many infinitesimal errors from each term of an infinite Riemann sum still add up to only an infinitesimal error?

Ok, so after extensive research on the topic of how we deal with the idea of an infinitesimal amount of error, I learned about the standard part function as a way to deal with discarding this ...
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On the HoTT Cauchy Reals

In the Homotopy Type Theory Book there is a construction given of a kind of Cauchy reals via higher inductive type and the authors remarked, that this construction is preferred to other notions (reals ...
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Absolute arithmetic convergence and place selection rules

? Edit I was wondering about absolute arithmetic convergence of relative frequencies and place selection functions and randomness in sequences. When I mean arithmetic absolute convergence i mean a ...
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digits of pi, the nature of its randomness, immunity to place selections (Von mises) reichenbach partial limits

The digits of $\pi$ uniform distribution non-standard analysis, and its partial limit I was wondering whether there has been any investigation into the distribution of the decimals of $\pi$ using non-...
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What is… A Parsimonious History?

Interpreting historical mathematicians involves a recognition of the fact that most of them viewed the continuum as not being made out of points. Rather they viewed points as marking locations on a ...
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60 views

Non standard complex analytic functions

I'm having trouble understanding what should mean analytic for hypercomplex functions. In deed I'm studying a book (Non standard analysis in practice by Diener) where they just say that the function ...
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Norm vs A-norm in non-Archimedean Functional Analysis

Let $K =(K,| \cdot |)$ be a non-Archimedean valued field. Let $E$ be a $K$-vector space. A norm on $E$ is a map $||\cdot||:E\to[0,\infty)$ such that: $||x||=0$ if and only if $x=0$, $||\lambda x||=...
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Is there a bijective map from $\mathbb{R}$ to ${}^{*}\mathbb{R}$? [duplicate]

Is there a one-to-one correspondance between the real numbers and the hyperreal numbers?
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Proving that $\mathbb{Q}_p$ is not formally real.

I've been looking for a concrete proof without results. The only hint that I have found says: 1) $\mathbb{Q}_2$ contains a square root of $-7$. 2) $\mathbb{Q}_p$ ($p>2$) contains a square root of ...
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1answer
34 views

Order in a quotient space of $\mathbb{R}^\mathbb{N}$ ($\mathbb{R}^\omega $)

Let $\mathcal{F}$ be a filter in $\mathbb{N}$ finer than Fréchet filter. In $\mathbb{R}^\mathbb{N}$ we define the equivalente relation : $(a_n) \equiv (b_n)$ iff $\{n | a_n = b_n\} \in \mathcal{F}$. ...
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52 views

The meaning of “EXACT laws of large numbers”

I have come across various papers that consider a stronger form of probability-relative frequency convergence theorem called the 'exact law of large numbers". I note that in particular such theorems ...
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Can the transfer principle apply to second-order logic if we transfer sets and relations to hyper-sets and hyper-relations?

The transfer principle doesn't apply to second-order logic. For example, if I take a standard statement. $$\text{A lower bounded set of Reals has a greatest lower bound}$$ Is false for the hyperreals:...
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66 views

Periodicity in the hyperreals

Suppose we had a sequence say $K=\langle1,8,5,1,8,5,1,8,5,\ldots\rangle$ periodic on these $3$ numbers and our ultrafilter contained the odd numbers. Then am I right in thinking that K could be $\...
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Is this proof of the four color theorem for infinite graphs legit?

So you got an infinite planar graph $G$. I will prove that it is four colorable. So, construct an infinite number of statements about graphs: The first is "is four colorable" Next, for each vertex $...
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The magic of existential transfer

Yesterday I finished grading the final exam in a course on infinitesimal calculus taught to 130 freshmen. One of the problems on the exam was to show that if a function $f$ is differentiable at $c\in\...
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653 views

Is it to the students' advantage to learn the language of infinitesimals? [closed]

A colleague of mine asked an interesting question reproduced below with his permission. It is reasonable to ask whether it is to the students' advantage to learn the language of infinitesimals - ...
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59 views

Proving a Function Continuous with Non-Standard Analysis

I am reading a text on non-standard analysis. I need to prove the following: Suppose that $f$ is non-decreasing on the real interval $[a,b]$ and that $f$ satisfies the intermediate value property. ...
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Probability theory with the hyperreals?

Forgive the undoubted ignorance of this question. I am out of my element both in probability and in nonstandard analysis. A mathematically curious layperson friend recently had a conversation with me ...
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Survey of varieties of non-standard analysis?

Is there a reliable, reasonably up-to-date, survey article doing a "compare and contrast" on varieties of non-standard analysis?
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Definite Integral of a infinitesimal

I did not study math, but have some foundations in it. I have been looking through some books on nonstandard analysis, and have (what I consider to be) a pretty simple question which I haven't been ...
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Constructing infinite field in which all subrings are subfields

A classmate posed a question in class as to if there existed an infinite field $F$ for which every subring $R \subseteq F$ was a subfield. We'd already determined that if $F$ was a finite field, then ...
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Characterisation of convergence of bounded sequences via ultra-filters

Let $\{a_n\}_{n\in\mathbb N}$ be a bounded sequence of real or complex numbers and $\mathscr F\subset\mathscr P(\mathbb N)$ be a non-principal ultra-filter. Then $a=\lim_{\mathscr F}a_n$ is well-...
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What are the disadvantages of non-standard analysis?

Most students are first taught standard analysis, and the might learn about NSA later on. However, what has kept NSA from becoming the standard? At least from what I've been told, it is much more ...
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What's an example of an infinitesimal?

If you want to use infinitesimals to teach calculus, what kind of example of an infinitesimal can you give them? What I am asking for are specific techniques for explaining infinitesimals to students, ...
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5answers
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Is arithmetic with infinite numbers fictitious?

In 1933 Skolem constructed models for arithmetic containing infinite numbers. In a 1977 article Stillwell emphasized the constructive nature of Skolem's approach; see here. Is this at odds with ...
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Does evaluating hyperreal $f(H)$ boil down to $f(±∞)$ in the standard theory of limits? [closed]

Does evaluating $f(H)$ at an infinite hyperreal $H$ when doing calculus in Robinson's (Keisler's) framework amount merely to assigning $f(\infty)$ in the standard theory of limits? This question has ...
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209 views

Can I have something larger than infinite? [duplicate]

My question is "Can I have something larger than infinite?" Sometimes, we add infinite numbers into our set of numbers by simply extending our set and adding infinite numbers to it. But can't you ...
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296 views

Concept behind the limit to infinity?

I can across transfinite numbers and came up with a thought. What if$$\lim_{x\to\infty}f(x)=f(T)$$where $T$ was a transfinite number? Generally, in calculus, I have noted that it is two different ...
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Justification for manipulations according to Leibniz-notation

Is there a way to justify the manipulations according to Leibniz-notation without nonstandard-analysis. E.g. $\frac{dy}{dx} = x \\ dy = x dx\\ \int dy = \int x dx\\ y = \frac{1}{2} x^2$
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Non-standard version of Frechet derivative

Non-standard analysis offers very convenient tools to prove facts about continuity or differentiability. I am looking for such tool in infinite-dimensional calculus. To be more precise, let $X$ and $...
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Is it a new type of induction? (Infinitesimal induction) Is this even true?

Suppose we want to prove Euler's Formula with induction for all positive real numbers. At first this seems baffling, but an idea struck my mind today. Prove: $$e^{ix}=\cos x+i\sin x \ \ \ \...
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Nonstandard analysis, Lie groups and universal enveloping algebras

The idea of nonstandard analysis is to combine finite quantities with infinitesimals. And, back in the day, Lie algebras were roughly considered the "infinitesimal elements" of Lie groups. Say we want ...
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Two different definitions of big O notation

I find there are two different definitions of big O notations for $f(n)=O(g(n))$ as $n\rightarrow\infty$: There exist $M>0$, and $N\in\mathbb{N}$, such that $|f(n)|\leq M|g(n)|$ for $n\geq N$. ...
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First order logic,Step in Derivation of Non standard real numbers

This is from first order logic, specifically a section detailing the construction of the non standard real numbers, after Los' theorem And we have that:\ \ Let $L = \{+, ×, <, 0, 1\}$ be the ...
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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 ...
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Star mapping in Non-standard analysis

I'm trying to understand the star mapping in non-standard analysis in particular for the Hyperreals. I know that $*: \mathbb R\to \mathbb{^* R}$ is a mapping such that $^*(x)=^*x$ where $^*x= (x,x,x,x,...
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Should we say $0.\bar9$ *can* equal $1$ also in the hyperreals?

Consider the sequence $R_n$ of repunits, defined as $\displaystyle\frac{10^n-1}{9}$. We have $$\frac{R_{n+1}}{R_n}=\frac{9}{9} \frac{10^{n+1}-1}{10^n -1}=\frac{10^{n+1}-1}{10^n -1}=\frac{\overbrace{99\...
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Mathematical descriptions of physical space

Bear with me as I'm a philosophy (not math) student. First some philosophical background, and then the math question. One philosophical view is that physical space is composed of infinitely many ...
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96 views

Is there such a thing as “hypertopology” (analogous to the hyperreals)?

The hyperreal number system adds infinities and infinitesimals, allowing Calculus to be done using these things instead of limits (sort of like when calculus was originally invented, but with rigor)....
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In NSA (ZFC+IST), what can we say about generators for $\mathbb{Z}/\nu\mathbb{Z}$ for unlimited $\nu$?

Recently I've been going through a short text on Nonstandard Analysis that uses the axiomatic approach of Nelson (Internal Set Theory - IST). Its study has led me to be curious about the properties of ...
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Nonstandard analysis and integral transforms

Can integral transforms be evaluated without limits(i.e Laplace transform) such as in non standard analysis? Can the improper integral be bounded by a hyperreal number? I am not very familliar with ...
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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 ...
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Non-standard analysis - infinitesimals and archimedean property

I got a question about infinitesimals in non-standard analysis. If I understand correctly, they are defined to be the number that is closest to zero. However, at the same time, they satisfy all the ...
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Is there in anyway possible to prove that 0.999 recurring does not equal to 1 [duplicate]

I know that the reason why 0.999 recurring equals to one because it's goes on forever, and the difference between 0.999 recurring and 1 is 0 since it's infinite. But is it possibly to prove otherwise? ...
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What happens if to introduce infinite and infinitesimel quantities this way?

What if to introduce $\varepsilon$ and $\omega=1/\varepsilon$ from the following equation as a definition? $$\left(1+\varepsilon\right)^{1/\varepsilon}=e$$ or such $\varepsilon$ that $e^\varepsilon=...