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).

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Nonstandard Natural Numbers via Internal Set Theory in Coq

I'm trying to translate http://www.stat.umn.edu/geyer/nsa/o.pdf to Coq, but I'm stuck right at the start on the simple axioms for the predicate "standard" over natural numbers. The pdf linked here ...
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Applications of Non-Standard Analysis to Number Theory and Topology?

S.E friends, I am wondering what might be some useful applications of the non-standard analysis to the number theory and topology? I am very interested in the set-theoretic topology and prime ...
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645 views

Are there non-standard counterexamples to the Fermat Last Theorem?

This is another way to ask if Wiles's proof can be converted into a "purely number-theoretic" one. If there is no proof in Peano Arithmetic then there should be non-standard integers that satisfy the ...
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2answers
382 views

What is the point of making dx an infinitesimal hyperreal?

It seems fairly common to describe $\mathrm{d}x$ in nonstandard analysis as an infinitesimal. But after thinking it through (and then skimming Keisler's text), I can't see the point and actually think ...
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How do we interpret and perform integrals using infinitesimals?

If $\mathrm{d}x$ is treated as a hyperreal infinitesimal we can easily do derivations. How do we interpret and perform integrals using infinitesimals? What is the $\mathrm{d}x$ in $\int x^3\,\...
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Is the set of hyperreal numbers a quotient ring?

It is easy to see that the set of real sequences $\mathbb{R}^{\mathbb{N}}$ is a ring. It suffices to define, for all $r,s\in\mathbb{R}^{\mathbb{N}}$, the operations $r\oplus s =(r_n+s_n)_{n\in\mathbb{...
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1answer
298 views

How to get an introduction to non-standard analysis?

I am a high school rising senior with an interest in mathematics, and I will be taking AP calculus AB next year. I have been doing research online, and recently came across hyperreal numbers, which I ...
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1answer
247 views

What is the difference between hyperreal numbers and dual numbers

Wikipedia has two different but unconnected pages for Hyperreal and Dual numbers. https://en.wikipedia.org/wiki/Hyperreal_number and https://en.wikipedia.org/wiki/Dual_number I cannot stop seeing ...
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372 views

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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2answers
3k views

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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1answer
2k views

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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1answer
154 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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254 views

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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72 views

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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846 views

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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1answer
104 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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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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105 views

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
40 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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1answer
172 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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2answers
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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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1answer
72 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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472 views

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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2answers
866 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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1answer
158 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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1answer
177 views

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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1answer
121 views

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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484 views

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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1answer
68 views

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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1answer
129 views

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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536 views

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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10answers
3k views

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 to the students? What I am asking for are specific techniques for explaining infinitesimals ...
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3answers
858 views

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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705 views

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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3answers
372 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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3answers
116 views

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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409 views

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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15answers
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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 \ \ \ \forall ...
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0answers
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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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1answer
73 views

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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1answer
61 views

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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2answers
63 views

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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1answer
137 views

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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233 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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323 views

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 ...