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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).

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What is the Galois group of the Hyperreal Numbers?

The Galois Group of $\mathbb{R}$ as an extension of $\mathbb{Q}$ is trivial. That’s because any field automorphism of $\mathbb{R}$ is order preserving, so since $\mathbb{Q}$ is dense in $\mathbb{R}$, ...
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Practical applications of non-standard probability

Recently I read a paper by Benci et al. describing an alternative to Kolmogorov's construction of probability where the probability measure $P$ takes values in a non-Archimedian field and we have $P(A)...
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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 ...
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Transfinitely iterating the Puiseux, Levi-Civita, or Hahn series constructions

There are many ways to take some real-closed field and generate a proper extension of it with elements that are infinite and infinitesimal relative to the original field. One well-known example is ...
Mike Battaglia's user avatar
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2 answers
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What is the “maximal hyperreal field”?

In many SE posts and the Wikipedia article on the surreal numbers I’ve seen references to a “maximal” hyperreal field that’s isomorphic to the surreals. If they’re isomorphic, then why is it that ...
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Explicitly defining a nonstandard extension of the reals

Kanovei & Shelah (2004) explicitly constructed a nonstandard extension of the reals. Now, I'm no expert on this but I gather that they first specified a set of free ultrafilters, and then used it ...
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Nonstandard Complex Analysis?

I recently discovered Nonstandard Analysis and am slowly working my way through Kelsier's textbook and Foundations companion. However while I have found plenty of stuff about real nonstandard ...
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Nonstandard analysis and hyperreals

Hyperreal numbers and nonstandard analysis are intimately connected. First, the nonstandard analysis consists of an extension of the axioms for the real numbers, which introduces a new predicate $\...
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Noetherian Rings in Nonstandard Framework

I have been trying to go through some algebraic geometry using the nonstandard framework. Noetherian rings are of course fundamental in this subject and it is characterized by the attribute that every ...
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How do we naturally extend the average from the Hausdorff Measure for functions with a domain without a gauge function?

Motivation: According to this question Some sets have a Hausdorff Dimension $\alpha$ but have a zero-dimensional Hausdorff Measure. These sets may have another dimension function, i.e. a function $h:[...
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Use of Hyperreal numbers

I've come across hyperreal numbers and was curious about something in measure theory. Non measurable sets can be constructed with AC (correct me, if they can also be constructed without AC), and the ...
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Is ordering done component wise in the hyperreals?

In the ultrafilter construction of the hypperreals, is ordering done component wise? I.e. if $a,b$ are hyperreal numbers then $a = [x_n]$ and $ b = [y_n]$ where $[x_n]$ and $[y_n]$ are equivalence ...
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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 ...
Kit Joyce's user avatar
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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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Is there an external reflection principle in IST?

Background The reflection principle is a theorem schema in ZF. Given a formula $ φ $ and a set $ M $ we obtain $ φ $ relativized to $ M $ by restricting all quantifiers of $ φ $ to range over $ M $. ...
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How to use non-standard analysis to prove Baire Category Theorem?

I'm caring about some questions of non-standard analysis. I have found the only book talking about Baire Category Theorem, which is the book of Siu-Ah Ng. But I think the proof in this book is not ...
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How to derive the triple product rule with Nonstandard Analysis?

$\frac{\partial x}{\partial y} \cdot \frac{\partial y}{\partial z} \cdot \frac{\partial z}{\partial x} = -1$ according to the triple product rule However, it would be 1, if derivatives behaved like ...
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Isomorphism of hyperreal fields viewed as extensions of the real field

Crossposted on MathOverflow: https://mathoverflow.net/q/368381/461 Let $A$ be the $\mathbb R$-algebra of all $\mathbb R$-valued functions on $\mathbb N$, that is $$ A=\mathbb R^{\mathbb N}=\prod_{n\in ...
Pierre-Yves Gaillard's user avatar
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Reference request: How is $0.99\cdots$ defined in nonstandard analysis?

In this answer to the question Is it true that $0.999999999\dots=1$?, Noah Snyder points out that Symbols don't mean anything in particular until you've defined what you mean by them. This ...
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Resources for Learning Hyperreal Numbers

I've somewhat recently discovered hyperreal numbers, but I haven't gotten the chance to thoroughly research them. What resources do you all recommend for undergrad level study of the hyperreal number ...
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Canonical hyperreal numbers

The hyperreal numbers are constructed by any free ultrafilter. We know that we can't exhibit a concrete example of a free ultrafilter on natural numbers (see here). Is it possible to give a canonical ...
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Nonstandard-Analysis: What are traits of sets that are "strange"?

By the power of the transfer principle, the principle of internal definition and the overspill principle, most sets in the nonstandard-superstructure behave rather tame (or rather, standard). However, ...
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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 $...
Christopher King's user avatar
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What is the basis of the hyperreal numbers?

Let us consider the Hyperreal numbers as a vector space over the real numbers. This vector space is quite interesting. Here are some interesting subspaces: The finite numbers The infinitesimal ...
Christopher King's user avatar
2 votes
1 answer
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Is this a valid basis for a transfinite number system?

I've been curious about transfinite number systems including infinite ordinals, hyperreals, and surreal numbers. The hyperreals in particular seem particularly appealing for introducing a hierarchy of ...
Aidan Simmons's user avatar
2 votes
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65 views

Does this basic property of hyperreal function hold?

Let $x=(a_1, a_2, a_3, ...) + \mathcal U \in {}^\ast \mathbb R := \displaystyle\prod_1^\infty \mathbb R/\mathcal U$ be a hyperreal number using the ultrapower construction and $f \colon \mathbb R\to \...
Markus Klyver's user avatar
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How sensible is the limit from above against infinity in nonstandard analysis?

In nonstandard analysis, we have hyperreal numbers that are greater than any real number. As such, we can create a sequence of infinite, hyperfinite hyperreal numbers which grows ever smaller. More ...
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Does non-standard analysis lead to different PDEs from those obtained through standard analysis?

Take the Navier-Stokes equations as an example. If we take a non-standard analysis approach, will the final form of the PDEs be different from what presented in classical books on fluid dynamics?
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Levi-Civita field vs Puiseux series: why is Cauchy completeness important?

The Levi-Civita field's main claim to fame is that it's the smallest real-closed field which is a proper extension of the reals and which is Cauchy-complete. That last bit is important, or else the ...
Mike Battaglia's user avatar
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Automating proofs via indicator functions?

It is a common technique in measure theory to prove something for indicator functions / elementary functions, generalize it to positive-valued functions and to measurable functions via $X = X^+ - X^-$....
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Internal subsets of nonstandard extensions.

I am studying the first chapter of L. O. Arkeryd et. al: Nonstandard Analysis. Theory and Applications. There it is shown that for the multiset $(\mathbb{X}, \mathcal{P}(\mathbb{X}))$ it is possible ...
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On The expansion of $e^x$.

http://www.17centurymaths.com/contents/introductiontoanalysisvol1.htm The chapter $7$ of this mentions a proof of expansion of $e^x$ without using idea of derivative but only using idea of ...
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A hyperreal field is a minimal field such that ...?

The standard presentation of hyperreals is difficult to understand. One typically motivates the hyperreals by a desire to have a consistent theory of infinitesimal elements, and then introduces the ...
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Probability theory on the transfinite

The usual fomalization of probability through $\sigma$-algebras and $\sigma$-additive measures can effectively model (countable) infinite chains of trials. This is usually done by defining a countable ...
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Nonstandard extension of nonstandard hull

Let $(X_i, d_i, e_i)$ be a sequence of pointed metric spaces, let $\prod _\omega (X_i, d_i, e_i)$ be the ultraproduct of said spaces with respect to a nonprincipal ultrafilter $\omega$, and let $(\hat{...
pseudocydonia's user avatar
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Bound for roots of a polynomial with coefficients in a non-Archimedean valued field

Is there any bound for the valuation of the roots of a given polynomial with coefficients in an algebraically closed non-Archimedean valued field? Any reference or insight would be appreciated.
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Nonstandard-Analysis: Showing L'Hospital

Let there be two functions $f,g:(a,b) \to\mathbb R$ that are differentiable in $(a,b)$ with either $$\text{Case 1:}\qquad\lim_{x\to b} f(x) = \lim_{x\to b} g(x) = 0$$ or $$\,\,\,\,\,\,\,\text{Case 2:}\...
Sudix's user avatar
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2 votes
0 answers
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Is this proof correct: A nonstandard-statement transfers exactly iff it's an internal set.

Let $\varphi\in \widehat {^*S}$ be a statement in a nonstandard-superstructure. The following proof is supposed to show that if and only if $\varphi$ is internal, there exists a corresponding ...
Sudix's user avatar
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2 votes
1 answer
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How can a universe include an infinite (sum) relation

Let the universe be $ \hat{V}$, which is constructed as: $ \hat{V} := V_0 \cup V_1 \cup V_2\cup ... $ where $V_0$ is the set of primary elements and $V_{v+1} := V_v \cup P(V_v)$ So, in other words,...
Sudix's user avatar
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2 votes
0 answers
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Value groups of ultrapowers of $\mathbb{R}$.

$\DeclareMathOperator{\Noo}{No}$ $\DeclareMathOperator{\ee}{e}$ Let $U$ be a non principal ultrafilter on $\mathbb{N}$, let $^*\mathbb{R}$ denote the corresponding hyperreal field. Let $^*\Gamma$ ...
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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\...
Vincenzo Oliva's user avatar
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99 views

Limit points in nonstandard analysis

Let $A\subseteq\mathbb{R}$, $p\in\mathbb{R}$. I proved that the following are equivalent: $\exists\left(x_{n}\right)_{n\in\mathbb{N}}\subseteq A\cap\left\{ p\right\} ^{c}$ such that $x_{n}\rightarrow ...
Sogol Thamaem's user avatar
2 votes
0 answers
72 views

non-commutative infinitesimal extension of $\mathbb R$

Background: The transfer principle in nonstandard analysis implies that any nonstandard model of the reals is a commutative (for additively and multiplicatively). It is also well-known that the set $\...
Ittay Weiss's user avatar
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2 votes
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Size of a geometric point

It is well known that the geometric points do not have any length, area, volume, or any other dimensional attribute, also geometric object (for example "line") is made up of a infinite number of ...
vito's user avatar
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2 votes
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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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Concurrent relation and enlargement

The superstructure $V({}^{\ast}X)$ [with respect to a monomophism $\ast : V(X) \to V({}^{\ast}X)$] is called an enlargement of $V(X) $ if for each set $A \in V(X)$ there is a set $B \in {}^{\ast} \...
Metta World Peace's user avatar
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Law of iterated expectation in an algebraic axiomatization of probability theory

In the second chapter of Radically Elementary Probability Theory [PDF], Edward Nelson gives an axiomatization of probability theory based on algebras of random variables, briefly discusses a couple of ...
pash's user avatar
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Berkovich analytification of Robinson fields

Let $\rho$ be an infinitesimal and let $^\rho \mathbb{R}$ be a (non-archimedean) Robinson valued field. Is there anything known about the topological structure of $\mathbb{A}^{1,an}_{^\rho \mathbb{R}}$...
Willem Noorduin's user avatar
1 vote
0 answers
70 views

Why isn't Riemann sum infinitesimal in nonstandard analysis?

I am trying to learn nonstandard analysis from Keisler's book. In the integration chapter it feels like the use of the Transfer Principle is some kind of magic that just requires us to believe results ...
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1 vote
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Are there guaranteed conglomerate sized models for a given first-order theory?

In a comment under What is the “maximal hyperreal field”? a commenter put “for any first-order theory 𝑇 with infinite models, one can prove the following in NBG set theory with the axiom of global ...
Lave Cave's user avatar
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