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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In what sense is $\Bbb R(x)$ an "instantiation" of the hyperreals?

I'm teaching myself about hyperreal numbers. My main motivation for doing so is that they include infinite numbers, whose existence I hear disputed & doubted often as "quantifying the ...
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How to make sense of the definition for the derivative using the standard part function?

If the standard part function of Δy/Δx is dy/dx. How do you show that using the increment theorem? Increment Theorem: Δy = dy + εΔx => Δy = f'(x)Δx + εΔx I tried it but I don't know how to go ...
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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 ...
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Why does this nonstandard model of Robinson Arithmetic fail to be a model of PA?

As talked about in this question, there is a nonstandard model of Robinson arithmetic of the form: $z_0 + z_1\omega + z_2\omega^2 + z_3\omega^3 + ... + z_n\omega^n$ in which $\omega$ is a formal ...
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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 ...
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"Real-closed" vs "transfer principle"

The hyperreals are "real-closed," which means that any first-order statement that is true of the reals is true of the hyperreals. However, they also satisfy a "transfer principle," ...
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Formula for probabilistically selecting a good server

There are $N$ nodes, each has response time $t_i$ (here and further I use $i$ as an index). I want probabilistically choose a node with low response time; if there are several nodes with low response ...
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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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Nonstandard proof of Lebesgue Number Lemma

I am currently going through Munkres's Topology but trying to prove things with nonstandard methods. The one I am struggling with right now is the Lebesgue Number Lemma, not too sure if my proof is ...
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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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Are all Hyperreal Infinitesimals representable by Monotonically Decreasing Sequences to 0?

I know there are many possible theoretical ways to built *R, including axiomatic and set-theoretic approaches. I am limiting my attention specifically to the Superstructure approach, perhaps best ...
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Uniform distributions in non-standard analysis

Can we use non-standard analysis to define uniform distributions over: $\mathbb{N}$, $\mathbb{Q}$, $\mathbb{R}$ and $\mathbb{R}^k$? What about $L^0$, $L^1$, $L^2$, $L^p$ and $L^\infty$? The meaning ...
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Non-Standard Analysis: applying the chain rule using differential notation?

I asked a question about applying the chain rule to the original theory of Infinitesimal Calculus (as Leibniz originally defined it). I provided an answer based on feedback. Does anyone know if this ...
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Some questions about the order on the hyperreals and the Ramsey theory on it.

Let $^* \mathbb R$ denote the set of hyperreal numbers, which is constructed with a nonprincipal ultrafilter on $\mathbb N$. Since we can order-embed $\omega_1$ into $^* \mathbb R$, by the Erdős-...
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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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Definition of Continuity in Non Standard Analysis (Internal Set Theory)

In Alain M. Robert's book, he affirms that continuity is implicitly defined by S-continuity and give the following example: Let $I$ be a standard interval and $\mathcal{F}$ be the space of numerical ...
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Do hypercontinuous fields exist?

"Hypercontinuity" is a cardinality of a continuous set's power set (set of all subsets). When talking about fields, I mean the cardinality of field's set. At first glance there is nothing ...
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Defining the Derivative using Internal Set Theory (Non Standard Analysis)

As an engineer, I have found NSA (Non Standard Analysis) to be much closer to our intuition than traditional calculus. Since there are basically two approaches to NSA, one using Hyperreals and another ...
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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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Best way to replicate logical operations using standard definitions

I want to replicate logical operations using standard function definitions. I found that the logical operation ...
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When is the transfer a subset not its inclusion Into Ultra-power?

Prompted by a really cool proof of the Hahn–Banach theorem that relies only on nonstandard-analysis and the ultrafilter lemma, I have decided to take some time to learn more about the non-standard ...
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In the absence of CH, there are just $2^{2^{\aleph_0}}$ ultrapowers of $\mathbb{R}$ of length $\omega$ up to isomorphism.

We consider structures of the language of ordered fields. In the absence of CH, there are just $2^{2^{\aleph_0}}$ ultrapowers of $\mathbb{R}$ of length $\omega$ up to isomorphism. This seems to be a ...
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When is 0.99999… ≠ 1?

There are many ways to see that $0.999\ldots=1$ over the Reals (or over $\Bbb Q$ or $\Bbb C$ for that matter) like "Is it true that 0.99999…=1?", and the reasoning is easy enough: If $x=0.\...
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Keisler measures obtained from hyperfinite samples

Let $M$ be an infinite model. Let $F$ be the set of maps $M\to\mathbb R^+\cup\{0\}$ with finite support (i.e. 0 almost everywhere). Let $\langle M,\mathbb R, F\rangle$ a 3-sorted expansion of $M$ in a ...
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Can every nonarchimedean ordered field be embedded in some hyperreal field?

Let $F$ be a nonarchimedean ordered field. Is there always a hyperreal field $^*\mathbb{R}$ such that there is an embedding of $F$ in $^*\mathbb{R}$? As far as I understand it, the answer here implies ...
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In what way does the nonstandard definition of microcontinuity differ from that of epsilon-delta continuity, and related quibbles

During a very cursory glance over the Wikipedia articles on non-standard calculus, I spotted the following definitions of continuity, uniform continuity and "microcontinuity", and I wonder ...
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Zero divisors in the hyperreal numbers

I am currently reading this introduction to hyperreal numbers. On the first page, to illustrate the problem with just taking hyperreal numbers to be sequences of reals, the following example is used: $...
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$\otimes$-products of global types in $\text{RCF}$

I'm struggling to understand a remark in Pierre Simon's book on NIP theories. Let $T$ be a complete theory and $\mathfrak{U}\models T$ a monster model. Suppose $p(x),q(y)\in S(\mathfrak{U})$ are ...
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Do these statements about analysis with dual numbers make any sense?

I am reading Color for the Sciences by Jan Koenderink, and in Ch. 3 he introduces the dual number system to define the space of possible power spectra for a beam of light. However, his statements do ...
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Different versions of Szemeredi's Regularity Lemma

I am having a hard time seeing how the statement of Szemeredi's Regularity Lemma used by Tao in this blogpost (Lemma 18) https://terrytao.wordpress.com/2010/11/27/nonstandard-analysis-as-a-completion-...
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Countable saturation and internal sequences

I am trying to solve this exercise on ultrapower constructions and nonstandard objects. The definition of the countable saturation property I am working with is: Suppose $ \lbrace B_n \rbrace_{n \in \...
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A positive number smaller than any positive infinitesimals

I've been searching everywhere for an answer, but to no avail. I know that the hyperreals are a quotient of the space $\mathbb{R}^\mathbb{N}$ of sequences of real numbers modulo a fixed free ...
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Is hyperreal field internally ismorphic to the real field?

According to this answer $^{*}\mathbb{R}$ is internally Dedekind complete. If I am not wrong, by uniqueness of $\mathbb{R}$, that makes $^{*}\mathbb{R}$ internally isomorphic to $\mathbb{R}$. But this ...
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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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Are there any significant/meaningful ultrapowers other than the hyperreals?

I have recently begun reading about non-standard analysis. According to this wikipedia article it is possible to construct an ultrapower $M^I/\mathcal{U}$ from any structure $M$ and index set $I$ with ...
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Cognitive view of Infinitesimals in Keisler's Book through Cauchy's idea of infinitesimals?

Edit - I wanted to acknowledge the rather repeated and possibly awkward use of the word 'Cognitive'. All that I can say is that when I asked this question, It was the first time I had started to learn ...
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Cauchy completeness of an ordered field

Let $\Omega(t)=\Bbb{R}(t)$ denote all rational functions with real coefficients. Supposing $x(t),y(t) \in \Omega(t)$, a total strict order $<$ is defined where $x(t)<y(t) \iff \exists \, T \in \...
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Can we have nonstandard/transfinite "decimal" expansions?

Can we take the decimal (or any arbitrary base) representation of a real number and just append some more digits beyond it? Is there a theory that covers this, maybe some kind of non-standard analysis?...
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Can numbers smaller than infinitesimals exist?

I have a good idea of infinitesimals to some extent.( A bit of non standard analysis) I am reading the book of keisler on non standard analysis and calculus. I am okay with them all but, if "a&...
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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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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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What is difference between derivative in standard and non standard analysis?

I am reading the book on complex analysis by Tristan Needham. In that book he explains derivative in an intuitive way as a quantity by which dx is expanded to get dy in both complex and real number ...
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Any Equivalence between Limit computed via L'Hospital's rule and 0/0 i.e $(\frac{\lim_{x \to c}f(x)}{\lim_{x \to c}g(x)})$?

In Standard Analysis, Does $$\lim_{x \to c}\frac{f(x)}{g(x)}=\frac{\lim_{x \to c}f(x)}{\lim_{x \to c}g(x)}$$ Given that $$\lim_{x \to c}f(x)=\lim_{x \to c}g(x)=0$$ Yes, we can get the computed answer ...
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On the infinitesimals. [duplicate]

I have completed my calculus course a few months back, I was introduced to the idea of infinitesimals as just a symbol in that course. Now I see that they are being used not only in math but in many ...
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Non-standard model of naturals and Löwenheim–Skolem

I'm taking a beginner course in mathematical logic. In the proof of some properties of non-standard natural numbers, the lecturer has used the downward Löwenheim–Skolem theorem, which I didn't ...
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Does any theory of infinite quantities provide info for $-1$ raised to infinite power, its absolute value, finite part and a series or integral form?

I've heard about surreal numbers, hyperreal numbers, Hardy fields, nonstandard analysis, cardinal ariththmetic, ordinal arithmetic, games, etc. My impression is that neither of them can exactly show ...
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What is a starfinite set?

In the book Field Arithmetic by Fried and Jarden, the following definition is given on p. 273: Consider an enlargement of a higher order structure that contains both $P$ and $K$. Call the elements of ...
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Looking for a comprehensive text that compares the development of calculus using limits vs infinitesimals

I am looking for a book that covers the development of calculus using the ideas of limits working within the reals and also infinitesimals by extending the reals to hyperreals. I have seen people talk ...
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Thinking about a uniform probability distribution over the natural numbers

I well know that in standard probability theory there is no way to define a uniform probability distribution over the natural numbers in the strict sense, but nonetheless, I'm trying to form an ...
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$f:X\to Y$ is an open map. Fix $x\in X$. Why is $\bigcap_{x\in U\in T_1}{}^*f({}^*U)\subseteq {}^*f(\bigcap_{x\in U\in T_1}{}^*U)$ true?

Suppose that $f:X\to Y$ is an open map. Fix $x\in X$. Why is $\bigcap_{x\in U\in T_1}{}^*f({}^*U)\subseteq {}^*f(\bigcap_{x\in U\in T_1}{}^*U)$ true? https://iugspace.iugaza.edu.ps/bitstream/handle/20....
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