Questions tagged [nonstandard-analysis]
Nonstandard analysis is a way of doing calculus and analysis with infinitesimals. The historical approach of Leibniz, Euler, and others to infinitesimal calculus was gradually replaced by epsilon, delta techniques in the context of a real continuum, in the 19th century. It was not until the 1960s that Abraham Robinson developed a theory of a hyperreal continuum that allows for a development of analysis procedurally akin to that of its founders.
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What's Reeb's take on naive integers?
Georges Reeb's "claim Q" is the statement that "naive integers don't fill up $\mathbb{N}$". To anyone familiar with model theory this could easily be interpreted as the existence of nonstandard models ...
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Who said "the naive counting numbers don't exhaust $\Bbb N$"?
In the context of Robinson's framework, or more precisely its reformulation by Ed Nelson, one of the practitioners in the field expressed the sentiment something like "the naive counting numbers don't ...
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What is the modern consensus on the difficulty of infinitesimals?
At a related thread at MSE an expert in reverse mathematics noted that "As the modern consensus is that only nonstandard models have infinitesimals, it will be quite challenging to give a concrete ...
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What is... a grossone?
Y. Sergeyev developed a positional system for representing infinite numbers using a basic unit called a "grossone", as well as what he calls an "infinity computer". The ...
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What are the advantages of the more abstract approaches to nonstandard analysis?
This question does not concern the comparative merits of standard (SA) and nonstandard (NSA) analysis but rather a comparison of different approaches to NSA. What are the concrete advantages of the ...
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Bibliographic request concerning an article by Bernstein and Robinson
Concerning the article "Bernstein, Allen R.; Robinson, Abraham.
Solution of an invariant subspace problem of K. T. Smith and
P. R. Halmos. Pacific J. Math. 16 1966 421-431" I am interested in
finding ...
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Pontryagin dual of the surreal numbers?
Has any work been done on the Pontryagin dual of the surreal numbers (suitably topologized)? I have not been able to find anything and am not sure if this is still unknown.
Alternatively, has this ...
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"Lebesgue-measurable" cardinals and real-closed fields
I understand the motivation behind measurable cardinals is to ask the question: "is there any set large enough to admit a non-trivial measure on all of its subsets?"
Hence, it's also worthwhile to ...
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Are there any standard analysis facts that can be proven or arrived only by means of non-archimedean extensions of reals and non-standard analysis?
I wonder whether non-standard analysis, non-archimedean extensions of reals such as surreal or hyperreal numbers can help us in obtaining standard-analytic results which are not possible to obtain by ...
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Obtaining graphics of functions in non-standard analysis [closed]
In the context of $R(\varepsilon)$ or more broad fields, Levi-Civita field or $No(\omega_1)$, how can we obtain the graphics of functions on the infinitesimal range?
For instance, it is alleged that ...
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Are hyperreal numbers isomorphic to formal power series?
I wonder whether hyperreal numbers isomorphic with formal Laurent series?
It seems that any hyperreal number can be represented in the form of Laurent series over $\omega$. For instance,
$e^{\omega}=...
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Which universities teach true infinitesimal calculus? [closed]
My colleague and I are currently teaching "true infinitesimal calculus" (TIC), in the sense of calculus with infinitesimals, to a class of about 120 freshmen at our university, based on the book by ...
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Is non-existence of the hyperreals consistent with ZF?
I know that it is possible to construct the hyperreal number system in ZFC by using the axiom of choice to obtain a non-principal ultrafilter. Would the non-existence of a set of hyperreals be ...
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Salvaging Leibnizian formalism?
Can one justify Leibniz's formalism in a suitable algebraic or topological context?
We have published some papers recently where we argue that Leibniz's formalism for the calculus wasn't ...
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Literature that helps explain what the theory of numerosities contributes with
Since 2003 a group of Italian mathematicians (Benci, Di Nasso and Forti) has developed a new measure for infinite sets that satisfies the Euclidian principle: The whole is greater than the part. The ...
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Berkovich Analytification of the transseries
I am looking for references to articles about the following subjects:
Connections from the field of (real) transseries to the field of surreal numbers (mentioned very briefly in the introduction of ...
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Survey of the history of calculus?
Boyer 1939 is a nice readable survey of the history of the calculus, but it's showing its age. Discussing the notion of instantaneous velocity, he has:
Mathematics knows no minimum interval of ...
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More information on Kruskal's treatment of Surreal numbers as an asymptotic behavior of a real valued function
The only way that I could think about Surreal numbers is how Conway defined them inductively, with the two axioms and so on. I can't find any information about Kruskal's point of view and would very ...
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Is a model of arithmetic contained in a model of arithmetic an initial segment?
It's easy enough to show that if $\mathbb{N}_1$ is a non-standard model of the Peano axioms, then there is a canonical embedding $\mathbb{N} \to \mathbb{N}_1$, and we have a theorem that if $x \in \...
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Surreal numbers, ultrapowers of $\Bbb R$, ordinal-valued functions and the slow-growing hierarchy
Philip Ehrlich's paper “The Absolute Arithmetic Continuum and the Unification of All Numbers Great and Small”, The Bulletin of Symbolic Logic 18 (1) 2012, pp. 1-45. claims as a theorem that, in NBG, $\...
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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 theory....
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A stronger version of supramenability?
A group $G$ is supramenable iff for all $\varnothing\ne A\subseteq G$ there is a finitely-additive left-$G$-invariant measure $\mu_A$ on $G$ with $\mu_A(A)=1$. I'm interested in a seemingly stronger ...
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Deducing Skolem's nonstandard integers from downward Lowenheim-Skolem?
If one has a nonstandard model $\mathcal{N}$ of PA and adjoins to the first-order theory the countable list of axioms $1<H,\, 2<H,\, 3<H, \ldots$ (satisfied in $\mathcal{N}$) for all the "...
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Why does CH imply that there is a unique ultrapower of $\mathbb{N}$?
I've read these words: "How many ultra products $∏_Uℕ$ exist up to isomorphism, where $U$ is a non-principal ultrafilter over $ℕ$? If continuum hypothesis(CH) holds, then obviously just one ..."
i ...
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Can nonstandard analysis be used to prove results in constructive or computable analysis?
Nonstandard analysis is a useful tool which can be used to prove a number of results in analysis.
Question
Can it also be used to prove results in computable or constructive analysis?
If so, what are ...
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Surreals and NSA: some foundational issues
Surreals and NSA: some foundational issues.
A.
Leaving aside the whole internal machinery of surreals (with funny questions like is $\omega$ an entire number and if yes is it odd or even, simple, a ...
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Euler's mathematics in terms of modern theories?
Some aspects of Euler's work were formalized in terms of modern infinitesimal theories by Laugwitz, McKinzie, Tuckey, and others. Referring to the latter, G. Ferraro claims that "one can see in ...
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dense orders are saturated
In a FOM msg of Fri May 21 19:59:44 EDT 1999,
http://www.cs.nyu.edu/pipermail/fom/1999-May/003149.html
Simpson gave a short proof of the following theorem:
Theorem. Let F be a real closed ordered ...
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Lapses of "the early proponents of the doctrine of limits"
I have a question that I have been wondering about for a long time without finding any answer. Concerning the period around 1900, Robinson commented in his 1966 book that "there is in the writings of ...
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differential geometry using Robinson's infinitesimals?
Is there a detailed treatment of differential geometry using Robinson's infinitesimals?
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Was the early calculus inconsistent?
This question does NOT concern the RIGOR, or lack thereof, of the early calculus. Rather the question is of its CONSISTENCY.
George Berkeley wrote in 1734 with reference to the early calculus that ...
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Nonstandard definition for the generator of a standard Ito diffusion
For a standard Brownian motion, the generator of the diffusion is
$$
L = \frac12 \frac{d^2}{dx^2}.
$$
Is there a nonstandard definition of this generator?
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Constructing black noise with non-standard analysis
With noise in the sense of i.i.d. random sequence,
a noise is black if it is not isomorphic to standard Gaussian white noise.
Tsirelson showed the existence of black noise through the scaling limit ...
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Translation of a non-standard analysis formulation
Usually, it is quite easy (if cumbersome) to translate a formula like "if $\varepsilon$ is infinitesimal, and if $f$ is differentiable at $a$, $f'(a)$ is the shadow of $\frac{f(a+\varepsilon)-f(a)}{\...
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tennenbaum phenomena for the reals?
Let $\mathfrak{M} = \langle R, +,\times,> \rangle$ be such that $R$ is the set of real numbers and $\mathfrak{M} \models RA^1$ (the first-order axioms for the reals). Do we have characterisations ...
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Star-transfer of powerset
What is the difference between ${^\sigma}\mathcal{P}(\mathbb{R})$, ${^\ast}\mathcal{P}(\mathbb{R})$, and $\mathcal{P}({^\ast}\mathbb{R})$? I know that $\mathcal{P}({^\ast}\mathbb{R})$ is the powerset ...
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Hyperfinite set containing the reals, with specified upper bound on internal cardinality?
Is this true? For any hyperfinite $n$ that isn't finite, there is a hyperfinite set $A$ such that $\mathbb R \subset A$ and $|A|\le n$ (that's the crucial part, of course)? Intuitively it seems ...
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Hyperreal finitely-additive measure on [0,1) assigning $b-a$ to $[a,b)$ or $(a,b]$ and infinitesimals to singletons
Is there a hyperreal-valued finitely additive measure on all the subsets of [0,1), or at least the Borel ones, that
assigns $b-a$ to $[a,b)$ and to $(a,b]$ for all $a\lt b,$ and
assigns an ...
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Nonstandard analysis in probability theory
I am quite new at nonstandard analysis, and recently I became aware of its use in probability theory mainly through the following two books:
Nelson (1987). Radically Elementary Probability Theory
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Isomorphisms between non-standard reals.
Let $U$ and $V$ be two non-principal ultrafilters over N, and $R_1$ and $R_2$ the non-standard extensions of R given by $R_1=R^N/U$ and $R_2=R^N/V$. Are they always isomorphic (I think not, but could ...
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Surreal numbers vs. non-standard analysis
What is the relationship between the surreal numbers and non-standard analysis?
In particular, is there a transfer principle for surreal numbers they way there is for NSA?
A specific situation in ...
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Non-ZFC set theory and nonuniqueness of the hyperreals: problem solved?
The reals are the unique complete ordered field. The hyperreals $\mathbb{R}^*$ are not unique in ZFC, and many people seemed to think this was a serious objection to them. Abraham Robinson responded ...
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Non standard Algebraic Topology
Let *$\mathbb R$ a field of non-standard real numbers (or any real closed field) equipped with its natural generalized metric $d(x,y)=|x-y|$. Equip *$\mathbb R^2$ and *$\mathbb R^3$ with the $\ell^1$-(...
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Nonstandard Methods ( or Model Theory ) and Arithmetic Geometry
I hear that the nonstandard methods are applied to many problems in various fields of mathematics such as functional analysis, topology, probability theory and so on.
So, I have become interested in ...
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Defining the slowest divergent series
This question might seem too fuzzy, and if so, I will be happy to withdraw it. Until then, here it is:
I know that a method of slowing a divergent series of positive reals is to replace the $n$-th ...
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Non standard algebraic geometry: shadows of varieties
In this question $\mathbb F$ is a field and $P({\mathbb F}^{n+1})$ is the projective space of dimension $n$ over $\mathbb F$. The term algebraic variety means a subset of $P({\mathbb F}^{n+1})$ ...
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In what ways did Leibniz's philosophy foresee modern mathematics?
Leibniz was a noted polymath who was deeply interested in philosophy as well as mathematics, among other things. From my mathematical readings I have the impression that Leibniz's stature as a ...
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Are all countable, nonstandard models of arithmetic given by ultrapowers?
Countable models of PA fall into two categories: the standard one $(\omega, S)$ and the nonstandard ones (all the rest). The only way I've seen to construct a nonstandard model is through taking an ...
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What is the spectrum of possible cofinality types for cuts in an ordered field? Or in a model of the hyperreals? Or in a nonstandard model of arithmetic?
I am interested to know the full range of possibilities for the cofinality type of cuts in an ordered field and in other structures, such as nonstandard models of arithmetic.
Definitions. ...
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Various flavours of infinitesimals
I'm not sure if this is a soft question, and whether it may be too broad or, on the contrary, too localized. Well, in Mathematics the concept of "infinitesimal" has been of extreme importance for ...
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