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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0answers
122 views

Is anything known about $\Delta_n$ bounding?

For a class $\Gamma \in \{ \Sigma_n, \Pi_n, \Delta_n \}$ in the arithmetical hierarchy, we can consider the induction, bounding, and least number principles for $\Gamma$: $\mathsf{I}\Gamma$ is $\big[ ...
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Can $\mathsf{RCA}_0$ prove that every nonempty c.e. set $A \subseteq \mathbb{N}$ has a least element?

In other words, can $\mathsf{RCA}_0$ prove that for every function $f\colon \mathbb{N} \to \mathbb{N}$, there is $b \in \mathbb{N}$ such that $$ \exists k \in \mathbb{N},\ f(k) = b\quad \land\quad \...
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Legitimacy of the shadow map serving as a field homomorphism for a specific hyperfinite field formed of a union of hyperfine lattices

I'm hoping to get some comment on the legitimacy of my approach to creating a hyperfinite ring formed of a union of modular groups in order to obtain a field homomorphism from this hyperfinite space ...
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1answer
107 views

What's the size of non standard monad for weak topology?

There have been several works characterizing weak topology by nonstandard analysis, which give rise to the following monad ($X$ is a Hilbert space): $$\mu(0) = \{y\in{}^{*}X: \forall x\in X ~~ \...
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Request for bibliographic information

Greetings to everyone on this forum (I am a new-comer). I would like to ask the experienced members for suggestions on (as) comprehensive and systematic (as possible) bibliographic sources regarding: ...
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1answer
124 views

Loeb measures and non-standard hull of Banach spaces

$\DeclareMathOperator\Fin{Fin}$I am trying to understand the notion of "liftings" of Loeb measurable functions to internal, internally measurable functions, and its connection to non-...
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421 views

Decidability of a first-order theory of hyperreals

The theory of real closed fields is decidable. The hyperreals satisfy that theory, so we can interpret statements in the theory of real closed fields as being about hyperreals. If we add a unary ...
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1answer
183 views

Unbounded $\omega_1$-sequence in $^*\mathbb{N}$

Let $\mathcal{F}$ be a non-principal ultrafilter on $\omega$. Let $^*\mathbb{N}$ = $\mathbb{N}^\omega/\mathcal{F}$ be an ultrapower. Let $\{n_\alpha\}_{\alpha\in\omega_1}$ be a strictly increasing ...
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132 views

Isomorphism of hyperreal fields viewed as extensions of the field of reals

I asked this question on Mathematics Stackexchange but got no answer. Question. Does $ZFC$ prove that there are non-principal ultrafilters $\mathcal U$ and $\mathcal V$ over $\mathbb N$ such that the ...
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402 views

Can nonstandard fields contain $\mathbb R$ in different ways?

Suppose $e : \mathbb R \to F$ is an elementary embedding in the language of ordered fields. Can there exist an elementary embedding $e' : \mathbb R \to F$ such that $e \not= e'$? Note that it would ...
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1answer
287 views

Cofinality of infinitesimals

Suppose $\kappa$ is an infinite cardinal and $U$ is a countably incomplete uniform ultrafilter over $\kappa$. Then $\mathbb R^\kappa/U$ is nonstandard. What is the cofinality of the set of ...
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193 views

Influence of cardinal characteristics on nonstandard analysis?

As I understand, nonstandard analysis usually proceeds by taking a ultrapower of the universe by some nonprincipal ultrafilter on $\mathbb N$. There are continuum many “integers” of this model, but ...
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Continuum hypothesis in nonstandard universe

In Vladimir Kanovei's book "Nonstandard Analysis, Axiomatically", some nonstandard set theory is introduced. It seems that, one of them, DNST, is useful. When we are talking about higher order ...
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1answer
173 views

Ultrapower of amenable group

Let $\Gamma$ be an amenable group. Consider its ultrapower $^*\Gamma$. It is known that $^*\Gamma$ need not be amenable. In fact, there is a stronger notion of uniform amenability for $\Gamma$ (...
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Countable roots of unity

I recently learned about non-standard analysis and have the following question. Take the rational numbers; there is a maximal cyclotomic extension (containing all roots of the multiplicative identity)....
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152 views

How to construct “inaccessible hypernatural”?

Consider that, take a sufficient large natural number $a_1$, then take a natural number $a_2$ sufficient large to $a_1$, then take $a_3$,... Now we have a function $n \mapsto a_n$ which grows very ...
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Embedding standard function spaces into superstructure

I have a question concerning the precise handling the usual function spaces like $L^2$ in the context of the superstructure. In their paper Benci, Vieri; Luperi Baglini, Lorenzo. Generalized ...
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121 views

Self homomorphisms of hyperreals fixing the reals

What do we know about the circumstances (whether having to do with the axioms of set theory or the model itself) under which a field $F$ of hyperreals (=ultrapower of $\mathbb R$ with respect to a non-...
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698 views

A definition of topology using monads (a.k.a. halos)

In nonstandard analysis, there is a way of studying topological spaces known as "monads" (more commonly known as halos, as it turns out). The monad of a point $x$ (written $\mu(x))$ is the set of all ...
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Has anything (other than what is in the obituary witten by M. Noether) survived of Paul Gordan's defense of infinitesimals?

Question. Has anything other than what can be guessed from this obituary written by Max Noether survived of the 'defense' of infinitesimals that Paul Gordan gave in his doctoral disputation on March 1,...
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1answer
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What is the Turing degree associated with an ultrafilter $U$?

I asked Turing degree of a turing machine with access to an (arbitrary) nonstandard integer, not thinking about the possiblity that this could depend on the model used. The question was not formulated ...
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What is $\mu$-approximablity in Loeb measure (conflicting statements in books)?

In Loeb measure, a set is Loeb measurable iff it is $\mu$-approximable, where $\mu$ (roughly speaking) is a finitely additive hypervalued measure over internal sets. But I found the definitions of $\...
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567 views

Turing degree of a turing machine with access to an (arbitrary) nonstandard integer

Let us consider Turing machines (or other Turing-complete model of computation) that, in addition to their regular input, are given some integer $H$, where $H$ is positive nonstandard. This means, in ...
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Was Cauchy prescient?

Cauchy proved a sum theorem for series of continuous functions in 1821, and published another article on the subject in 1853. Michael Segre, writing in Archive for History of Exact Sciences, claimed ...
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1answer
452 views

Non standard extension of real numbers via nonprincipal ultra filters

Assume That $U,V$ are two filters on the natural number $\mathbb{N}$. We say that $U$ is equivalent to $V$ if there is a bijection $\phi: \mathbb{N} \to \mathbb{N}$ such that $\tilde{\phi}(U)=...
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1answer
624 views

Did Bishop make those comments in his oral presentation?

The 1975 published version of a 1974 talk at a workshop by Errett Bishop contains the following comment: "A more recent attempt at mathematics by formal finesse is non-standard analysis. I gather ...
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421 views

Are the definable hyper-reals, using quantifiers only over the standard reals and natural numbers, the same as the algebraic numbers?

This question arose today at Yevgeny Gordon's talk, "Will nonstandard analysis be the analysis of the future?" at the CUNY Logic Workshop. Here is my way of asking it. Consider the ordered real field ...
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620 views

Automorphisms of the hyperreals over the rationals and nontrivial automorphism groups

A classic result says the automorphism group of $\mathbb{R}$ (over $\mathbb{Q}$) is trivial. The proof is simple: every automorphism preserves squares, and hence fixes the positive reals, so it must ...
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385 views

How is compactness related to countable saturation?

By Cantor's intersection theorem every decreasing nested sequence of nonempty compact sets has a common point. A superficially similar result holds that every decreasing nested sequence of nonempty ...
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1answer
149 views

Archimedean completeness of some fields

I need a reference (different from Hahn's 1907 paper) for the following result. Theorem: If $G$ is a totally ordered abelian group, then the field $\mathbb{R}((G))$ is archimedean complete. $\...
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1answer
130 views

Compactness and omega models

If $T$ is a first order set theory having finitely many axioms, suppose the consistency of $T$ is already known and that $T$ proves existence of naturals, now suppose that $S$ is a schema and that $T+...
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2answers
200 views

What are the minimal requirements for the definable hyperreal field plus transfer?

It is interesting that to prove the transfer principle for the definable hyperreal field, one requires no more choice than for proving, for instance, the countable additivity of the Lebesgue measure. ...
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1answer
265 views

Is there a model of ZF+ACC where transfer fails for the definable hyperreals?

In 2003 Kanovei and Shelah constructed a definable hyperreal field. The ultrapower used exploits a fairly large index set so that it is clear that the usual proof of Los and transfer does not go ...
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1answer
208 views

Transfer with minimal choice

Let FUF postulate the existence of a Free UltraFilter on $\mathbb{N}$ and ACC the axiom of countable choice. Consider the superstructure on $\mathbb{R}$ and its inclusion in the bounded ultrapower. ...
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1answer
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Can there be a numerical system in which logarithms can be expressed in terms of exponentials in closed form?

The invention of complex numbers allowed to express trigonometric functions through hyperbolic ones in closed form. Is there possible an extension of real/complex numbers in which logarithms and ...
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858 views

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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1answer
583 views

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? [closed]

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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5answers
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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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3answers
788 views

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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1answer
467 views

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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484 views

“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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490 views

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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2answers
1k views

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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2answers
2k views

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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2k views

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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2answers
1k views

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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1answer
473 views

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 ...