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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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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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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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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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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Points of the sheaf topos over Blass' category
There is a site $\textsf{Blass}$ used for (constructive) non-standard analysis, whose objects are sets equipped with a filter, and morphisms are continuous functions defined up to a small set. (It is ...
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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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In constructive set theory, is it consistent for there to be a ring that models smooth infinitesimal analysis?
In a constructive set theory such as CZF, it is consistent to assume that every function $f : \mathbb R \to \mathbb R$ is continuous. However, it is not consistent to assume that every such function ...
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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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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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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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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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Comparison of model-theoretic and axiomatic approaches to NSA
This question is motivated by the discussion in the comments to this
post. The question concerns
a comparison of model-theoretic (extension) approaches to nonstandard
analysis, and axiomatic (...
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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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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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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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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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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 ...