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Results tagged with nonstandard-analysis
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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.
2
votes
0
answers
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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 (syntact …
- 16.6k
4
votes
1
answer
437
views
Paris-Harrington via overspill?
I saw in an old logic paper that the Paris-Harrington theorem can be proved via Overspill. The presentation is unfortunately too technical for me to follow. Does somebody have any insight into this? …
- 17.7k
17
votes
1
answer
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Did Lagrange change his mind about infinitesimals?
Lagrange is famous for his attempt to found analysis algebraically using power series expansions, an approach that, as we know today, is limited to analytic functions. Lagrange is also known as the in …
- 13.2k
6
votes
5
answers
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Standard models of N and R: An Alice/Bob approach
This is a question about a comment in a recent publication by Roman
Kossak. Kossak wrote:
"Nonstandardness in set theory has a different nature. In
arithmetic, there is one intended object of study …
- 2,281
6
votes
1
answer
350
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Quantifier complexity of definition of compactness
This question is inspired by the post on quantifier complexity of
continuity. We work with metric spaces M
considered as two-sorted first-order structures (M,$\mathbb R$,d,+,⋅,<)
where $d:M^2→\mathbb …
- 12.4k
38
votes
6
answers
3k
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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 ab …
- 16.6k
7
votes
2
answers
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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 …
- 1,446
35
votes
9
answers
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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 mathematical value of this see …
- 63.9k
2
votes
1
answer
256
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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. I …
- 35.4k
12
votes
3
answers
881
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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 …
- 188k
1
vote
0
answers
158
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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 solut …
- 16.6k
6
votes
1
answer
727
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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 …
- 16.6k
5
votes
2
answers
486
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 …
- 16.6k
7
votes
3
answers
5k
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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 …
- 16.6k
7
votes
2
answers
1k
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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 inconsisten …
- 16.6k