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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.
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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 …
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6
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1
answer
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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
17
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1
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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 …
- 16.6k
8
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2
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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 Ke …
8
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1
answer
708
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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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3
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3
answers
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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 …
- 16.6k
4
votes
1
answer
437
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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? …
- 16.6k
2
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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 …
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9
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2
answers
917
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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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7
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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 …
6
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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 …
- 16.6k
21
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9
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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 s …
- 16.6k
2
votes
1
answer
313
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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 throu …
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3
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0
answers
473
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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 "standard" na …
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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