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Results tagged with nonstandard-analysis
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user 18939
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.
8
votes
Accepted
Interpreting Conway's remark about using the surreals for non-standard analysis
Conway was of course correct in saying that NSA is irrelevant to the surreals, but like Mike I found Conway’s further remarks about NSA
puzzling and I am not sure what he had in mind. What I think Co …
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3
votes
Surreal numbers, ultrapowers of $\Bbb R$, ordinal-valued functions and the slow-growing hier...
Alec, I'm afraid your construction does not work. You claim you obtain the surreals by forming the Dedekindean Completion of what you call the surrationals. However, since No--the surreals-- is not De …
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2
votes
Accepted
Archimedean completeness of some fields
There are many such proofs in the literature. The completeness for Hahn fields follows from the completeness for Hahn groups, since every Hahn field is a Hahn group. For a simple proof of the latter, …
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11
votes
Accepted
Survey of the history of calculus?
A History of Analysis, edited by Hans Niels Janhke, American Mathematical Society (2003) is a superb, all together scholarly collection of essays that cover a wide range of topics in the history of an …
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9
votes
Non-ZFC set theory and nonuniqueness of the hyperreals: problem solved?
As part of his question, Bell Crowell correctly observes:
"Section 9 of the Ehrlich paper discusses the relationship between R∗ and No within NBG. He presents Keisler's axioms for the hyperreals, whi …
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4
votes
Surreals and NSA: some foundational issues
Installment 2.
Having addressed a historical point in the first installment of my response to Vladimir (see below), I now turn to the first of his two interesting questions. To begin with, Theorem 20 …
6
votes
dense orders are saturated
Vladimir,
The results relating saturated real-closed fields and No go back to the following two papers of mine.
“Absolutely Saturated Models,” Fundamenta Mathematicae, 133 (1989), pp. 39-46.
“An Al …
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37
votes
How helpful is non-standard analysis?
In 1986 C. Ward Henson and H. J. Keisler published “On the Strength of Nonstandard Analysis” (The Journal of Symbolic Logic, Vol. 51, No. 2 (Jun., 1986), pp. 377-386), which is a seminal contribution …
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7
votes
differential geometry using Robinson's infinitesimals?
I'm not aware of much. But two works worth noting are:
K.G. Schlesinger. Generalized Manifolds. Chapman & Hall/CRC, 1997.
I.O. Hamad. Generalized curvature and torsion in nonstandard analysis.
PhD t …
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17
votes
Was the early calculus inconsistent?
Contrary to Andrej Bauer’s contention, seventeenth-century calculus looks very little like SDG. Unlike in SDG, the integrals were construed as infinite sums, the intermediate value theorem was assumed …
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28
votes
Accepted
Surreal numbers vs. non-standard analysis
In the final section of my paper “The Absolute Arithmetic Continuum and the Unification of All Numbers Great and Small” (The Bulletin of Symbolic Logic 18 (2012), no. 1, pp. 1-45, I not only point out …
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