"There are three main tools in nonstandard analysis. One is the transference principle, which roughly states that the same assertions of the formal language are true in the standard universe as in the nonstandard universe. It is typically used by proving a desired result in the nonstandard universe, and then, noting that the result is expressible in the language, concluding that it holds in the standard universe as well. Another technique is concurrence. This is a logical technique that guarantees that the extended structure contains all possible completions, compactifications, and so forth. The third technique is internality. A set s of elements of the nonstandard universe is internal if s itself is an element of the nonstandard universe; otherwiise, s is external. A surprislingly useful method of proof is one by reductio ad absurdum in which the contradiction is that some set one knows to be external would in fact be internal under the assumption being refuted."
Quote Details
Added by wikiquote-import-bot
Unverified quote
0 likes
Original Language: English
Available Languages (1)
Sources
Martin Davis: (originally published in 1977)
https://en.wikiquote.org/wiki/Nonstandard_analysis
Revision History
No revisions have been submitted for this quote.
Categories
Nonstandard analysis
5 quotes on TrueQuotesView all quotes by Nonstandard analysis →
Related Quotes
"Nonstandard analysis has proved to be a natural framework for studying the local properties of Banach spaces. The cen…"
"We describe and analyze a parametrization of fractal ‘‘curves’’ (i.e., fractal of topological dimension 1). The nondi…"
"It was in 1966 that A. Robinson's book ... on nonstandard analysis ... appeared. In it, a first rigorous foundation o…"
"... in the author's opinion this is the most important advantage of nonstandard analysis over standard analysis: To h…"
"Numerical analysis is often considered neither beautiful nor, indeed, profound. Pure mathematics is beautiful if your…"
"One of the most important theorems in calculus is the Mean Value Theorem (MVT), which is used to prove many theorems …"
"In the 1950s and 1960s, the founding fathers of the field discovered that inexact arithmetic can be a source of dange…"
"Mathematical techniques to achieve numerical solutions for partial differential equations began to appear about the t…"
"The subject of numerical analysis has ancient roots, and it has had periods of intense development followed by long p…"
"Problems relative to the uniform propagation, or to the varied movements of heat in the interior of solids, are reduc…"