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Translation of nonstandard definitions to standard ones. (English) Zbl 0574.03051
It is well-known that certain concepts from analysis like limit, derivative, continuity, etc. can be expressed either by using non- standard notions like ”is infinitely close” or ”infinitely large” or by using standard notions involving the familiar epsilon-delta formulation. We quote from the author’s abstract. ”An algorithm translating nonstandard definitions of notions to standard version is given. Counterexamples proving that our algorithm is in a certain sense the best one are described. It appears that in general this translation is much more complicated than in the case when the notion of a limit (and other similar notions) is translated by epsilon-delta method.”
A more precise statement of the translation result is as follows: Let st be a predicate whose meaning is ”to be a standard element of c”. For each normal formula \(\Phi\) (x,st,z), there is a set formula \(\Psi\) (x,y,z) and a set F of standard functions such that for all internal sets x,z, \(\Phi\) (x,st,z) is equivalent with (\(\exists f\in F)(\forall a\in P_{fin}(c))\Psi (x,f(a),z)\).
Reviewer: P.Clote
03H05 Nonstandard models in mathematics
26E35 Nonstandard analysis
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