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nomadreid
Gold Member
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The questions are based on the exposition in http://jfera.web.wesleyan.edu/docs/nonstandard.pdf
R* is identified with the set of sequences from RN. Identifying each r in R with the corresponding constant sequence, R is obviously a subset of R* and there are elements of R* which can be considered infinitesimal, such as the decreasing harmonic sequence and so forth, and using an ultrafilter one has equivalence classes for each real number, and so forth. Fine. What I am not sure of is whether it would not be sufficient to restrict R* to the collection of convergent (from the right?) sequences modulo the ultrafilter. By sufficient I mean in order to carry through the necessary operations in nonstandard analysis.
Second question: If the answer is yes, then could one define them either as
(a) the set spanned (under termwise additio0n and multiplication, modulo the ultrafilter) by
R U the set of such convergent sequences, or
(b) the set spanned by R U {the harmonic sequence}?
Third question (regardless of the answer to the first one): are (a) and (b) above equivalent?
Thanks.
R* is identified with the set of sequences from RN. Identifying each r in R with the corresponding constant sequence, R is obviously a subset of R* and there are elements of R* which can be considered infinitesimal, such as the decreasing harmonic sequence and so forth, and using an ultrafilter one has equivalence classes for each real number, and so forth. Fine. What I am not sure of is whether it would not be sufficient to restrict R* to the collection of convergent (from the right?) sequences modulo the ultrafilter. By sufficient I mean in order to carry through the necessary operations in nonstandard analysis.
Second question: If the answer is yes, then could one define them either as
(a) the set spanned (under termwise additio0n and multiplication, modulo the ultrafilter) by
R U the set of such convergent sequences, or
(b) the set spanned by R U {the harmonic sequence}?
Third question (regardless of the answer to the first one): are (a) and (b) above equivalent?
Thanks.
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