Three questions about non-standard analysis

[nomadreid]

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 R^N. 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.

 

[jvicens]

I would like to clarify the concept of infinitesimals and ultrafilters in nonstandard analysis before addressing the questions posed. In nonstandard analysis, the set of real numbers R is extended to include infinitesimals, which are numbers that are infinitely small but not equal to zero. This extension allows for a more precise analysis of limits and continuity, as well as a more intuitive understanding of the behavior of functions.

Now, to address the first question - it is indeed sufficient to restrict R* to the collection of convergent sequences modulo the ultrafilter. This is because in nonstandard analysis, infinitesimals are defined as limits of convergent sequences that are infinitely close to zero. Therefore, by considering only convergent sequences, we are already including all the necessary infinitesimals.

Moving on to the second question, both (a) and (b) are valid ways to define infinitesimals in nonstandard analysis. However, (a) is a more general definition as it includes all convergent sequences, while (b) is a specific example of a convergent sequence (the harmonic sequence). So, while (a) and (b) are not equivalent, (b) is a subset of (a).

Lastly, regardless of the answer to the first question, (a) and (b) are not equivalent. This is because (a) includes all convergent sequences, while (b) only includes the harmonic sequence. Therefore, (a) is a larger set than (b).

In conclusion, the choice between (a) and (b) depends on the specific context and purpose of the analysis. Both definitions are valid and useful in different situations.