Nonstandard Analysis is usually used to introduce infinitesimals into the real numbers in an attempt to make arguments in analysis more intuitive.
The idea is that you construct a superset which contains the reals and also some infinitesimals, prove that some statement holds of , and then use a general “transfer principle” to conclude that the same statement holds of .
Implicit in this procedure is the idea that is the real world, and therefore the goal is to prove things about it. We construct a field with infinitesimals, but only as a method for eventually proving something about .
We can do precisely the same thing with instead of with . But, in Weak Theories of Nonstandard Arithmetic and Analysis, Jeremy Avigad observed that if we don’t care about transferring the results back down to , then we can get all the basic results of calculus and elementary real analysis just by working with , and without ever having to construct the reals.