Julius Ross, February 22, 2006


Title: Non-standard analysis: infinitesimals and the hyperreals.

It is well known (to many, if not all) that the real numbers R can
be constructed from the rational numbers.  In this talk we will see
how we can further extend R to get an ordered field which includes
"infinitesimals" i.e. positive numbers that are infinitely close to
zero.  Using this we can give new definitions to familiar concepts
(e.g. limits) and simpler proofs (e.g. of the chain rule).

I will introduce the transfer principle which ,for instance, allows
one to translate between the statement "P holds for sufficiently
small real numbers \epsilon" and "P holds for some infinitesimal
\epsilon".  If time allows I will sketch how these ideas give a
conceptual proof of the Jordan curve theorem.