Nonstandard analysis (NSA) is a branch of mathematical logic that reinterprets analysis using infinitesimals (infinitely small numbers). Rigorously defined infinitesimals allow us to do away with the epsilons and deltas, and to give more intuitive definitions and easier proofs of basic theorems from calculus.
In this talk, I will focus on the internal set theory (IST), Edward Nelson's approach to the NSA. IST is an extension of set theory obtained by adding to the ZFC a new predicate ('standard'), together with three axioms governing the use of the new predicate. In this theory, infinitesimals are real numbers whose absolute value is smaller than any positive standard real. At the end of the talk, I will present some non-standard definitions of basic calculus concepts, and time permitting, give non-standard proofs of some basic theorems from calculus.
This talk only assumes familiarity with set theory and calculus.