Nonstandard analysis

Summary

Nonstandard analysis is a way of doing analysis with genuine infinitesimal and infinite numbers. Abraham Robinson founded it in the 1960s, using model theory to build an ordered field of hyperreal numbers that extends the real numbers. The transfer principle lets first-order statements pass between the reals and the hyperreals. The standard part function rounds a finite hyperreal to the nearest real. With these tools, limits, derivatives and integrals get short definitions close to the intuition of Leibniz. The theory proves the same theorems about real numbers as ordinary analysis. An axiomatic version is Nelson's internal set theory.