Non-standard analysis

Non-standard analysis is that branch of mathematics that formulates analysis using a rigorous notion of infinitesimal, where an element of an ordered field F is infinitesimal if and only if its absolute value is smaller than any element of F of the form 1/n, for n a natural number. Ordered fields that have infinitesimal elements are also called non-Archimedean. More generally, non-standard analysis is any form of mathematics that relies on non-standard models and the transfer principle.

Related Topics:
Mathematics - Analysis - Infinitesimal - Ordered field - Absolute value - Non-Archimedean - Non-standard models - Transfer principle

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Non-standard analysis was introduced in the early 1960s by the mathematician Abraham Robinson. Robinson's original approach was based on so-called non-standard models of the field of real numbers. His classic foundational book on the subject Non-standard Analysis was published in 1966. The book has been reissued in paperback by Princeton University Press (see reference below) and is widely available in popular bookstores.

Related Topics:
Abraham Robinson - 1966

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Several technical issues must be addressed to develop a calculus of infinitesimals. For example, it is not enough to construct an ordered field with infinitesimals. See the article on hyperreal numbers for a discussion of some of the relevant ideas.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~