An Introduction to Nonstandard Methods in the Plane

This article outlines ways in which nonstandard methods can be applied to continuum theory in the plane. 1. Nonstandard Models Nonstandard analysis, first developed by Abraham Robinson in the early 1960’s [10], makes use of the fact that there are other models besides the usual ones that satisfy all the same mathematical statements that can be made in First Order Logic. The basic idea is to exploit the additional properties that these models have, and yet often use the fact that they must share many properties with their standard counterparts. The additional properties roughly include “actualized”versions of limits. Whereas the standard reals contain arbitrarily small positive elements the nonstandard reals contain positive elements of “infinitesimal size” i.e. elements that are greater than zero but less than every standard positive real. A simple dense canal in the plane that has longer and longer segments approaching a certain arc will have a portion in the nonstandard plane that stays within an infinitesimal of the entire arc. A set such as a pseudoarc that is constructed using an intersection of nested sets can be characterized as the set of standard points near a single nonstandard object. Many times working with these idealized limits allows for simpler and more intuitive proofs than corresponding standard proofs once some basic nonstandard machinery is available. More importantly, the use of nonstandard models can allow us an entirely new perspective on some complicated standard objects, relating and characterizing sets in ways that allow for new insights. Our focus here will be entirely on the use of these methods as a way to obtain standard results about continuum theory in the plane. While it is possible to develop nonstandard methods purely using ultrafilters without any reference to logic, there is a fundamental benefit to starting with just a little perspective based on the nature of mathematical statements built up in the framework of first order logic. When we are studying a subject such as group theory in which we start with axioms and consider the properties of models (groups in this case) that satisfy various collections of statements we might use a very simple “language”in which we have a symbol “∗”for the group operation and the mathematical statements we are interested in are built up using that symbol along with the standard boolean connectives (∨,∧,→,¬) and the quantifiers (∀,∃). Any finite group is completely characterized by the set of statements that it satisfies, of course, but if we are interested in a particular infinite group G, basic results