Non-standard Analysis - First Consequences

First Consequences

The symbol *N denotes the nonstandard natural numbers. By the extension principle, this is a superset of N. The set *N − N is nonempty. To see this, apply countable saturation to the sequence of internal sets

The sequence {A_n}_{n ∈ N} has a nonempty intersection, proving the result.

We begin with some definitions: Hyperreals r, s are infinitely close if and only if

A hyperreal r is infinitesimal if and only if it is infinitely close to 0. For example, if n is a hyperinteger, i.e. an element of *N − N, then 1/n is an infinitesimal. A hyperreal r is limited or bounded if and only if its absolute value is dominated by (less than) a standard integer. The bounded hyperreals form a subring of *R containing the reals. In this ring, the infinitesimal hyperreals are an ideal.

The set of bounded hyperreals or the set of infinitesimal hyperreals are external subsets of V(*R); what this means in practice is that bounded quantification, where the bound is an internal set, never ranges over these sets.

Example: The plane (x,y) with x and y ranging over *R is internal, and is a model of plane Euclidean geometry. The plane with x and y restricted to bounded values (analogous to the Dehn plane) is external, and in this bounded plane the parallel postulate is violated. For example, any line passing through the point (0,1) on the y-axis and having infinitesimal slope is parallel to the x-axis.

Theorem. For any bounded hyperreal r there is a unique standard real denoted st(r) infinitely close to r. The mapping st is a ring homomorphism from the ring of bounded hyperreals to R.

The mapping st is also external.

One way of thinking of the standard part of a hyperreal, is in terms of Dedekind cuts; any bounded hyperreal s defines a cut by considering the pair of sets (L,U) where L is the set of standard rationals a less than s and U is the set of standard rationals b greater than s. The real number corresponding to (L,U) can be seen to satisfy the condition of being the standard part of s.

One intuitive characterization of continuity is as follows:

Theorem. A real-valued function f on the interval is continuous if and only if for every hyperreal x in the interval *,

Similarly,

Theorem. A real-valued function f is differentiable at the real value x if and only if for every infinitesimal hyperreal number h, the value

exists and is independent of h. In this case f'(x) is a real number and is the derivative of f at x.