Section 2.8: Nonstandard Analysis
In 1961 Abraham Robinson introduced a new method for treating limits... This method combines the intuitive advantages of working with infinitely small quantities with modern standards of rigor. The basic idea is to utilize a nonstandard model of the theory of the real numbers.
Let the language include the following parameters.
- , an -place predicate symbol, for every -ary relation on .
- , an -place function symbol, for every -ary operation on .
- , a constant symbol, for every .
is a structure in this language, where , , , and .
Let . By the compactness theorem, there is a structure and such that is satisfied in when is assigned .
- is an isomorphism of into .
Given the isomorphism , we can find a structure such that a) is a substructure of , b) , in particular, , and c) there is such that satisfies when is assigned .
- , so no special notation needed.
- is the restriction of to . is the restriction of to .
If a property of or can be expressed by a sentence of the language, then or , correspondingly, has the property.
- is an ordering relation.
- is a field.
The set of finite elements of is .
The set of infinitesimals of is .
- and are closed under , , as well as multiplication by a number from .
is infinitely close to , , iff is infinitesimal.
- is an equivalence relation on .
- If and , then . If are finite, then .
- If and at least one of them is finite, then there is such that .
there is unique
- For there is a unique decomposition into the standard part of , , and an infinitesimal .
- is a homomorphism of the ring onto with kernel , hence, is isomorphic to .
is dense in
: for every
- and both have cardinality at least .
- . iff is finite.
- Bolzano–Weierstrass Theorem. If is bounded and infinite, then there is that is infinitely close to, but different from a point in .
- A least upper bound property does not hold for .
, then (considered as a relation on
- If is one-to-one, then for every , .
A function converges at to iff , implies . In particular, for any . is continuous at iff implies .
exists, then for every
- If exists, then is continuous at .
- If and exist, then exists and .
- Convergence of sequences. converges to iff for every infinite .