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.
Construction of *𝕽
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
.
Asterisk Notation
,
.
 , so no special notation needed.
 is the restriction of to . is the restriction of to .
Properties
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 .

For every
there is unique
such that
. If
, then
iff
.
 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 .
Applications
Subsets and relations

is dense in
: for every
there is
such that
.
 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 .

If
, then (considered as a relation on
)
.
 If is onetoone, then for every , .
Convergence
A function
converges at
to
iff
,
implies
. In particular,
for any
.
is continuous at
iff
implies
.

Derivative. If
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 .