Ultraproduct
An
ultraproduct is a
mathematical construction, which is used in
abstract algebra to construct new fields from given ones, and in
model theory, a branch of
mathematical logic. In particular, it can be used in a "purely semantic" proof of the
compactness theorem of
first-order logic. Certainly the most important case is the construction of the
hyperreal numbers by taking the ultraproduct of countably infinitely many copies of the field of
real numbers.
The general construction uses an index set I, a field Fi for each element i of I, and an ultrafilter U on I (the usual choice is for I to be infinite and U to contain all cofinite subsets of I).
Algebraic operations on the cartesian product
are defined in the usual way (such that (
a +
b)
i =
ai +
bi ), and an
equivalence relation is defined by
a ~
b if and only if
and the
ultraproduct is the
quotient set with regard to ~. The ultraproduct is therefore sometimes denoted by
One may define a finitely additive
measure m on the index set
I by saying
m(
A) = 1 if
A ∈
U and = 0 otherwise. Then two members of the Cartesian product are equivalent precisely if they are equal
almost everywhere on the index set. The ultraproduct is the set of equivalence classes thus generated.
Other relations can be extended the same way: a R b if and only if
In particular, if every
Fi is an
ordered field, then so is the ultraproduct.
Los' theorem
Los' theorem states that any first-order formula is true in the ultraproduct if and only if the set of indices i such that the formula is true in Fi is a member of U.
Examples
The hyperreal numbers are the ultraproduct of one copy of the real numbers for every natural number, with regard to an ultrafilter containing all cofinite sets of natural numbers. Their order is the extension of the order of the real numbers.
Analogously, you could define nonstandard complex numbers by taking the ultraproduct of copies of the field of complex numbers.