Ultraproduct
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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
- <math>\prod_{i \in I} F_i<math>
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
- <math>\left\{ i \in I: a_i = b_i \right\}\in U,<math>
and the ultraproduct is the quotient set with regard to ~. The ultraproduct is therefore sometimes denoted by
- <math>\prod_{i\in I}F_i/U.<math>
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
- <math>\left\{ i \in I: a_i\,R\,b_i \right\}\in U.<math>
In particular, if every Fi is an ordered field, then so is the ultraproduct.
Łoś' theorem
Łoś' 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.