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
Other relations can be extended the same way: a R b if and only if
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.
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.Examples