Theorems in the foundations of mathematics | Mathematical logic | Model theory | Nonstandard analysis | Universal algebra

Ultraproduct

The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structures. All factors need to have the same signature. The ultrapower is the special case of this construction in which all factors are equal. For example, ultrapowers can be used to construct new fields from given ones. The hyperreal numbers, an ultrapower of the real numbers, are a special case of this. Some striking applications of ultraproducts include very elegant proofs of the compactness theorem and the completeness theorem, Keisler's ultrapower theorem, which gives an algebraic characterization of the semantic notion of elementary equivalence, and the Robinsonโ€“Zakon presentation of the use of superstructures and their monomorphisms to construct nonstandard models of analysis, leading to the growth of the area of nonstandard analysis, which was pioneered (as an application of the compactness theorem) by Abraham Robinson. (Wikipedia).

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Set theory | Quotient set | Nonstandard analysis | Measurable cardinal | Ultrafilter (set theory) | Almost everywhere | Archimedean property | Model theory | Compactness theorem | Relation (mathematics) | Filter (set theory) | Direct limit | Direct product | Large cardinal | Abraham Robinson | Mathematics | Field (mathematics) | Hyperreal number | Cartesian product | Quotient | Signature (logic) | Equivalence relation | Structure (mathematical logic) | Mathematical logic | Abstract algebra | Measure (mathematics) | Ordered field | Transfer principle