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 Robinson–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.
The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory...
hypothesis holds. Ultraproducts are used as a general technique for constructing models that realise certain types. An ultraproduct is obtained from the...
\varphi } holds in the ultraproduct ∏ i ⊆ Σ M i / U . {\textstyle \prod _{i\subseteq \Sigma }{\mathcal {M}}_{i}/U.} So this ultraproduct satisfies all formulas...
constructions. For example, an ultraproduct modulo a principal ultrafilter is always isomorphic to one of the factors, while an ultraproduct modulo a free ultrafilter...
Reduced product, a construction that generalizes both direct product and ultraproduct Reduced residue system, a set of φ(n) integers such that each integer...
Kripke's T-schema Transfer principle Truth predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably...
Kripke's T-schema Transfer principle Truth predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably...
Kripke's T-schema Transfer principle Truth predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably...
Kripke's T-schema Transfer principle Truth predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably...
Kripke's T-schema Transfer principle Truth predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably...
Kripke's T-schema Transfer principle Truth predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably...
Kripke's T-schema Transfer principle Truth predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably...
is an initially subamenable graph, or equivalently a subgroup of an ultraproduct of finite-rank symmetric groups such that every two elements of the group...
Kripke's T-schema Transfer principle Truth predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably...
compact Hausdorff spaces, and in model theory in the construction of ultraproducts and ultrapowers. Every ultrafilter on a compact Hausdorff space converges...
Kripke's T-schema Transfer principle Truth predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably...
Kripke's T-schema Transfer principle Truth predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably...
reduced product is a construction that generalizes both direct product and ultraproduct. Let {Si | i ∈ I} be a nonempty family of structures of the same signature...
Kripke's T-schema Transfer principle Truth predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably...
Kripke's T-schema Transfer principle Truth predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably...
Kripke's T-schema Transfer principle Truth predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably...
Kripke's T-schema Transfer principle Truth predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably...
Kripke's T-schema Transfer principle Truth predicate Truth value Type Ultraproduct Validity Computability theory Church encoding Church–Turing thesis Computably...
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