Structure (mathematical logic)
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| Structure (mathematical logic) | |
|---|---|
| Name | Structure (mathematical logic) |
| Field | Mathematical logic |
| Introduced | Alfred Tarski |
| Related | Model theory, First-order logic, Set theory, Category theory |
Structure (mathematical logic) A structure in mathematical logic is a set equipped with distinguished relations, functions, and constants that interpret the nonlogical symbols of a formal language. It underlies Model theory and connects formal systems developed by Kurt Gödel, Alfred Tarski, Emil Post, Alonzo Church, and David Hilbert to concrete mathematical objects studied by Évariste Galois, Carl Friedrich Gauss, Henri Poincaré, Richard Dedekind, and Émile Borel. Structures provide semantics for sentences considered by researchers at institutions such as Princeton University, University of Cambridge, Massachusetts Institute of Technology, University of Göttingen, and University of Oxford.
Definition and Examples
A structure for a signature is a carrier set together with interpretations of each symbol in the signature; examples include the ordered field of real numbers used by Isaac Newton and Carl Friedrich Gauss, the group of integers under addition connected to Niels Henrik Abel and Évariste Galois, and the graph theoretic structures studied by Paul Erdős and Alfréd Rényi. Model-theoretic examples often reference the algebraic closure of a field as in the work of Emmy Noether and Alexander Grothendieck, the natural numbers with successor studied by Giuseppe Peano, and metric structures appearing in the research of André Weil and John von Neumann. Classical structures include Boolean algebras related to George Boole and Alonzo Church, vector spaces connected to Hermann Grassmann and David Hilbert, and topological structures linked to Henri Lebesgue and Maurice Fréchet.
Languages and Signatures
A signature (or language) specifies constant symbols, function symbols, and relation symbols as formalized by Alfred Tarski and employed in treatments by Jerzy Łoś and Saharon Shelah; signatures used at research centers like Institut des Hautes Études Scientifiques and Princeton University vary from finite algebraic signatures to richer relational vocabularies in investigations by Alfred North Whitehead and Bertrand Russell. First-order languages central to the work of Kurt Gödel and Alfred Tarski are contrasted with higher-order languages explored by Henri Poincaré and Nicolas Bourbaki, while many-sorted languages appear in applications connected to Emmy Noether and John von Neumann. The choice of signature affects definability studied by Alfred Tarski, categoricity analyzed by Mal'cev, and completeness theorems of Kurt Gödel and David Hilbert.
Homomorphisms, Embeddings, and Isomorphisms
Morphisms between structures—homomorphisms, embeddings, and isomorphisms—are central in lines of inquiry by Saunders Mac Lane in Category theory and by Alfred Tarski in Model theory; group homomorphisms trace to Niels Henrik Abel and Évariste Galois, ring homomorphisms to Richard Dedekind and Emmy Noether, and field embeddings to Évariste Galois and Alexander Grothendieck. Isomorphisms characterize structural sameness as in classification programs by William Rowan Hamilton and Sophus Lie, while elementary embeddings appear in foundational work by Kurt Gödel and set-theoretic applications by Paul Cohen and Kurt Gödel. Categorical equivalences studied by Saunders Mac Lane relate to model-theoretic notions used by Saharon Shelah and Per Martin-Löf.
Substructures and Generated Substructures
A substructure is a subset closed under the interpreting functions and containing the constants, a notion employed in algebraic studies by Emmy Noether, Richard Dedekind, and Évariste Galois; generated substructures arise in constructions by Émile Borel and combinatorial approaches of Paul Erdős. In algebraic geometry, schemes of Alexander Grothendieck can be analyzed via substructures, and lattice of substructures connects to work by Marshall Stone and Garrett Birkhoff. Finitely generated structures are studied in combinatorial group theory by Max Dehn and Wilhelm Magnus, while free structures trace to the foundational contributions of Nikolai Lobachevsky and Hermann Grassmann.
Elementary Substructures and Elementary Equivalence
Elementary substructures satisfy the same first-order sentences as the larger structure, a concept central to proofs by Alfred Tarski and compactness arguments associated with Jerzy Łoś and Mal'cev; elementary equivalence undergirds classification theory developed by Saharon Shelah and stability theory influenced by Michael Morley and Robert Vaught. The Löwenheim–Skolem theorems, proven in contexts involving Thoralf Skolem and Leopold Löwenheim, regulate cardinalities of elementary substructures, and applications range from set-theoretic constructions by Paul Cohen to algebraic model-theoretic results by Lou van den Dries and David Marker.
Ultraproducts and Elementary Extensions
Ultraproduct constructions, formalized by Jerzy Łoś and used by Alfred Tarski and Saharon Shelah, build new structures via ultrafilters studied by Henri Cartan and André Weil; Łoś's theorem ensures that ultraproducts preserve first-order properties, with landmark applications by Kurt Gödel and Paul Cohen in independence proofs. Ultraproducts yield elementary extensions and saturation concepts exploited in stability theory by Saharon Shelah and in nonstandard analysis initiated by Abraham Robinson and developed by Edward Nelson; ultrapowers underpin transfer principles used by Robinson and model-theoretic algebra pursued by Lou van den Dries and Martin Davis.