Editing Model Theory (section)

== <span style="color: #FFFFFF;">Remembering</span> ==
* '''Model Theory''' — The study of the classes of mathematical structures from the perspective of mathematical logic.
* '''Language ($L$)''' — A set of symbols (constants, functions, relations) used to build statements.
* '''Structure ($\mathcal{M}$)''' — A "World" consisting of a set of objects and definitions for the symbols in the language.
* '''Model''' — A structure that makes all the axioms of a theory true ($\mathcal{M} \models T$).
* '''Satisfaction ($\models$)''' — The relationship between a structure and a statement (e.g., "The integers satisfy the statement 1+1=2").
* '''Isomorphism''' — A mapping between two models showing they are identical in structure even if their labels are different.
* '''Elementary Equivalence''' — Two models are equivalent if they satisfy all the same statements in a given language.
* '''Löwenheim–Skolem Theorem''' — A theorem showing that if a theory has an infinite model, it must have models of every possible infinite size.
* '''Categoricity''' — A property where a theory has only one possible model (up to isomorphism).
* '''Stability Theory''' — A branch of model theory that classifies models based on how many "types" of elements they contain.
* '''Quantifier Elimination''' — A method to simplify statements by removing "For all" or "There exists" symbols.
* '''Ultraproduct''' — A way to combine an infinite number of models into one "Super-model."
* '''Non-Standard Model''' — A model that satisfies a theory but looks very different from the "usual" one (e.g., a model of arithmetic with "infinite" numbers).
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