Theories and Models

Mathematical Logic · Axiom Academy

Explore the fundamental concepts of model theory: how formal theories describe mathematical structures, and how structures satisfy or refute logical statements. A theory collects all the truths we can derive from some basic assumptions. Think of it as the "rule book" that governs a mathematical universe. The key property is closure: if we can prove something from the theory, it's already part of the theory. Not all theories can be captured by a finite or even computable set of axioms. An axiomatizable theory has a "starting point"—a set of axioms from which everything else follows. This is what makes mathematics practical: we don't need to memorize every theorem, just the axioms! 3. Models: Structures That Satisfy Theories A model breathes life into abstract syntax. It's a concrete mathematical structure—with elements, operations, and relations—that makes all the sentences of T true. One theory can have many different models, revealing different "flavors" of the same logical structure. 4. Mod(T): The Class of All Models Mod(T) captures the "semantic content" of a theory. Two theories with the same model class are semantically equivalent, even if they have different axiomatizations. This class perspective is central to modern model theory. 5. Th(A): The Theory of a Structure

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