Abstract model theory

In mathematical logic, abstract model theory is a generalization of model theory that studies the general properties of extensions of first-order logic and their models.^[1]

Abstract model theory provides an approach that allows us to step back and study a wide range of logics and their relationships.^[2] The starting point for the study of abstract models, which resulted in good examples was Lindström's theorem.^[3]

In 1974 Jon Barwise provided an axiomatization of abstract model theory.^[4]

References

^ Institution-independent model theory by Răzvan Diaconescu 2008 ISBN 3-7643-8707-6 page 3
^ Handbook of mathematical logic by Jon Barwise 1989 ISBN 0-444-86388-5 page 45
^ Jean-Yves Béziau Logica universalis: towards a general theory of logic 2005 ISBN 978-3-7643-7259-0 pages 20–25
^ J. Barwise, 1974 "Axioms for abstract model theory" , Annals of Mathematical Logic 7:221–265

Traditional	Classical logic Logical truth Tautology Proposition Inference Logical equivalence Consistency Equiconsistency Argument Soundness Validity Syllogism Square of opposition Venn diagram
Propositional	Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued logic 3 finite ∞
Predicate	First-order list Second-order Monadic Higher-order Fixed-point Free Quantifiers Predicate Monadic predicate calculus

Set theory

Set hereditary Class (Ur-)Element Ordinal number Extensionality Forcing Relation equivalence partition Set operations: intersection union complement Cartesian product power set identities
Types of sets	Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter Recursive Fuzzy Universal Universe constructible Grothendieck Von Neumann
Maps and cardinality	Function/Map domain codomain image In/Sur/Bi-jection Schröder–Bernstein theorem Isomorphism Gödel numbering Enumeration Large cardinal inaccessible Aleph number Operation binary
Set theories	Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck Von Neumann–Bernays–Gödel Ackermann Constructive

Formal systems (list),
language and syntax

Alphabet Arity Automata Axiom schema Expression ground Extension by definition conservative Relation Formation rule Grammar Formula atomic closed ground open Free/bound variable Language Metalanguage Logical connective ¬ ∨ ∧ → ↔ = Predicate functional variable propositional variable Proof Quantifier ∃ ! ∀ rank Sentence atomic spectrum Signature String Substitution Symbol function logical/constant non-logical variable Term Theory list
Example axiomatic systems (list)	of arithmetic: Peano second-order elementary function primitive recursive Robinson Skolem of the real numbers Tarski's axiomatization of Boolean algebras canonical minimal axioms of geometry: Euclidean: Elements Hilbert's Tarski's non-Euclidean Principia Mathematica