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interpretation model theory wikipedia jump to content main menu main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 definition 2 bi interpretability 3 example 4 references 5 further reading toggle the table of contents interpretation model theory 1 language 日本語 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia concept in model theory not to be confused with interpretation logic in model theory interpretation of a structure m in another structure n typically of a different signature is a technical notion that approximates the idea of representing m inside n for example every reduct or definitional expansion of a structure n has an interpretation in n many model theoretic properties are preserved under interpretability for example if the theory of n is stable and m is interpretable in n then the theory of m is also stable note that in other areas of mathematical logic the term interpretation may refer to a structure 1 2 rather than being used in the sense defined here these two notions of interpretation are related but nevertheless distinct similarly interpretability may refer to a related but distinct notion about representation and provability of sentences between theories definition edit an interpretation of a structure m in a structure n with parameters or without parameters respectively is a pair n f displaystyle n f where n is a natural number and f displaystyle f is a surjective map from a subset of n n onto m such that the f displaystyle f preimage more precisely the f k displaystyle f k preimage of every set x m k definable in m by a first order formula without parameters is definable in n by a first order formula with parameters or without parameters respectively clarification needed since the value of n for an interpretation n f displaystyle n f is often clear from context the map f displaystyle f itself is also called an interpretation to verify that the preimage of every definable without parameters set in m is definable in n with or without parameters it is sufficient to check the preimages of the following definable sets the domain of m the diagonal of m 2 every relation in the signature of m the graph of every function in the signature of m in model theory the term definable often refers to definability with parameters if this convention is used definability without parameters is expressed by the term 0 definable similarly an interpretation with parameters may be referred to as simply an interpretation and an interpretation without parameters as a 0 interpretation bi interpretability edit if l m and n are three structures l is interpreted in m and m is interpreted in n then one can naturally construct a composite interpretation of l in n if two structures m and n are interpreted in each other then by combining the interpretations in two possible ways one obtains an interpretation of each of the two structures in itself this observation permits one to define an equivalence relation among structures reminiscent of the homotopy equivalence among topological spaces two structures m and n are bi interpretable if there exists an interpretation of m in n and an interpretation of n in m such that the composite interpretations of m in itself and of n in itself are definable in m and in n respectively the composite interpretations being viewed as operations on m and on n example edit the partial map f from z z onto q that maps x y to x y if y 0 provides an interpretation of the field q of rational numbers in the ring z of integers to be precise the interpretation is 2 f in fact this particular interpretation is often used to define the rational numbers to see that it is an interpretation without parameters one needs to check the following preimages of definable sets in q the preimage of q is defined by the formula φ x y given by y 0 the preimage of the diagonal of q is defined by the formula φ x 1 y 1 x 2 y 2 given by x 1 y 2 x 2 y 1 the preimages of 0 and 1 are defined by the formulas φ x y given by x 0 and x y the preimage of the graph of addition is defined by the formula φ x 1 y 1 x 2 y 2 x 3 y 3 given by x 1 y 2 y 3 x 2 y 1 y 3 x 3 y 1 y 2 the preimage of the graph of multiplication is defined by the formula φ x 1 y 1 x 2 y 2 x 3 y 3 given by x 1 x 2 y 3 x 3 y 1 y 2 references edit goldblatt robert 2006 11 2 formal language and semantics topoi the categorial analysis of logic 2nd ed mineola n y dover publications isbn 978 0 486 31796 0 oclc 853624133 hodges wilfrid 2009 functional modelling and mathematical models in meijers anthonie ed philosophy of technology and engineering sciences handbook of the philosophy of science vol 9 elsevier isbn 978 0 444 51667 1 further reading edit philosophy portal ahlbrandt gisela ziegler martin 1986 quasi finitely axiomatizable totally categorical theories annals of pure and applied logic 30 63 82 doi 10 1016 0168 0072 86 90037 0 hodges wilfrid 1997 a shorter model theory cambridge cambridge university press isbn 978 0 521 58713 6 section 4 3 poizat bruno 2000 a course in model theory springer isbn 978 0 387 98655 5 section 9 4 v t e mathematical logic general axiom list cardinality first order logic formal proof formal semantics foundations of mathematics information theory lemma logical consequence model theorem theory type theory theorems list paradoxes gödel s completeness incompleteness theorems tarski s undefinability banach tarski paradox cantor s theorem paradox diagonal argument compactness halting problem lindström s löwenheim skolem russell s paradox logics 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 cardinality function map domain codomain image in sur bi jection schröder bernstein theorem isomorphism gödel numbering enumeration large cardinal inaccessible aleph number operation binary 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 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 true 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 proof theory formal proof natural deduction logical consequence rule of inference sequent calculus theorem systems axiomatic deductive hilbert list complete theory independence from zfc proof of impossibility ordinal analysis reverse mathematics self verifying theories model theory interpretation function of models model atomic equivalence finite prime saturated spectrum submodel non standard model of non standard arithmetic diagram elementary categorical theory model complete theory satisfiability semantics of logic strength theories of truth semantic tarski s kripke s t schema transfer principle truth predicate truth value type ultraproduct validity computability theory church encoding church turing thesis computably enumerable computable function computable set decision problem decidable undecidable p np p versus np problem kolmogorov complexity lambda calculus primitive recursive function recursion recursive set turing machine type theory related abstract logic algebraic logic automated theorem proving category theory concrete abstract category category of sets history of logic history of mathematical logic timeline logicism mathematical object philosophy of mathematics supertask mathematics portal v t e logic history major fields computer science computational logic formal semantics natural language inference philosophy of logic proof semantics of logic syntax logics classical informal critical thinking reason mathematical non classical philosophical theories argumentation metalogic metamathematics set foundations abduction analytic and synthetic propositions antecedent consequent contradiction paradox antinomy deduction deductive closure definition description dichotomy entailment linguistic form induction logical truth name necessity and sufficiency premise probability proposition reference statement substitution truth validity lists topics mathematical logic boolean algebra set theory other logicians rules of inference paradoxes fallacies logic symbols category outline portal wikiproject changes retrieved from https en wikipedia org w index php title interpretation_ model_theory oldid 1301081015 categories model theory interpretation philosophy hidden categories articles with short description short description matches wikidata wikipedia articles needing clarification from november 2022 this page was last edited on 18 july 2025 at 00 32 utc page was rendered with parsoid text is available under the creative commons attribution sharealike 4 0 license additional terms may apply by using this site you agree to the terms of use and privacy policy wikipedia is a registered trademark of the wikimedia foundation inc a non profit organization privacy policy about wikipedia disclaimers contact wikipedia legal safety contacts code of conduct developers statistics cookie statement 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