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Unformatted text preview: 1 An Introduction to Model Theory Prepared by: Kendra Cooper Contents • Introduction • Basics of Model Theory • A Second Look at the Definition of a Model • Sentences, Axioms, and Theorems • Example: Ordered Field of Real Numbers • Example: Propositional Logic • Completeness and Soundness Theorems • Problems Introduction • Model Theory shows how to apply logic to the study of structures in pure mathematics • Model Theory has been applied to first order logic (predicate logic), ordered fields of real numbers, … • The fundamental tenet of Model Theory is that mathematical truth is relative o A statement may be true or false, depending on how and where it is interpreted & This is a consequence of the language used to express mathematical ideas Interpretation examples: & The birds fly south & The birds eat copious quantities of nectar & How do you know if the sentence is true or false? o What birds are you considering? What time of the year? & Canadian geese, in the Fall? Spring? & Hummingbirds? & Need additional information to determine if a sentence is true or false? o Interpret 2 & Do not need any additional information to determine if a sentence is true or false? o Sentence is called “fully interpreted” Basics of Model Theory A mathematical model, or structure, for a language is defined by the following: • A universe, or underlying set • A collection of functions • A collection of relations • A collection of distinguished elements Example The model, or structure, for the real numbers can be defined with these components: Domain (aka universe): R binary functions: +, x relations: < distinguished elements: 0 and 1 We defined a model for a language – what is a language? Definition of a Language A language L is a set consisting of all the logical symbols with perhaps some constant, function and/or relational symbols included. The formulae of L are made up from this set. Logical symbols Conjunction Disjunction Implication Negation Biconditonal ∀ , ∃ quantifiers infinite collection of variables indexed by the natural numbers N: v0 ,v1 , v2, … parentheses ( , ) the equal sign symbol = Constant symbols often denoted by the letter c c0 ,c1 , c2, … 3 Function symbols often denoted by the letter F each function symbol is an mplaced function symbol for some natural number m F1(c1, v1, …) Relation symbols often denoted by the letter R each relational symbol is an nplaced relation symbol for some natural number n R1(c1, v1, …) Terms (1) a variable is a term (2) a constant symbol is a term (3) if F is an mplaced function symbol and t1…tm are terms, then F(t1…tm) is a term (4) a string of symbols is a term if and only if it can be shown to be a term by a finite number of applications of (1), (2) and (3)....
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 Spring '09
 Nhut
 Logic, Model theory, Firstorder logic, Modus ponens, Model Theory A

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