handout011 - Math 5020 Handout 1.1 Quantifier Elimination...

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Unformatted text preview: Math 5020 Handout 1.1 Quantifier Elimination and Elementary Submodels The aim of this handout is to show that ( Q , < ) ( R , < ). For this we first prove a result of quantifier elimination. Throughout the rest of this handout we consider the language L = { < } where < is a binary relation symbol. Let T be the L-theory of dense linear orders without endpoints , i.e., T consists of sentences x ( x < x ) (irreexive) x y z ( x < y y < z x < z ) (transitive) x y ( x < y y < x x = y ) (linear order) x y ( x < y z ( x < z z < y )) (dense order) x y ( y < x ) (no left endpoint) x y ( x < y ) (no right endpoint) Theorem (Quantifier elimination for dense linear orders without endpoints) For every L-formula there is a quantifier-free L-formula such that T (we say that is equivalent in T to ). Definition A quantifier-free L-formula is said to be a clause if it is of the form 1 k where each j , j = 1 , . . . , k , is atomic. (The key point is that no i,j is negated atomic!) A quantifier-free formula is said to be in positive DNF if it is of the form 1 n where each i is a clause. Lemma A For every quantifier-free L-formula there is a quantifier-free L-formula in positive DNF such that T (we say that is equivalent in T to ). Proof Without loss of generality we may assume that is already in DNF (because there is always some in DNF so that ). Thus is of the form 1 n with each i a conjunction of atomic and negated-atomic formulas. If we can show that each i is equivalent in T to a formula in positive DNF, then it follows that is equivalent in T to a formula of the same form. Thus without loss of generality, we may assume is of the form 1 k where each i is either atomic or negated-atomic. Our objective is to show that is equivalent in T to a formula in positive DNF. We prove this by induction on k . First consider the case k = 1. Thus 1 is atomic or negated atomic. If it is atomic then there is nothing to prove (since it is already in positive DNF). Suppose is negated atomic. Then we have two cases: either is of the form x < y or it is of the form x = y . Note that T x < y ( y < x x = y ) and T x = y ( x < y y < x ) since any model of T is a linear order. Thus is equivalent in T to the disjunction of two atomic formulas.to the disjunction of two atomic formulas....
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This note was uploaded on 04/11/2011 for the course MATH 5020 taught by Professor Staff during the Spring '11 term at North Texas.

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handout011 - Math 5020 Handout 1.1 Quantifier Elimination...

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