handout011

# handout011 - Math 5020 Handout 1.1 Quantifier Elimination...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

## 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.

### Page1 / 4

handout011 - Math 5020 Handout 1.1 Quantifier Elimination...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online