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5.4limForm071110

# 5.4limForm071110 - 1 5.4 Limited formulas limited...

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1 5.4. Limited formulas, limited indiscernibles, x-definability, normal form. We fix M = (A,<,0,1,+,-,•, ,log,E,c 1 ,c 2 ,...) as given by Lemma 5.3.18. DEFINITION 5.4.1. Let L(E) be the first order predicate calculus with equality, using <,0,1,+,-,•, ,log,E, where E is 1-ary. The c's are not allowed in L(E). We will always write t E instead of E(t). We follow the convention that ϕ (v 1 ,...,v k ) represents a formula of L(E) whose free variables are among v 1 ,...,v k . This does not require that v k be free or even appear in ϕ . Recall that all variables are of the form v n , where n 1. In this section, we will only be concerned with what we call the E formulas of L(E). DEFINITION 5.4.2. The E formulas of L(E) are inductively defined as follows. i) Every atomic formula of L(E) is an E formula; ii) If ϕ , ψ are E formulas then ( ¬ ϕ ), ( ϕ∧ψ ), ( ϕ∨ψ ), ( ϕ→ψ ), ( ϕ↔ψ ) are E formulas; iii) If ϕ is an E formula and n 1, then ( v n E)( ϕ ), ( v n E)( ϕ ) are E formulas. DEFINITION 5.4.3. We take ( v n E)( ϕ ), ( v n E)( ϕ ) to be abbreviations of ( v n )(v n E ϕ ), ( v n )(v n E ϕ ). Although general formulas of L(E) will arise in this section, content will be focused on their relativizations, which are E formulas of L(E). DEFINITION 5.4.4. Let ϕ (v 1 ,...,v k ) be a formula of L(E) and v be a variable not among v 1 ,...,v k . We let ϕ (v 1 ,...,v k ) v be the result of bounding all quantifiers in ϕ (v 1 ,...,v k ) to E [0,v]. I.e., we replace each quantifier

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2 ( u) by ( u E [0,v]) ( u) by ( u E [0,v]). These bounded quantifiers are (can be) expanded in the well known way to create an actual formula in L(E). We now define a very important 6-ary relation. DEFINITION 5.4.5. We define A(r,n,m, ϕ ,a,b) if and only if i) r,n,m,a,b N\{0}, n < m; ii) ϕ = ϕ (v 1 ,...,v r ) is a formula of L(E); i.e., all free variables of ϕ are among v 1 ,...,v r ; iii) Let x 1 ,...,x r E [0,c n ]. Then ϕ (x 1 ,...,x r ) c_n aCODE(c m ;c n ,...,c m-1 ,x 1 ,...,x r )+b E. LEMMA 5.4.1. Let r,n 1 and ϕ (v 1 ,...,v r ) be a quantifier free formula of L. There exist a,b such that A(r,n,n+1, ϕ ,a,b). Proof: Let r,n, ϕ be as given. Note that ϕ c_n = ϕ . By Lemma 5.3.18 vii), let a,b N\{0} be such that the following holds. Let n 1 and x 1 ,...,x r E [0,c n ]. ( y E)(y |c n ,x 1 ,...,x r | ↑↑ y |x 1 ,...,x r | ϕ (c n ,x 1 ,...,x r ,y)) ( y E)(y |c n ,x 1 ,...,x r | ↑↑ ρ (c n ,x 1 ,...,x r ,y)) aCODE(c n+1 ;c n ,x 1 ,...,x r )+b E. ϕ (x 1 ,...,x r ) aCODE(c n+1 ;c n ,x 1 ,...,x r )+b E. Hence A(r,n,n+1, ϕ ,a,b). QED Note that in the proof of Lemma 5.4.1, we have written the second equivalent in a form that justifies the application of Lemma 5.3.18 vii). The formula ρ used can be read off easily from the first equivalent. We will be using this style of exposition throughout this section. LEMMA 5.4.2. Let r 1 and ϕ be t E, where t(v 1 ,...,v r ) is a term of L. There exist a,b such that the following holds.
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5.4limForm071110 - 1 5.4 Limited formulas limited...

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