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5.3CntbleNon071110

5.3CntbleNon071110 - 1 5.3 Countable nonstandard models...

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1 5.3. Countable nonstandard models with limited indiscernibles. LEMMA 5.3.1. There exist positive integers σ 1 , τ 1 , σ 2 , τ 2 ,..., each divisible by 96, such that for all n 1 and x,y N, σ n x + τ n = σ m y + τ m (n = m x = y). σ n , τ n 96n. Proof: For n 1, let σ n = 96(p n !) and τ n = 96p n , where p n is the n-th prime. Suppose 96(p n !)x + 96p n = 96(p m !)y + 96p m . Then p n !x + p n = p m !y + p m . If n m then p n clearly divides the left side and the first term of the right side. Hence p n divides p m . Therefore n = m. If m n then also n = m. Hence n = m. Therefore x = y. QED DEFINITION 5.3.1. We fix σ n , τ n , n 1, as given by Lemma 5.3.1. Recall the standard pairing function P (Definition 3.2.1). We have P(n,m) n,m. We use the extension P(x,y,z) = P(P(x,y),z). We have P(n,m,r) n,m,r, and P is strictly increasing in each argument. The following Lemma adjoins r sets E 1 ,...,E r N to our basic standard countable structure (N,<,0,1,+,-,•, ,log). LEMMA 5.3.2. Let r 3. There exists a countable structure (N,<,0,1,+,-,•, ,log,E 1 ,...,E r ) such that the following holds. i) E 1 ... E r N\{0}; ii) |E 1 | = r and E r is finite; iii) For all x < y from E 1 , x < y; iv) Let 1 i γ (r), 1 j < r, 0 a,b < r, and x α (r,E j ;1,r). Then ( v 2 ,...,v r E j+1 )(v 2 ,...,v r ax+b ϕ [i,r](x,v 2 ,...,v r )) σ P(i,a,b) x+ τ P(i,a,b) E j+1 ; v) For all 1 j r-1, 2 α (r,E j ;1,r)+1, 3 α (r,E j ;1,r)+1 E j+1 ; vi) E 1 α (r,E 2 ;2,r) = ; vii) Let 1 i β (2r), x 1 ,...,x 2p E 1 , y 1 ,...,y r α (r,E 2 ), where (x 1 ,...,x r ) and (x r+1 ,...,x 2r ) have the same order type and min, and y 1 ,...,y r min(x 1 ,...,x r ). Then t[i,2r](x 1 ,...,x r ,y 1 ,...,y r ) E 3 t[i,2r](x r+1 ,...,x 2r ,y 1 ,...,y r ) E 3 . Proof: Let r 3, and let r’ >> r. We define γ (r)+3r-ary g BAF, rng(g) 48N, as follows. Let x# =

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2 (y 1 ,...,y γ (r) ,z 1 ,...,z r ,w 1 ,...,w r ,x,x 2 ,...,x r ) N γ (r)+3r . Let i be greatest such that y 1 = ... = y i+1 . Let a be greatest such that z 1 = ... = z a+1 . Let b be greatest such that w 1 = ... = w b+1 . (It will prove to be convenient to write x here instead of x 1 .) case 1. 0 < |x#| (3+a+b)x |x 2 ,...,x r | ax+b ϕ [i,r](x,x 2 ,...,x r ). Set g(x#) = σ P(i,a,b) x+ τ P(i,a,b) . case 2. Otherwise. Set g(x#) = 96|x#|+48. In case 1, g(x#) 96P(i,a,b)x 96P(1,a,b)x 96max(1,a,b)x 32(1+a+b)x 8(3+a+b)x 8|x#| > |x#|. Also g(x#) σ P(i,a,b) |x#|+ τ P(i,a,b) ( σ P(i,a,b) + τ P(i,a,b) )|x#|. In case 2, |x#| < g(x#) 192|x#|. Hence g ELG SD BAF. Clearly rng(g) 48N. So we can apply Lemma 5.2.12 with r’. Let D 1 ,...,D r’ N be given by Lemma 5.2.12. For all 1 < i r, set E i = D i . Set E 1 to be the first r elements of D 1 . Claims i),ii),iii),v),vi) for E 1 ,...,E r follow immediately from clauses i),ii),iii),v),vii) for D 1 ,...,D r’ in Lemma 5.2.12. For claim iv), let 1 i γ (r), 1 j < r, 0 a,b < r, and x α (r,E j ;1,r). We claim that σ P(i,a,b) x+ τ P(i,a,b) 48 α (r’,E j ;1,r’). To see this, write |d 1 ,...,d k | x = t[i,r](d 1 ,...,d k ) r|d 1 ,...,d k |, where k 1 and d 1 ,...,d k E j . Since 96| σ P(i,a,b) ,96| τ P(i,a,b) , let p = σ P(i,a,b) /48 and q = τ P(i,a,b) /48. Then we have px+q α (r’,E j ;1,r’) since r’ >> r and p,q > 0. This establishes the claim.
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5.3CntbleNon071110 - 1 5.3 Countable nonstandard models...

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