4.2Proof071110 - 1 4.2. Proof using Strongly Mahlo...

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1 4.2. Proof using Strongly Mahlo Cardinals. Recall Proposition A from the beginning of section 3.1. This is the Principal Exotic Case. PROPOSITION A. For all f,g ELG there exist A,B,C INF such that A . fA C . gB A . fB C . gC. Recall the definitions of N, ELG, INF, ., fA, in Definitions 1.1.1, 1.1.2, 1.1.10, 1.3.1, and 2.1. In this section, we prove Proposition A in SMAH + . It is convenient to prove a stronger statement. PROPOSITION B. Let f,g ELG and n 1. There exist infinite sets A 1 ... A n N such that i) for all 1 i < n, fA i A i+1 . gA i+1 ; ii) A 1 fA n = . LEMMA 4.2.1. The following is provable in RCA 0 . Proposition B implies Proposition A. In fact, Proposition B for n = 3 implies Proposition A. Proof: Let f,g ELG. By Proposition B for n = 3, let A B C N be infinite sets, where fA B . gB, fB C . gC, and A fC = . Note that C,gC are disjoint. Hence C,gB are disjoint. In addition, A,fA are disjoint, and A,fB are disjoint. We now verify the inclusion relations. Let x A fA. If x fA then x B gB C gB. If x A then x C C gB. Let x A fB. If x fB then x C gC. If x A then x C C gC. QED Recall the definition of f ELG from section 2.1: there are rational constants c,d > 1 such that for all but finitely many x dom(f), c|x| f(x) d|x|. We wish to put this in more explicit form. Assume f,c,d are as above. Let t be a positive integer so large that 1 + 1/t < c,d < t, and for all x dom(f), |x| > t c|x| f(x)
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2 d|x|. Let b be an integer greater than t and max{f(x): |x| t}. Then for all x dom(f), |x| > t f(x) b|x|. |x| t f(x) b. |x| b f(x) b 2 . Hence f ELG if and only if there exists a positive integer b such that for all x dom(f), |x| > b (1 + 1/b)|x| f(x) b|x|. |x| b f(x) b 2 . We now fix f,g ELG, where f is p-ary and g is q-ary. According to the above, we also fix a positive integer b such that for all x N p and y N q , i. if |x|,|y| > b then (1 + 1/b)|x| f(x) b|x| (1 + 1/b)|y| g(y) b|y|. ii. if |x|,|y| b then f(x),g(y) b 2 . We also fix n 1 and a strongly p n-1 -Mahlo cardinal κ . We begin with the discrete linearly ordered semigroup with extra structure, M = (N,<,0,1,+,f,g). The plan will be to first construct a structure of the form M* = (N*,<*,0*,1*,+*,f*,g*,c 0 *,. ..), where the c*’s are indexed by N. This structure is non well founded and generated by the constants 0*,1*, and the c*’s. The indiscernibility of the c*’s will be with regard to atomic formulas only. The first nonstandard point in M* will be c 0 *. While it is obvious that we cannot embed M* back into M, we
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4.2Proof071110 - 1 4.2. Proof using Strongly Mahlo...

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