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6.1Further071310

# 6.1Further071310 - 1 CHAPTER 6 FURTHER RESULTS 6.1...

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1 CHAPTER 6. FURTHER RESULTS 6.1. Propositions D-H. 6.2. Effectivity. 6.3. A Refutation. 6.1. Propositions D-H. Our treatment of Propositions A,B,C culminated with Theorems 5.9.9, 5.9.11, and 5.9.12 at the end of Chapter 5. In this section, we consider five Propositions D-H that have the same metamathematical properties as Propositions A,B,C. We will also consider some variants of Propositions D-H that do not share these properties, or whose status is left open. Recall the main theorems of Chapter 5 (in section 5.9), which are Theorems 5.9.9, 5.9.11, and 5.9.12. Examination of the proofs of these three Theorems reveal that Theorem 5.9.11 with 1-Con(SMAH) is the key. If ACA’ proves the equivalence of a statement with 1-Con(SMAH) then all of the other properties provided by these three Theorems quickly follow. Accordingly, we establish these same three Theorems for Propositions D-H by showing that they are also each equivalent to 1-Con(SMAH) over ACA’. We begin with Proposition D (see below), which is a sharpening of Proposition B. Proposition D immediately implies Propositions A-C over RCA 0 . Note that Propositions A-C are based on ELG. Examination of the proof of Proposition B in Chapter 4 shows that we can separately weaken the conditions on f,g in different ways. Also, we can place an inclusion condition on the starting set A 1 . As usual, we use | | for the sup norm, or max. This results in Proposition D below. DEFINITION 6.1.1. We say that f is linearly bounded if and only if f MF, and there exists d such that for all x dom(f),

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2 f(x) d|x|. We let LB be the set of all linearly bounded f. DEFINITION 6.1.2. We say that g is expansive if and only if g MF, and there exists c > 1 such that for all but finitely many x dom(f), c|x| g(x) We let EXPN be the set of all expansive g. Recall the definitions of MF, SD (Definition 1.1.2), and ELG, EVSD (Definitions 2.1, 2.2). PROPOSITION D. Let f LB EVSD, g EXPN, E N be infinite, and n 1. There exist infinite 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 = ; iii) A 1 E. Note that ELG LB EVSD EXPAN, and so Proposition D immediately implies Proposition B. Proposition D is the strongest Proposition that we prove in this book (from large cardinals). Recall that Propositions A-C are official statements of BRT. More accurately, Proposition B is really an infinite collection of statements of BRT. Proposition D not a statement (or statements) of BRT for two reasons. a. There is no common set of functions used for f,g (asymmetry). b. The set E is used as data, rather than just f,g. Features a,b both suggest very natural expansions of BRT. Feature a suggests “mixed BRT”, where one uses several classes of functions instead of just one. One can go further and use several classes of sets as well.
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6.1Further071310 - 1 CHAPTER 6 FURTHER RESULTS 6.1...

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