2.1.Meth030109

2.1.Meth030109 - 1 CHAPTER 2 CLASSIFICATIONS 2.1....

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1 CHAPTER 2 CLASSIFICATIONS 2.1. Methodology. 2.2. EBRT, BRT in A,fA. 2.3. EBRT, IBRT in A,fA,fU. 2.4. EBRT in A,B,fA,fB, on (SD,INF). 2.5. EBRT in A,B,fA,fB, on (ELG,INF). 2.6. EBRT in A 1 ,...,A k ,fA 1 ,...,fA k , on (MF,INF). 2.7. IBRT in A 1 ,...,A k ,fA 1 ,...,fA k , on (SD,INF),(ELG,INF), (MF,INF). In this Chapter, we treat several significant BRT fragments. For most of these BRT fragments, we show that every statement is either provable or refutable in RCA 0 . For the remainder of these BRT fragments, we show that every statement is either provable in RCA 0 , refutable in RCA 0 , or provably equivalent to the Thin Set Theorem of section 1.4 over RCA 0 . Thus in this Chapter, we do not run into any independence results from ZFC. In the classification of Chapter 3, we do run into a statement independent of ZFC, called the Principal Exotic Case, which is the focus of the remainder of the book. In this Chapter, we focus on five BRT settings (see Definition 1.11). These fall naturally, in terms of their observed BRT behavior, into three groups (see Definitions 1.1.2, and 2.1): (SD,INF), (ELG SD,INF). (ELG,INF), (EVSD,INF). (MF,INF). The inclusion diagram for these five sets of multivariate functions is ELG SD SD ELG EVSD MF
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Here each item in any row is properly contained in any item in any lower row. Multiple items on any row are incomparable under inclusion. (SD,INF), (ELG,INF), and (MF,INF) are the most natural of these five BRT settings. The remaining two BRT settings are closely associated, and serve to round out the theory. MF (multivariate functions), SD (strictly dominating), and INF (infinite) were defined in section 1.1 in connection with the Complementation Theorem and the Thin Set Theorem. DEFINITION 2.1. Let f MF. We say that f is of expansive linear growth if and only if there exist rational constants c,d > 1 such that for all but finitely many x dom(f), c|x| f(x) d|x| where |x| is the maximum coordinate of the tuple x. Let ELG be the set of all f MF of expansive linear growth. DEFINITION 2.2. Let f MF. We say that f is eventually strictly dominating if and only if for all but finitely many x dom(f), f(x) > |x|. We write EVSD for the set of all f MF that are eventually strictly dominating. 2.1. Methodology. In this section, we use notation and terminology that was introduced in section 1.1. Recall the definitions of BRT fragment. Definition 1.18. BRT environment. Definition 1.19. BRT signature. Definition 1.21. flat BRT fragment. Definition 1.34. In Definition 1.39, the flat BRT fragments were divided into these four mutually disjoint categories: 1) EBRT in σ on (V,K), where σ does not end with . 2) EBRT in
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This note was uploaded on 08/05/2011 for the course MATH 366 taught by Professor Joshua during the Fall '08 term at Ohio State.

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2.1.Meth030109 - 1 CHAPTER 2 CLASSIFICATIONS 2.1....

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