2.1.Meth020911

# 2.1.Meth020911 - 1 CHAPTER 2 CLASSIFICATIONS 2.1...

This preview shows pages 1–3. Sign up to view the full content.

1 CHAPTER 2 CLASSIFICATIONS 2.1. Methodology. 2.2. EBRT, IBRT 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 1.1.11). 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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2 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. In this Chapter, the two asymptotic BRT settings (ELG,INF), (EVSD,INF), have the same behavior, whereas the two non asymptotic BRT settings (SD,INF), (ELG SD,INF), also have the same behavior. In this Chapter, the behavior of (ELG,INF), (EVSD,INF) differs from the behavior of (SD,INF), (ELG SD,INF). In this Chapter, (MF,INF) behaves differently from the other four settings. 2.1. Methodology. In this section, we use notation and terminology that was
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 14

2.1.Meth020911 - 1 CHAPTER 2 CLASSIFICATIONS 2.1...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online