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