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CHAPTER 6.
FURTHER RESULTS
6.1. Propositions DH.
6.2. Effectivity.
6.3. A Refutation.
6.1. Propositions DH.
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 DH that
have the same metamathematical properties as Propositions
A,B,C. We will also consider some variants of Propositions
DH 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 1Con(SMAH) is the key. If ACA’ proves the
equivalence of a statement with 1Con(SMAH) then all of the
other properties provided by these three Theorems quickly
follow.
Accordingly, we establish these same three Theorems for
Propositions DH by showing that they are also each
equivalent to 1Con(SMAH) over ACA’.
We begin with Proposition D (see below), which is a
sharpening of Proposition B. Proposition D immediately
implies Propositions AC over RCA
0
.
Note that Propositions AC 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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f(x)
≤
dx.
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),
cx
≤
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 AC 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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 Fall '08
 JOSHUA
 Math, Logic, Countable set, Lemma, RCA0

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