1
4.4. Proof using 1-consistency.
In this section we show that Propositions A,B can be proved
in ACA’ + 1-Con(SMAH). Here 1-Con(T) is the 1-consistency
of T, which asserts that “every
Σ
0
1
sentence provable in T
is true”. 1-Con(T) is also equivalent to “every
Π
0
2
sentence
provable in T is true”.
By Lemma 4.2.1, Proposition B implies Proposition A in RCA
0
.
Hence it suffices to show that Proposition B can be proved
in ACA’ + 1-Con(SMAH).
DEFINITION 4.4.1. We write ELG(p,b) for the set of all f
∈
ELG of arity p satisfying the following conditions. For all
x
∈
N
p
,
i. if |x| > b then (1 + 1/b)|x|
≤
f(x)
≤
b|x|.
ii. if |x|
≤
b then f(x)
≤
b
2
.
Note that from Definition 2.1, f
∈
ELG if and only if there
exist positive integers p,b such that f
∈
ELG(p,b). Also
note that each ELG(p,b) forms a compact subspace of the
Baire space of functions from N
k
into N.
DEFINITION 4.4.2. Let p,q,b
≥
1. A p,q,b-structure is a
system of the form
M* = (N*,0*,1*,<*,+*,f*,g*,c
0
*,.
..)
such that
1. N* is countable. For specificity, we can assume that N*
is N.
2. (N*,0*,1*,<*,+*) is a discretely ordered commutative
semigroup (see definition below).
3. +*:N*
2
→
N*, f*:N*
p
→
N*, g*:N*
q
→
N*.
4. f* obeys the above two inequalities for membership in
ELG(p,b), internally in M*.
5. g* obeys the above two inequalities for membership in
ELG(q,b), internally in M*.
6. Let i
≥
0. The sum of any finite number of copies of c
i
*
is < c
i+1
*.
7. The c*’s form a strictly increasing set of
indiscernibles for the atomic sentences of M*.
Note that the conditions under clauses 4-7 are all
universal sentences.