2
(
∀
u) by (
∀
u
∈
E
∩
[0,v])
(
∃
u) by (
∃
u
∈
E
∩
[0,v]).
These bounded quantifiers should be expanded in the usual
way to create an actual formula in L(E).
We now define a very important 6ary relation.
DEFINITION 5.4.5. We define A(r,n,m,
ϕ
,a,b) if and only if
i) r,n,m,a,b
∈
N\{0}, n < m;
ii)
ϕ
=
ϕ
(v
1
,...,v
r
) is a formula of L(E); i.e., all free
variables of
ϕ
are among v
1
,...,v
r
;
iii) Let x
1
,...,x
r
∈
E
∩
[0,c
n
]. Then
ϕ
(x
1
,...,x
r
)
c_n
↔
aCODE(c
m
;c
n
,...,c
m1
,x
1
,...,x
r
)+b
∈
E.
LEMMA 5.4.1. Let r,n
≥
1 and
ϕ
(v
1
,...,v
r
) be a quantifier
free formula of L. There exist a,b such that
A(r,n,n+1,
ϕ
,a,b).
Proof: Let r,n,
ϕ
be as given. Note that
ϕ
c_n
=
ϕ
.
By Lemma 5.3.18 vii), let a,b
∈
N\{0} be such that the
following holds. Let n
≥
1 and x
1
,...,x
r
∈
E
∩
[0,c
n
].
(
∃
y
∈
E)(y
≤
c
n
,x
1
,...,x
r

↑↑
∧
y
≤
x
1
,...,x
r

∧
ϕ
(c
n
,x
1
,...,x
r
,y))
↔
(
∃
y
∈
E)(y
≤
c
n
,x
1
,...,x
r

↑↑
∧
ρ
(c
n
,x
1
,...,x
r
,y))
↔
aCODE(c
n+1
;c
n
,x
1
,...,x
r
)+b
∈
E.
ϕ