2
variables in
ϕ
, and the number of occurrences of the
symbols
01+-•
↑
()=<
¬
∧∨→↔
v
1
v
2
,...,v
r
log
in
ϕ
. Note that for all n
≥
0, {
ϕ
: #(
ϕ
)
≤
n} is finite.
DEFINITION 5.2.3. For all r
≥
1, let
β
(r) be the number of
terms t in L with #(t)
≤
r. We fix a doubly indexed sequence
t[i,r] of terms in L, which is defined if and only if r
≥
1
and 1
≤
i
≤
β
(r). For each r
≥
1, the sequence t[i,r], 1
≤
i
≤
β
(r), enumerates the terms t with #(t)
≤
r, without
repetition.
DEFINITION 5.2.4. For all r
≥
1, let
γ
(r) be the number of
quantifier free formulas
ϕ
in L with #(
ϕ
)
≤
r. We fix a
doubly indexed sequence
ϕ
[i,r] of quantifier free formulas
in L, which is defined if and only if r
≥
1 and 1
≤
i
≤
γ
(r). For each r
≥
1, the sequence
ϕ
[i,r], 1
≤
i
≤
γ
(r),
enumerates the quantifier free formulas
ϕ
with #(
ϕ
)
≤
r,
without repetition.
We adhere to the convention of displaying all free
variables (and possibly additional variables). Thus
t(v
1
,...,v
n
) and
ϕ
(v
1
,...,v
m
) respectively indicate that all
variables in the term t are among the first n variables
v
1
,...,v
n
, and all variables in the quantifier free formula
ϕ
are among the first m variables v
1
,...,v
m
.
Note that all terms t[i,r] have variables among v