9
Proof of the Lemma
1.
Claim: there are only finitely many
inequivalent FO
k
formulas over R
[Proof: in class ?? ]
a
1
,
a
2
, …,
a
m
Hence:
R
²
8
x
(p(
x
)
$
a
i
(
x
))
R
²
8
x
(
:
a
i
$
:
a
j
)
R
²
8
x.(
9
x
j
.
a
i
$
a
l
)
Let
y
k
be their conjunction; clearly
R
²
f
k
Closure axioms