PL 3014  Metalogic
1
Proposition 4.40
Preliminaries
Suppose we enlarge
L
by adding new constants
b
0
,
b
1
, .. to form
L
+
.
Let
S
be an extension of
K
.
Now
construct an extension
S
+
of
S
by including as axioms all axioms of
S
and all instances of
S
axioms that
contain any of the new constants
b
0
,
b
1
, ... .
Example
:
Axiom (K5)
(
∀
x
i
)
A
(
x
i
)
→
A
(
t
), where
t
is a term
free for
x
i
in
A
(
x
i
), is an axiom of
S
+
, as is the particular instance (
∀
x
1
)
A
1
1
(
x
1
)
→
A
1
1
(
b
1
).
L
emma 1
:
If
S
is consistent, so is
S
+
.
Proof
:
Suppose
S
is consistent and
S
+
is not.
Then
:
There's a
wf
B
such that
S
+
B
and
S
+
(
∼
B
).
Note
:
These
S
+
proofs can be converted into
S
proofs.
Just replace all occurrences of
b
constants with
a

constants that do not occur in the
S
+
proofs.
(There will always be such
a
constants available
since there is a countable infinity of them, and there can only be a finite number of
wf
s, and hence
occurrences of
b
constants, in any
S
+
proof.)
Result
:
S
B
and
S
∼
(
B
).
But
S
was assumed consistent.
Hence
S
+
must also be consistent.
Prop. 4.40.
Let
S
be a consistent extension of
K
.
Then there is an interpretation of
L
in which every
theorem of
S
is true.
Outline of Proof:
I.
Enlarge
L
to
L
+
by adding new constants
b
0
,
b
1
, ... .
Extend
S
to
S
+
as above.
Construct a
particular
consistent extension
S
∞
of
S
+
.
Then, by Prop. 4.39, there must be a complete consistent extension of
S
∞
, call it
T
.
II. Use
T
to construct an interpretation
I
of
L
+
.
Prove that for every
closed
wf
A
of
L
+
,
T
A
iff
I
A
.
III. Show that for
any
(open or closed)
wf
B
of
L
, if
S
B
, then
I
B
.
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PL 3014  Metalogic
2
Part I.
Let
S
be a consistent extension of
K
.
S
∞
will be the extension of
S
+
that has as its axioms the
union of the sets of axioms of a particular sequence of extensions
S
0
,
S
1
, ..., of
S
+
.
This sequence is
constructed in 4 steps:
1.
List all
wf
s of
L
+
that contain
one
free variable:
F
0
(
x
i
0
),
F
1
(
x
i
1
),
F
2
(
x
i
2
), ...
2.
Choose a subset {
c
0
,
c
1
, ... } of the
b
constants that are free for the
x
i
0
,
x
i
1
, ... in the list.
Require:
(i)
c
0
doesn't appear in
F
0
(
x
i
0
).
(ii) For
n
> 0,
c
n
∉
{
c
0
, ...,
c
n
−
1
} and
c
n
doesn't appear in
F
0
(
x
i
0
), ...,
F
n
(
x
in
).
3.
Let
G
k
be the
wf
∼
(
∀
x
ik
)
F
k
(
x
ik
)
→
∼
F
k
(
c
k
).
4.
Construct the sequence
S
0
,
S
1
, ... as follows:
(i) Let
S
0
=
S
+
.
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 Spring '06
 JonathanBain
 Logic, Metalogic, Sn, inductive hypothesis

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