Henkin’s Model and Metatheorem 45.14
Branden Fitelson
04/10/07
Henkin’s Model
. Let
T
be a consistent, negationcomplete, and closed first order theory. Henkin’s model
M
is a a denumerable interpretation for
T
such that for each WFF
A
of
T
,
A
is true on
M
iff
‘
T
A
. The
existence of such a model
M
undergirds metatheorem 45.14.
Characterizing
M
will involve doing five
things:
(1) specifying
M
’s (denumerable) domain
D
, (2) saying for each constant symbol
c
of
T
which
object
d
in the domain
M
assigns to
c
, (3) saying for each
n
place function symbol
f
which
n
ary function
f
is assigned to
f
by
M
, (4) saying for each
n
place predicate symbol
F
which
n
ary property
F
(
i.e.
, which set
of ordered
n
tuples of closed terms of
T
, since we identify properties with their
extensions
) is assigned by
M
to
F
, and (5) saying for each propositional symbol
p
of
T
, what truthvalue is assigned to
p
by
M
. Here is
Henkin’s
M
, followed by a proof of 45.14 (arguably the most important metatheorem of the entire course).
1. The domain
D
of
M
is the set of closed terms of
T
.
This set contains all the constant symbols
a
0
, a
00
, a
000
, . . . , b
0
, b
00
, b
000
, . . . , c
0
, c
00
, c
000
, . . .
of
T
(the
b
’s and
c
’s are effectively enumerable sets of
new constant symbols that may be added to
Q
for
Q
+
purposes and/or for the purpose of ensuring
T
is closed).
D
also contains all the closed terms with function symbols:
f
*0
a
0
, f
**0
b
0
a
00
, . . .
of
T
.
Important Digression on Symbols, Abstract Objects, Types, and Tokens
. It is important
to note that the symbols of
T
are
abstract objects
, and they are
types not tokens
.
You
should not confuse a token of a symbol with the symbol itself. For instance, when I write
a token inscription “
a
0
” (the physical inscription between the quotation marks preceding
this parenthetical remark), I have not written down the symbol itself. It is not tokens of
symbols of
T
that get assigned to objects by
M
, but rather the symbols themselves. For
instance, when I say that the numeral “1” gets interpreted as the number one (which is
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 Spring '07
 FITELSON
 Logic, Tn

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