A SIMPLE TREATMENT O F CHURCH'S THEO REM
ON T HE DECISIO N PROBLEM
Thomas SCHWARTZ
The proof I w ill give of Church's undecidability theorem for
quantification theory is simpler than any other known to me,
and it is more direct: the non-effectiveness o f quantificational
validity is not inferred from some other unsolvability result.
Church's Theorem is deduced from his Thesis (CTh) that every
computable function in the set Z of natural numbers is recursive.
To define recursiveness we set u p a formal system whose
vocabulary consists o f o, S, t h e parentheses and comma,
the variables x, y, z, w, w
function
letter k f
i
f, S
wand the At, are function symbols.
.2 Terms are defined thus: variables and o are terms, and i f
T1, T k are terms, so are S
,
e t c .
( o, S(o), S(S(o))) etc. are numerals. For n in Z, the numeral n
a
n
d
is
r defined by setting = - o and k + I S ( k ) .
f An oequation
r is - = flanked by terms.
i
a If E lis a set
l of equations, an E-letter is a function letter in E.
) a n d
j
,
e belongs to the set E of equations or is deW T I means
,
krivable from
, E by these rules:
•
•
t
h resultsefr om substituting numerals f o r variables in
RI.
If B
•A, from, A to infer B.
k
tR2. kI f A(
.
)
.
fr om replacing one occurrence o f a i n
a
r b) results
y
A(a) by b, from a b and Ma) to infer A
( bA recursion is a finite set E o f equations such that ( I ) for
all
) . numerals a, b, if E 1–a b then a = b, and (2) for all E-letters k l
E
i f i
(at 1n
If E is a recursion and h a k-ary E-letter, T7Eis the function
1
defined
on Zk thus: TzE(n
d
i,I lAHfunction cri is recursive if f q = 71
and
- E
A
l,. n
s o m e
)_ E_ f o r
h
-n r e c u r s i o n
n
i)l e t t
i.t E f
f
er
E
h
h
-i
.
i
l
n
kZ
),
—
t
154
T
H
O
M
A
S
S CHW A RTZ
If f lc iZ , the characteristic function F o f His defined on Z
by setting n ( n ) = 0 if nen and f n(n) =- I i f not trEn •
Clearly, if H is effective, 1'11 is computable, and so, by CTh,
r i s recursive.
I w ill show there is n o effective test o f (quantificational)
validity for the language L whose well-formed formulas (wffs)
are generated in the usual way from the parentheses and comma,
the universal quantifier V and conditional sign t h e terms introduced above and the 2-place predicate a n d 1-place predicate P.
" 1=A" means A is a valid wff.
LEMMA. I f n is an effective subset of Z, there is an A such
that for all n in Z, =
i f f tlE n.
Proof. By CTh there is a recursion E with E-letters h
such
i that hi; = F f o r some p = I ,
,
Let L be L shorn of all function letters but E-letters and interpreted in Z by assigning 0 to o, the successor function to S,
identity to n to P and h t o h
of universal closures of the wffs in E
i (Let
i A be
= a conjunction
1 ,
plus
these
wffs:
m
)
.
(1) h
(3)
p ( x z — >y z ) — ÷ (z x - - > z y ) ,
(4gi)
( x ) ( x z — >y z ) — ( g ( w
(w
lo 1
E
,P and
w i = / , r (r = the number of arguments of g).
yi( E and_ A have 1these two properties:
,x (I) (a)x I f El—T=0,
-r=a is true for all values q
,
variables
t
w
t
)
ii Proof.
1 By induction on v plus the number / of occurren, , q (a)
ces
,, v. of
. .E-letters
, w
o
f in i t s
(
rt1 , If
=. -v = /a=0,
. t and a are numerals; thus, since E is a recursion,
t2
(,. = 0f ), s o T a i s t r u e .
)
and 1>0, t a has a part h
3
zx If) v =0
4
rA i
)g
=
A
( 7 El— h
i, Then
y) isf 1
-= -( F
to , r F
>
ru 7k k )
r,
.e ) ..
L
e
t
r z E
a
( x
i
la l_ (- r
f- zi
TREATMENT O F CHURCH'S THE O RE M
1
5
5
Let r e s u l t from replacing one occurrence of h
F i
F
A Since
(7
is. A
) 1
), by R2, E - r ' := a', so by inductive hypothesis, t ' = a' is
i But
true,
hence so is r a .
n a n
t dIf v > 0 and T
* a,- E i - r * = a * by R
= =F
thesis,
so r o is true for q
I
a b- - - ao *
i ,0(3)
(I)-(4gi)
are obviously true for all values of their variables,
tr se hos u l
b
and
by
(a),
so
are the wffs in E. So A is true.
,
y rnt *s - - - 0 *
(II) (a)
rs Iof 1 = A F i af
- t* a br
au n ed
. (m( 3)
r bts ( ub b)y =
e
by R2, I = A--->(T
(T cr(a)
i1-rs nt i d t u c
A a t1Hence,
a
it iv(y)nife 1=
,u and
a
=b
T
O
by
R2,
I = A- *1
*
T
O
a
n
!
) hfg o yl l
r
Proof.
(a) By
- induction
t
/ on the length of r(a).
=- /p=d
o
e' . w o s
C
=
o
If
a
=
t(a),
b
=
T(b),
so I = A- 4t( a) =t( b) , whence 1-= A->
f
o
l
l
o
A
c -fa
r
(i t(-rwa) .=a>r
(
b)
s m
o
-b -ifcIf a is ar properopart of t(a), r(a) has the form g( d
a
=
sh
i
b y ,d,) and r(b) the form g( d
>m
yd- - a)
1
)
( atq
,2 d) . t
R
a
r By
rt inductive hypothesis, 1 = A--(e(a)=0->c•(13)=a), s o b y
bnif A- 4 t ( ,a ) =a - ( b ) 1 . - _ . a ) .
2
)(4gi),
u
•
•
•
,
,. ad((3)
eo A - > ( a ( a ) =
A no(1)))
by (3).
i-r
- dThe
set
of A includes E and is closed under
( bconsequences
)=t)
f(I - - > aof
÷RItb
(since
R
I
is
valid)
and
R2 (by ( lly ) ) . So if Et- e,
t
y
T
a
( (i i f am i l ,) n , , Fri(n) = TzEp(n) = 0, whence E i - h
But
t
and thus 1 = A-->P(a) by (1). Conversely,
-so(snI = A---->h„01)=0,
p
== A - - 4 o
if
b
(r R1) == 0 A
, - 4 (3 )t z - - - ),
The class of v alid wffs of L is not effective.
a( i fiTheorem.
a
1
)
) o( a ) - 4 t
Proof.
7 ) Suppose the theorem false. Treating each superscript
l7 subscript
o(and
digit as a separate symbol, L has a finite vocabul sso we get an effective enumeration Ao, A
-ilary,
-tl o r
-u, wAe2
e t c .
o
f
>bL s '
s
ry f
(( r 1
b( o 3
)) m ,
=s a
156
T
H
O
M A S
SCHWARTZ
expressions by listing L's symbols in some chosen "alphabetic"
order, then the 2-symbol expressions i n lexicographic order,
next the 3-symbol expressions etc. Thus, since v alidity is effective, we can effectively tell for each n in Z whether or not
= A
So
n by the Lemma, there is a k such that f o r a l l n i n Z ,
1=
-—
- -A
ff
k> Pnot = A A
-(-4F t
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=t
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—
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)1
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