LOGIC COLLOQUIUM '84
J.B. Paris, A . J . Wilkie. and G.M. Wiliners (Editors)
0 ElseL>ierScience Publishers 0. V. (North-Holland). I986
I65
On t h e l e n g t h of p r o o f s o f f i n i t i s t i c c o n s i s t e n c y s t a t e m e n t s
i n f i r s t order theories
t
Pave1 Pudlgk
Mathematical I n s t i t u t e
Czechoslovak Academy o f S c i e n c e s
Prague
1.
Introduction
By t h e second i n c o m p l e t e n e s s theorem o f Gb;del, a s u f f i c i e n t l y r i c h t h e o r y
cannot prove i t s own c o n s i s t e n c y .
5
whose l e n g t h i s
T
i f one can
of t h e s t a t e m e n t , s a y , " t h e r e is no proof of
f i n d a f e a s i b l e proof in T
falsehood i n
T h i s l e a v e s open t h e q u e s t i o n ,
We s h a l l show some bounds t o t h e
l e n g t h of such p r o o f s i n some f i r s t o r d e r t h e o r i e s .
The main r e s u l t s (Theorems 3.1 and 5.5) can be roughly s t a t e d a s f o l l o w s :
Let
diction in
E
>
be a r e a s o n a b l e f o r m a l i z a t i o n of " t h e r e i s no proof of c o n t r a -
Con ( x )
T
0
and
whose l e n g t h i s
T
k
E
x
."
Then f o r r e a s o n a b l e
T
there exist
such t h a t
w
(1)
any proof of
ConT(c)
(2)
t h e r e e x i s t s a proof of
in
T
2 nE
has l e n g t h
Con,(;)
in
T
;
with length
(n
k
I t had been known t h a t some lower bounds c o u l d be d e r i v e d . t t
.
In f a c t we
were i n s p i r e d by a paper of M y c i e l s k i [ l o ] and we u s e an i d e a of h i s .
The
p r e s e n t knowledge of fragments of a r i t h m e t i c , which is mainly due t o P a r i s and
W i l k i e , e n a b l e d u s t o reduce t h e a s s u m p t i o n s a b o u t
mere containment of Robinson's a r i t h m e t i c
.
in t h e lower bound t o
The upper bound is based on a
It u s e s a l s o a t e c h n i q u e of w r i t i n g s h o r t
p a r t i a l d e f i n i t i o n of t r u t h .
formulas, c f .
Q
T
[ 5 ] , C h a p t e r 7.
' T h i s paper was f i n i s h e d w h i l e t h e a u t h o r was s u p p o r t e d by t h e NSF g r a n t
1-5-34648 a t t h e U n i v e r s i t y of Colorado, B o u l d e r , CO U.S.A.
' + A f t e r t h e paper had been t y p e d , I l e a r n e d t h a t H. Friedman had proved a lower
bound of t h e form
nE
,
E
>
0
.
P.PUDLAK
166
Our r e s u l t s may be i n t e r e s t i n g b e c a u s e of t h e f o l l o w i n g r e a s o n s .
lower and t h e u p p e r bound a r e o n l y p o l y n o m i a l l y d i s t a n t .
o f t h e lower bound,
(see Section 4 ) .
( 1 ) The
( 2 ) Some c o r o l l a r i e s
( 3 ) R e l a t i o n t o some problems i n
P e r h a p s t h e most
c o m p l e x i t y t h e o r y ( s e e Theorem 3.2 and S e c t i o n 6 ) .
i n t e r e s t i n g a p p l i c a t i o n o f t h e lower bound is a more t h a n e l e m e n t a r y speed-up
f o r t h e l e n g t h o f p r o o f s in GB r e l a t i v e t o ZF, (Theorem 4 . 2 ) .
S e v e r a l i m p o r t a n t p a p e r s t h a t are r e l a t e d t o o u r p a p e r a r e l i s t e d i n
references.
The p a p e r s E h r e n f e u c h t and M y c i e l s k i
Mostowski [ 9 ] ( l a s t c h a p t e r ) , M y c i e l s k i
141, Gandy 161, GEdel 171,
[ l o ] , P a r i k h 111) , [ 1 2 ] , S t a t m a n
[ 1 6 ] , [ 1 7 ] and Yukami 119) d e a l w i t h q u e s t i o n s a b o u t t h e l e n g t h o f p r o o f s .
Esenin-Volpin
In
[ 3 ] , Gandy [ 6 ] , M y c i e l s k i [ l o ] and P a r i k h [ l l ] t h e r e a d e r c a n
f i n d t h e o u t l i n e s o f some f i n i t i s t i c p r o j e c t s .
I n t h i s p a p e r w e c o n s i d e r a measure which is d i f f e r e n t from t h e m e a s u r e s
used i n most o f t h e p a p e r s m e n t i o n e d above.
number o f f o r m u l a s ( i . e .
Instead of counting j u s t t h e
p r o o f l i n e s ) , we i n c l u d e t h e l e n g t h of f o r m u l a s i n t o
More p r e c i s e l y , w e assume t h a t p r o o f s a r e coded by s t r i n g s i n
t h e complexity.
a f i n i t e a l p h a b e t and t h e l e n g t h of a p r o o f is t h e l e n g t h o f t h e c o r r e p o n d i n g
string.
T h i s is t h e m o s t r e a l i s t i c measure.
We d o not know w h e t h e r a s i m i l a r
lower bound h o l d s a l s o f o r t h e number o f f o r m u l a s i n t h e p r o o f .
Recently J.
K r a j i r e k g a v e a n i d e a f o r a l o w e r bound o f t h e number o f f o r m u l a s i n t h e p r o o f
T
which is of t h e form
of
ConT(;)
2.
Fragments o f a r i t h m e t i c
in
constant-log n
.
The w e a k e s t f r a g m e n t o f a r i t h m e t i c t h a t w e s h a l l u s e is R o b i n s o n ' s
arithmetic
Q
.
The l a n g u a g e of
S(x) = S(y) + x = y ;
S(x+y) ; x - 0 =
0#
0 ; x.S(y)
Q
0
x .
s(y); x #
= x*y
+
c o n s i s t s of
-b
0,S,
+,
; t h e axioms a r e
By ( x = S ( y ) ) ; x + 0 = x ; x + S(y) =
IAo
denotes
Q
p l u s t h e scheme of
Proofs and Finitistic Consistency Statements in First Order Theories
induction for bounded arithmetical formulas, i.e.
quantifiers are of the form
Q
3x
t
, Vx
(it is sufficient to assume that
plus an axiom expressing
IAo
,
i t
formulas where all
some term in the language of
t
is just a variable).
t
vxy 3 z ( z = xy)
167
.
LA
+ exp
is
Exponentiation can be
introduced naturally without using a function symbol for it (namely, Bennett
[l] has shown that exponentiation can be defined by a bounded formula).
All
the standard theorems o f number theory and finite combinatorics are provable in
IA
+ exp
,
(cf.
"21).
Syntax can be arithmetized in a natural way even in
some weaker theories, see [131.
information.
The reader can consult these papers for some
Let us only remark that one can prove the scheme of induction
also for exponentially bounded formulas (i.e. formulas with quantifiers of the
form
3 x c t Vx i t
IAo + exp
,t
term in the language of
S(0)
,2
- r + i.
(a1+
1) +?((.az
.
Denote by
1=
Q
1
n
Let
plus exponentiation) in
,
a. 6 {0,1) and
Then the term
will be denoted by
and called the
the numeral as a term of the form
a term is too long.
length of a numeral
1.1
SS
+ 1)
+Lo(...))
nth numeral.
...S(2)
The usual definition of
is not suitable here, since such
denotes the integral part of
2 is proportional to
symbols for the formalizations of
+,
0 ,
In
(...I
.
,
xy
logz(n+l)
.
Hence the
We shall not introduce new
etc.
If such a symbol is
not in the language of the theory in question, the terms constructed from them
should be understood as abbreviations.
P. PUDLAK
168
it is i m p o r t a n t t o
When we c o n s i d e r t h e l e n g t h of p r o o f s in some t h e o r y ,
s p e c i f y t h e set o f axioms.
axiomatization
A
T h e r e f o r e w e s h a l l d i s t i n g u i s h two c o n c e p t s :
i s an a r b i t r a r y set of s e n t e n c e s , while a t h e o r y
d e d u c t i v e l y c l o s e d set o f s e n t e n c e s .
T
an
is a
The d i s t i n c t i o n is more i m p o r t a n t i f
d o e s not have a f i n i t e a x i o m a t i z a t i o n ,
since i f
T
T
has a f i n i t e axiomatiza-
t i o n , t h e n t h e l e n g t h s of t h e s h o r t e s t p r o o f s i n f i n i t e a x i o m a t i z a t i o n s d i f f e r
o n l y by an a d d i t i v e c o n s t a n t (and we u s u a l l y u s e o n l y f i n i t e a x i o m a t i z a t i o n s ) .
We s h a l l w r i t e
t o d e n o t e t h a t t h e r e e x i s t s a p r o o f of
+
in
A
whose l e n g t h ( i n c l u d i n g t h e
i n .
l e n g t h of f o r m u l a s ) is
The aim o f t h i s s e c t i o n is t o show t h a t , in s p i t e o f t h e f a c t t h a t
much weaker t h a n
p r o v a b l e in
IAo + e x p
+ exp
IA
,
every numerical i n s t a n c e of a
h a s a s h o r t p r o o f in
.
Q
n1
is
(1
sentence
This i s r o u g h l y t h e c o n t e n t
of t h e f o l l o w i n g lemma.
Lemma 2.1
For e v e r y e x p o n e n t i a l l y bounded f o r m u l a
+), t h e r e e x i s t s a polynomial
v a r i a b l e of
IAo + exp
then,
for every
m E w
then
CutI
If
x
is t h e o n l y f r e e
such t h a t i f
,
d +(m)
F i r s t we prove a n o t h e r u s e f u l lemma.
.
p
(where
+ Vx +(XI
Qi
x, ZX, 22x,...
+(x)
I(x)
Let
.
,:2
2 7 , 2;
,...
denote the
is a f o r m u l a w i t h t h e s i n g l e f r e e v a r i a b l e
denotes the following sentence
x
,
Proofs and Finitistic Consistency Statements in First Order Theories
If
A I - CutI
,
t h e n we s a y t h a t
I
is a
CUt
in
I69
A
Lemma 2.2
I
Let
be a c u t i n
such t h a t , f o r every
and
A
k,n
E
Q C A
.
Then t h e r e e x i s t s a p o l y n o m i a l
p
w
.
A ,*Ice)
Proof:
I
Given a c u t
one c a n c o n s t r u c t a n o t h e r c u t
c l o s e d u n d e r a d d i t i o n and f o r e v e r y
[13].
x
I'
from
In f a c t it is p o s s i b l e t o f i n d a f o r m u l a
a r i t h m e t i c p l u s a unary p r e d i c a t e
QICutR
(i)
S t a r t i n g with
I
+
[Cut
JR
R
6 Vx(JR(x)
i n s t e a d of
R
I'
ZX
such t h a t
I'
is
e x i t s and i s i n
J,(x)
I
,
cf.
i n t h e l a n g u a g e of
such t h a t
+
6 3 y ( y = 2'
JR(2.x)
and a p p l y i n g
J
k-times
.
6 R(y)))]
we get a cut
Ik
such
(ii)
(iii
U s i n g a t e c h n i q u e f o r w r i t i n g s h o r t f o r m u l a s which is d e s c r i b e d in [5], C h a p t e r
7, we can f i n d
of
Ik
JR s u c h t h a t
R
o c c u r s in i t e x a c t l y o n c e .
w i l l increase only l i n e a r l y with
k
.
Now i t f o l l o w s from ( i ) t h a t t h e
l e n g t h s o f p r o o f s o f ( i i ) and ( i i i ) w i l l be o n l y p o l y n o m i a l i n
fact that
Ik
is a c u t and ( i i i ) we c a n c o n s t r u c t a p r o o f of
whose l e n g t h is p o l y n o m i a l i n
( i i ) we o b t a i n t h e lemma.
0
In(
.
Thus t h e l e n g t h
k
.
I (n)
k -
Using t h e
in
A
Combining t h i s p r o o f w i t h t h e proof of
P.PUDLAK
170
P r o o f o f Lemma 2 . 1 :
If
i s a bounded f o r m u l a , t h e n by C o r o l l a r y 8.8 o f [ 1 3 ]
+(x)
I A o + e x p k Px + ( x )
i f f f o r some c u t
I
c l o s e d under
+
and
*
By an i n e s s e n t i a l m o d i f i c a t i o n of t h e proof we g e t t h e same theorem a l s o f o r
e x p o n e n t i a l l y bounded f o r m u l a s .
I(;)
.
2.2,
(where we set
Thus to p r o v e
+(E) i t is s u f f i c i e n t t o p r o v e
The l a t t e r one h a s a p r o o f w i t h l e n g t h p o l y n o m i a l i n
k = 0)
.
(ml
by Lemma
0
I n o r d e r t o be a b l e t o a r i t h m e t i z e s y n t a x i n some t h e o r y , w e have t o
assume t h a t t h e t h e o r y c o n t a i n s some fragment o f a r i t h m e t i c .
us to reduce t h i s assumption t o
Q
.
Lemma 2 . 1 e n a b l e s
This i s b e c a u s e ( 1 ) t h e u s u a l s y n t a c t i c a l
c o n c e p t s a r e n a t u r a l l y f o r m a l i z e d by e x p o n e n t i a l l y bounded f o r m u l a s ,
b a s i c p r o p e r t i e s o f them a r e p r o v a b l e i n
IA
+ exp
,
(2) t h e
( 3 ) the sentences t h a t
we s h a l l c o n s i d e r w i l l be e x p o n e n t i a l l y bounded s e n t e n c e s o f t h e form
+(_I)
P u t o t h e r w i s e , t h e b a s i c p r o p e r t i e s of f o r m u l a s , p r o o f s e t c . whose l e n g t h is
assumed t o be
n
axiomatization
interprets
and
GB)
.
Q
A
,
have p r o o f s p o l y n o m i a l i n
contains
Q
(nl
.
The a s s u m p t i o n t h a t a n
can be weakened by assuming t h a t
( t h i s is r e a l l y n e c e s s a r y i n c a s e o f set t h e o r i e s ,
A
only
e.g.
ZF
.
Proofs and Finitistic Consistency Statements in First Order Theories
3.
171
The lower bound
In this section we shall prove the lower bound on the length of proofs of
finitistic consistency statements.
The main theorem will be stated using
finitistic counterparts of the well-known derivability conditions for the Znd
G"oe1
incompleteness theorem.
Then we shall argue that they are met by natural
The relation that we shall consider is
arithmetizations.
proof of length
5x
axiomatization.
In this section, however,
'I.
It will be denoted by
, where
PA(x,y)
some standard countradiction, say
denoted by
ConA(n)
0
=
A
is not determined by
PA
an arbitrary formula satisfying the derivability conditions.
A
in Theorem 3.1.
1.
r i
stress this fact, we omit the subscript
is provable by a
"y
Thus
P(n,
-I
1)
is an
A
,
it is
In order to
Let
1
, which
denote
will be
later, is a finitistic consistency statement.
It is convenient to assume that formulas and proofs are strings in the
element alphabet
{O,l}
.
The GEdel numbers of formulas and proofs are the
numbers with corresponding diadic expansions.
This allows us to use
also to denote the length of formulas and proofs.
GEdel number
n
,
is proportional to
(01
.
,
(thus
x1
.
,...,xk
I.-I
a formula with the
Again the length of
0
We shall also use the notation
for an arithmetization of the function
...,I&)
0 is
If
will be denoted by '0'
then
tWO
are free in
(nl,
...,n.k)
3(:,,...,
this arithmetization has the following property:
CC
Gsdel number of
+(n-1 '
We shall assume that
there exists a polynomial
such that
Why we can make such an assumption will be explained later.
p
P.PUDLAK
172
Theorem 3.1
Let
and l e t
A
be a c o n s i s t e n t a x i o m a t i z a t i o n ,
p1 , p 2 , p 3 , q 1 , q 2
A I-
E
>
,
let
P(x,y)
be a f o r m u l a
b e p o l y n o m i a l s such t h a t
(0)
Then t h e r e e x i s t s
Q E A
0
x
5 x'
6 P(x',y)
+
P(x,y) ;
s u c h t h a t f o r "0 n E w
Proof:
I n o r d e r t o s i m p l i f y n o t a t i o n , we s h a l l write
... +
t o d e n o t e t h a t f o r some p o l y n o m i a l
...
1 . .
p
,
... .
p(n)
By D i a g o n a l i z a t i o n Lemma, t h e r e e x i s t s a f o r m u l a
Q I-
Thus
(i)
Q
D(x) ++
&
D(m)
- ++
h e n c e t h e same is t r u e a l s o f o r
A
A
we c a n e a s i l y d e r i v e f o r e v e r y
(ii)
1
not
.
P(x,'D(x)'
1
D(x)
)
such t h a t
.
P(E,~D(I$~ )
,
Now, from ( l ) , ( i ) and t h e c o n s i s t e n c y of
m
A*
~ ( 5,)
Proofs and Finitistic Consistency Statements in First Order Theories
Let
S(m)
denote
P(m, D(m) )
.
173
Since
is a p r o p o s i t i o n a l t a u t o l o g y , we g e t from ( 0 ) and (1)
Now, s e v e r a l a p p l i c a t i o n s of ( 3 ) and (0) y i e l d
f o r some p o l y n o m i a l
and t h e d e f i n i t i o n o f
q3
,
J s ( ~ ) ( is p r o p o r t i o n a l t o
(since
S(2)
(m))
.
BY ( 2 )
we have
which t o g e t h e r w i t h ( i ) i m p l i e s
A p p l y i n g (1) t o an i m p l i c a t i o n of ( i ) w e g e t
f o r a polynomial
q4
i m p l i c i t l y d e t e r m i n e d by ( i )
c i e n t l y l a r g e , we have by ( 0 )
By ( 3 ) and by t h e d e f i n i t i o n of
Hence,
for
m
S(m)
sufficiently large,
.
Thus, i f
m
is s u f f i -
P. PUDLAK
174
Thus we get, for m
sufficiently large,
A&
(V)
-'P(q2(m),
rvS(m)'
+
D(m)
.
Now ( i i i ) , (iv) and (v) implies that for some polynomials
p4
and
q5
and
every sufficiently large m
Thus by ( i i )
does
not hold
for any sufficiently large m
easy computation and condition (0)
.
.
The theorem now follows using an
0
There are several ways in which one can argue that the natural
arithmetization meets the conditions ( 0 ) - ( 3 ) .
particular arithmetization.
We shall not construct any such
( F o r some fragments of arithmetic such an
arithmetization is constructed i n [ 1 3 ] and can easily be generalized for other
axiomatizations).
Instead we shall describe some more general properties which
look natural and imply the conditions of the Theorem 3.1.
We start by observing that from the finitistic point of view it is too
little to know that an axiomatization is recursive.
Therefore we shall
consider here NP axiomatizations (which means that the set of axioms can be
accepted by a nondeterministic polynomial time Turing machine).
every finite axiomatization is NP
.
In particular
Now we shall introduce a finitistic
counterpart of the concept of numerability.
175
Proofs and Finitistic Consistency Statements in First Order Theories
Definition
Let
R s wk
p(xl,
.
...,xk)
We say that
polynomial p
be a formula, let
p
and every
A
be an axiomatizat i o n and let
polynomially numerates
...,nk
A1
0
in A
if for some
E w
nl,
R(n l....,nk)
R
P((nl(
,... , Ink])
Phi,...,n+) .
Theorem 3 . 2
Let
R
S
wk
A
.
be a consistent NP axiomatization such that
Q
S
A
and let
Then the following are equivalent:
(1)
R
is
(2)
R
is polynomially numerable in Q ;
(3)
R
is polynomially numerable in
NP ;
A
.
Now it is clear that the additional property of the formula
Y =
4(:l,...,:k)
is just the polynomial numerability. By Theorem 3 . 2
a formula exists, since the
such
k+l-ary relation
m = "the number of
$(z1,.
..,n-k ) "
is NP.
The proofs of
( 2 ) => (1)
and
( 3 ) => (1)
are trivial.
To prove the
converse implications we need first to arithrnetize the concept of an NP set.
In [13] this was done using
another possibility.
so
called
R;
formulas. Here we briefly sketch
P. PUDLAK
176
Theorem 3.3
T h e r e e x i s t s a n e x p o n e n t i a l l y bounded f o r m u l a
e v e r y NP s u b s e t
of
w
polynomially numerates
R
compute
R
for a given
k
,
Q
in
NP
k E w
there exists
.
U N P( t , x )
such t h a t
such t h a t , f o r
UNP(k,x)
(More ove r, t h e r e is a f a s t a l g o r i t h m t o
T u r i n g m a c hine d e f i n i n g
R .)
P r o o f - s k e t c h:
First consider
+ e xp
IA
i n s t e a d of
Q
.
I n t h i s t h e o r y we f o r m a l i z e
t h e c o m p u t a t i o n s of a u n i v e r s a l n o n d e t g r m i n i s t i c T u r i n g m a c h i n e .
w i l l mean t h a t t h e u n i v e r s a l n o n d e t e r m i n i s t i c T u r i n g m a c h i n e w i t h t h e
IJNP (t , x)
p ro g ra m
"clock"
Thus
a c c e p t s t h e i n p u t word
t
so t h a t
Turing machines.
x
5
i t runs i n t i m e
.
We a l s o augment t h e m a c h i n e w i t h a
This enables us to take
The i d e a is r o u g h l y a s f o l l o w s .
there exists a matrix
M
and s t i l l i t is u n i v e r s a l f o r
(xlt + t
A word
UNP(t,x)
x
NP
e x p o n e n t i a l l y bounded.
is a c c e p t e d w i t h a p r o g r a m
t
if
in some f i n i t e a l p h a b e t s u c h t h a t
(1)
t h e f i r s t row c o n s i s t s o f
t , x a n d a s t r i n g of
(2)
M
local
(3)
i-1,j-1'
m.1 - 1 , j ' m i - l , j + l ) ;
"'ij to
t h e l a s t row c o d e s some a c c e p t i n g c o n f i g u r a t i o n ( s a y d e t e r m i n e d by t h e
s a t i s f i e s f i n i t e l y many
0's ;
c o n d i t i o n s (which d e s c r i b e r e l a t i o n of
o c c u r r e n c e of some p a r t i c u l a r s y m b o l ) .
F i n a l l y , t h e m a t r i x is c ode d by some
Let
(i)
k
e x p a n s i o n o f a n a t u r a l number.
be t h e number whic h c o d e s a n NP T u r i n g m a c h i n e f o r
IAo + e x p f U N P ( k , n )
p
R
.
Then
=> R ( n )
since every sentence provable i n
polynomial
11 a d i c
IAo + e x p
is t r u e .
To p r o v e , f o r some
,
( i i ) R(n) => IAo + e x p Ip(Jn0 UNP(k,")
w e h a ve t o p r o v e t h e e x i s t e n c e o f a n a c c e p t i n g c o m p u t a t i o n ( t h e m a t r i x
M) v i a
Proofs and Finitistic Consistency Statements in First Order Theories
a p o l y n o m i a l l y long p r o o f .
1
.
m
which codes t h e a c c e p t i n g
M) and check t h e c o n d i t i o n s ( I ) , ( 2 ) , ( 3 ) f o r t h e
computation ( t h e m a t r i x
numeral
It is enough t o t a k e
1I1
1.
There a r e p o l y n o m i a l l y many i n
(i.e.,
a l s o i n t h e l e n g t h of
i n p u t ) such c o n d i t i o n s , hence w e a r e done.
The proof f o r
Q
can be o b t a i n e d by a n a l y z i n g t h e above proof and
W
e o m i t t h e d e t a i l s s i n c e t h e proof f o r
a p p l y i n g Lemma 2 . 1 .
+ exp
IA
was
0
only sketched.
Here w e were i n t e r e s t e d o n l y i n t h e f a c t t h a t n o n d e t e r m i n i s t i c polynomial
But it i s c l e a r t h a t a more
t i m e c o r r e s p o n d s t o polynomial l e n g t h p r o o f s .
e x p l i c i t r e l a t i o n between t h e s e two measures can be found.
One c a n a l s o bound
t h e l e n g t h of formulas o c c u r r i n g i n p r o o f s u s i n g t h e s p a c e bound of t h e Turing
machine.
P r o o f s o f t h e r e m a i n i n g i m p l i c a t i o n s o f Theorem 3 . 2 :
is a d i r e c t consequence of Theorem 3 . 3 .
( 1 ) => ( 2 )
To prove
i t is enough t o show ( i ) and ( i i ) from t h e proof above f o r
since
Q rA
and we have ( i i ) f o r
consistent, Q E A
,
Q
already.
A
.
(1) = > ( 3 )
( i i ) is true,
( i ) holds since for
A
e v e r y e x p o n e n t i a l l y bounded p r o v a b l e s e n t e n c e is t r u e .
0
P r o p o s i t i o n 3.4
Let
A
be an a x i o m a t i z a t i o n .
t i o n of t h e r e l a t i o n
"z
is an
Suppose
A-proof
of
is a polynomial numeration of t h e r e l a t i o n
Then
P (x,y)
A
PrfA(x,y)
y" i n
1.
A
5x .
is a polynomial numera-
, suppose
Let
that
PA(x,y)
s a t i s f i e s t h e c o n d i t i o n s ( 0 ) and ( 1 ) of Theorem 3.1.
'ID(
be
x'
P. PUDLAK
178
The proof f o l l o w s i n m e d i a t e l y f r o m t h e d e f i n i t i o n .
axiomatization,
t h e n t h e assumption t h a t
P r f A and
If
'1z(
<
A
x'
is an NP
a r e polynomial
numerations is q u i t e n a t u r a l , s i n c e by Theorem 3.2 t h e r e a r e such f o r m u l a s .
The second d e r i v a b i l i t y c o n d i t i o n is u s u a l l y proved by f o r m a l i z i n g t h e
T h i s is t h e c a s e a l s o h e r e .
proof o f t h e f i r s t one.
theorem,
We c a n u s e t h e f o l l o w i n g
( c f . Theorem 6.4 of 1131, where such a theorem is proved f o r NF'
f o r m a l i z e d by
f o r m u l a s i n a weaker t h e o r y ) .
Rr
Theorem 3.5
For a s u i t a b l e polynomial numeration
of
y
of l e n g t h
5x
"
P (x,y)
Q
and a polynomial
I A +~ e x p C U N P ( t , x )
+
o f " t h e r e e x i s t s a Q-proof
q
.
r
~ ~ ( g ( l +x tl )~, U N P C E , ~ ~ ~
T h i s theorem can be proved by f o r m a l i z i n g a p a r t of t h e proof of Theorem
3.3.
Using Theorem 3.5 w e c a n p r o v e t h e d e r i v a b i l i t y c o n d i t i o n (2) f o r
we have t h e f o l l o w i n g :
(1) PrfA
(2) A
and
PA
s a t i s f y t h e a s s u m p t i o n s of P r o p o s i t i o n 3.4;
proves t h a t
PrfA
is
NP; more p r e c i s e l y , f o r some
AE P r f A ( z , x )
where
<...> is
(3) A
++
UNP(k,<z,x>)
k
,
the usual pairing function;
proves t h a t
( 4 ) PA c o n t a i n s
'IzI i x '
NF' ( i n t h e same way);
PQ ; more p r e c i s e l y
A
f o r some p o l y n o m i a l s
is
p m , PQ (n,m)
--
+
PA(i(z)
,z)
p,q ;
( 5 ) PQ s a t i s f i e s t h e d e r i v a b i l i t y c o n d i t i o n s
(0),(1),(3)
.
E
w
PA
if
Proofs and Finitistic Consistency Statements in First Order Theories
179
We omit the proofs.
The derivability condition (3) is the simplest one.
instance, that if
concatenation of
6
is a proof of
d
d, e,
IJI
and
is a proof of
P(x,y) & P(x,y'+'
Hence using Lemma 2.1 and the assumption
.
z)
+
Q
S
P(3x,z)
A
the value of
length of
in
t
,
t
For
t
+
,
0
then the
I A + exp
Thus we have i n
.
we get (3).
Finally we prove an easy generalization of Theorem 3.1.
set of closed arithmetical terms.
+
is a proof of
e
JI
We can assume, for
Let
L
be some
E L , let I be ;
, where
the structure of natural numbers; let
L(t)
n
is
denote the
(It( would be ambiguous).
Theorem 3.6
Let
A 7 Q
polynomial
p
be a consistent axiomatization and suppose that there exists a
such that for every
t
E L
Assuming the derivability conditions of Theorem 3.1 there exists
that for "0 term
(ii) A+
t
6
ConA(t)
E
1
0
such
.
Proof:
n > 0
>
L
(In ( i ) we can write the bound also in the form
Let
6
be so small that
9
A + t = I
p'((t
=I()).
P.PUDLAK
180
would imply
E
(V)
,t ConA(&)
A
where
is from Theorem 3.1.
b
Take
K
so l a r g e and
,
0
5 t'
,
6
<
6
5 11
so small
that
(vi)
t(t)
5
t6
t
6
>
K => p ( l o g t , t ( t ) )
and
K6
<
1
.
Now c o n s i d e r t h e f o l l o w i n g t h r e e c a s e s .
(a)
L(t)
>
t6
.
(b)
t
(c)
t(t)
5K .
Then t h e proof of
Con ( t )
A
mst have t h e l e n g t h a t l e a s t
Then ( i t ) is i m p o s s i b l e , s i n c e t h e bound is
5 t6
and
t
>
.
K
Then by
<
1
.
(i) and ( v i ) we g e t ( i i i ) . I f ( i i ) were
t r u e in t h i s c a s e , t h e n w e would g e t a l s o ( i v ) and h e n c e ( v ) , which is
i m p o s s i b l e by Theorem 3.1.
4.
0
A p p l i c a t i o n s o f t h e l o w e r bound
If
A
c o n t a i n s a s u f f i c i e n t l y s t r o n g fragment of a r i t h m e t i c , t h e n
A +
ConA h a s a speed-up by a n a r b i t r a r y r e c u r s i v e f u n c t i o n f o r s e n t e n c e s t h a t a r e
provable i n both t h e o r i e s .
[91.
This t h e o r e m g o e s b a c k t o Ggdel [ 7 ] and Mostowski
L a t e r r e s u l t s of t h i s k i n d were proved e . g .
[41 and Statman 1171.
Gandy 161 h a s shown t h a t i f w e c o n s i d e r o n l y c l o s e d
i n s t a n c e s of e l e m e n t a r y p r e d i c a t e s ,
t h e n t h e speed-up
n e x t c o r o l l a r y shows t h a t s u c h a speed-up
Con ( t )
A-
,
f o r some t e r m s
We d e n o t e
by E h r e n f e n c h t and M y c i e l s k i
Vx Con ( x )
A
t
is s t i l l v e r y l a r g e .
The
is a c h i e v e d on s e n t e n c e s o f t h e form
.
by
ConA
.
Recall that
ConA(x)
is - P A ( x , r l l )
.
Proofs and Finitistic Consistency Statements in First Order Theories
181
C o r o l l a r y 4.1
Let
A 8 Q
be a consistent axiomatization.
c o n d i t i o n s o f Theorem 3 . 1 f o r
Then f o r some c o n s t a n t s
(1)
A + ConA I c . ( k + l )
>
0
Con
A
PA(x,y)
0
0
(2-)
k
and
c
Assume t h e d e r i v a b i l i t y
and t h a t
and e v e r y
k E w
.
’
Proof:
The f i r s t p a r t is t r i v i a l .
The s e c o n d p a r t f o l l o w s from Theorem 3 . 6 .
t h i s end we s h o u l d p r o v e c o n d i t i o n ( i ) o f Theorem 3.6 f o r t h e t e r m s
which is an e a s y e x e r c i s e .
0
0
0
1
2-,2-,...,
I n f a c t it is n o t d i f f i c u l t t o p r o v e i t f o r any
c l o s e d t e r m of t h e a l p h a b e t
{c,S,+,*,2x}
.
0
Such a speed-up c a n b e a c h i e v e d a l s o by a c o n s e r v a t i v e e x t e n s i o n ( c f .
C o r o l l a r y 4.5 of 1 1 4 1 ) .
Theorem 4 . 2
There e x i s t s
(1)
E
>
0
GB p0 ConZF(%)
and a p o l y n o m i a l
0
p
such t h a t f o r every
;
Proof:
I t is well-known t h a t t h e r e is a c u t
I
in
GB
such t h a t
k E w
To
P. PUDLAK
182
( T h i s is e s s e n t i a l l y due t o R. S o l o v a y , c f .
the f i r s t part.
(141).
A p p l y i n g Lemma 2.2 w e g e t
The second p a r t is a c o n s e q u e n c e o f t h e p r e c e d i n g c o r o l l a r y .
0
Theorem 4 . 3
A S Q
Let
b e a c o n s i s t e n t a x i o m a t i z a t i o n and assume t h e d e r i v a b i l i t y
c o n d i t i o n s o f Theorem 3 . 1 .
(1)
if
is a c u t in
I
Then we have:
,
A
then
A + 'Tx(I(x) & -ConA(x))
is c o n s i s t e n t ;
(2)
if
D(x)
.A
is
(i.e.,
a r e t r u e , then t h e r e e x i s t s
bounded a r i t h m e t i c a l ) f o r m u l a and
k E w
D(c),D(L),
...
such t h a t
A + 3 x ( D ( x ) & -Con
k
(x ) )
is c o n s i s t e n t .
Proof:
(1)
If
A +
Vx(I(x)
f o r some polynomia
for large
(2)
n
Let
Con,(x))
+
,
t h e n in t h e same way a s a b o v e w e would
P . b
i c h is i m p o s s i b l e by Theorem 3 . 1 ,
be a
.A
since
p(ln()
.
D(x)
t h e r e is a p o l y n o m i a l
P1
f o r m u l a and l e t
s u c h t h a t for e v e r y
Hence we c a n c o n s t r u c t a p o l y n o m i a l
p2
D(z),D(L),
...
be t r u e .
n E w
with the property that i f
Then
n
Proofs and Finitistic Consistency Statements in First Order Theories
A $ v ~ ( D ( ~ )+ ConA(xk ) )
(i)
183
,
then
A- I
(ii)
Take
k
l a r g e so t h a t f o r
(iii)
E
p2 h,n , k )
ConA(”).
o f Theorem 3.1 we have
p2(m,n,k)/nk.E
+
0
for
n
+
*
Now s u p p o s e t h a t t h e t h e o r y of (2) is i n c o n s i s t e n t f o r t h i s
m
k
,
i.e.
f o r some
w e have ( i ) . Then w e g e t ( i), h e n c e by ( i i i ) w e have
A
for
n
s u f f i c i e n t l y large.
k*E
& Con
k
(n )
,
But t h i s is p r o h i b i t e d by Theorem 3 . 1 .
( 1 ) h a s been proved in [141 ( i n a d i f f e r e n t way).
0
I t was employed t h e r e
t o show a speed-up by a n a r b i t r a r y e l e m e n t a r y f u n c t i o n o f t h e o r d i n a r y l o g i c
over t h e l o g i c a l c a l c u l i without c u t - r u l e s .
( 2 ) is an improvement of a theorem
of t h e same p a p e r .
5.
The u p p e r bound
To be a b l e t o d e r i v e some n o n t r i v i a l u p p e r bounds t o t h e l e n g t h of p r o o f s
o f f i n i t i s t i c c o n s i s t e n c y s t a t e m e n t s we have t o assume more t h a n we d i d i n
s e c t i o n 3.
We need t h a t f i n i t e p i e c e s o f i n f o r m a t i o n a b o u t t h e u n i v e r s e a r e
coded in n a t u r a l numbers.
The s e q u e n t i a l t h e o r i e s , which we i n t r o d u c e d in
[ 1 5 ] , have t h i s p r o p e r t y .
The f o l l o w i n g d e f i n i t i o n is d i f f e r e n t from but
e q u i v a l e n t t o t h e o r i g i n a l one of ( 1 5 1 .
Definition
A theory
T
is c a l l e d s e q u e n t i a l i f i t s a t i s f i e s t h e f o l l o w i n g
conditions:
(1)
T
is a t h e o r y w i t h e q u a l i t y ,
P. PUDLAK
184
(2)
Q
is i n t e r p r e t a b l e i n
T
relativized to
,
N(x)
(N(x) is some f o r m u l a of
T) 9
(3)
t h e r e e x i s t s a f o r m u l a , which we d e n o t e by
a t o t a l function
T
k Vx,y,t
Intuitively,
e l e m e n t of
x[t]
of two v a r i a b l e s
3z(N(t)
+
<
(Bs
,
x[t] = y
x,t
t h a t deEines i n
such t h a t
.
t(z[sI = x[s]) h z [ t ] = y))
( 3 ) means t h a t we have a d e f i n i t i o n of “ y
x” such t h a t for a given
is t h e
we can always r e p l a c e t h e
t
T
t-th
t-th
e l e m e n t by an a r b i t r a r y one and a l l t h e e l e m e n t s which p r e c e d e it w i l l be
preserved.
Examples of s e q u e n t i a l t h e o r i e s .
(1)
In
PA
we can t a k e e.g.
x [ t ] = yt
where
x =
t h e series of p r i m e s .
(2)
In
GB
’t
lI pt
,
p1,p2
,...
is
we can d e f i n e
X[T] =
X[T
(3)
-
t=l
O t h e r examples a r e
=
8
if
T
{W(<~,T>
is a p r o p e r c l a s s ,
E
x)
IAo, IAo + exp
if
,
T
is a s e t
.
ZF, A l t e r n a t i v e S e t Theory.
I t seems t o be t o o d i f f i c u l t even t o s t a t e t h e u p p e r bound i n such
g e n e r a l i t y a s t h e lower bound.
T h e r e f o r e w e s h a l l b e more e x p l i c i t a b o u t t h e
l o g i c a l c a l c u l u s and i t s f o r m a l i z a t i o n .
W
e s h a l l consider a f i r s t order
l a n g u a g e w i t h f i n i t e l y many p r e d i c a t e symbols
which is
=
,
w i t h l o g i c a l symbols
7 ,
+,
8,
Pd
,
d = 1,2,
.. . , e ,
and w i t h v a r i a b l e s
one o f
assume t h a t t h e f u n c t i o n symbols a r e t r e a t e d a s r e l a t i o n s y m b o l s . )
.
vo’vI’”*
(Thus when w e s p e a k a b o u t t h e o r i e s which c o n t a i n f u n c t i o n s y m b o l s , e . g . (7
,
we
The l o g i c a l
c a l c u l u s w i l l be t h e one p r e s e n t e d i n [ a ] , ( i t h a s 5 axiom schemas and t h e
Proofs and Finitistic Consistency Statements in First Order Theories
r u l e s o f modus ponens and g e n e r a l i z a t i o n ) .
strings i n
Th u s
{O,l}
P (x,y)
A
ConA(x)
185
F o r m u l a s and p r o o f s a r e a g a i n
and t h e s t r i n g s a r e a r i t h m e t i z e d v i a d i a d i c e x p a n s i o n s .
( t h e f o r m a l i z a t i o n of
"y
has an
A
proof of length
5 x")
is u n i q u e l y d e t e r m i n e d by t h e n u m e r a t i o n o f t h e a x i o m a t i z a t i o n
c.2
We e x t e n d t h e n o t a t i o n
t o a r b i t r a r y s t r i n g s o f symbols.
A
and
.
The
c o n c a t e n a t i o n w i l l b e d e n o t e d just by j u x t a p o s i t i o n .
S i n c e t h e c o m p l e t e p r o o f s o f t h e Lemmas which f o l l o w
would b e e x t r e m e l y
long and u n i n t e r e s t i n g , w e s h a l l p r o v e o n l y some t y p i c a l c a s e s , w h i c h s h o u l d
d e m o n s t r a t e s u f f i c i e n t l y our proof t e c h n i q u e s .
In t h e f o l l o w i n g t h r e e l e m m a s w e a s s um e t h a t
A
is s e q u e n t i a l .
In o r d e r
t o s i m p l i f y o u r n o t a t i o n l e t u s assume t h a t t h e l a n g u a g e c o n t a i n s j u s t a
s i n g l e , s a y b i n a r y , p r e d i c a t e symbol
g =i f <==>df
Fm ( x ) <==>
df
"x
.
P
vt
*
Further, l e t
i(g[tl = f [ t l ) ;
is a f o r m u l a of l e n g t h
5
n",
n E w .
Lemma 5 . 1
There e x i s t s a polynomial
formula
(1)
Sat ( x , f )
a nd t h e r e a r e
fin(x) + {Satn(x,f)
++
hn(x)
+
such t h a t f o r e v e r y
A-proofs
n E w
of length
p(n)
++
[ 3 i , j ( x = r ~ ( v i , v . P & ~ ( f [ i ~ , f [ j v~ j )
J
v $y,z(x =
(2)
p
yrJz
& (Satn(y,f) + Satn(z,f)))
v 3 y ( x =r-;ly
6 --rSatn(y,f)) v
v 3i(x
y & vg(g = i f
= :
V
'
(Sat ( x , f )
++
Satn+l(x,f))
+
.
v
Satn(y,g)))]} ;
there exists a
of
P.PUDLAK
186
Proof:
Sato
.
<0
c a n be a n a r b i t r a r y f o r m u l a , s i n c e t h e r e a r e no f o r m u l a s o f l e n g t h
Denote by
t h e r i g h t hand s i d e o f t h e e q u i v a l e n c e i n ( 1 ) .
Z(Satn)
o r d e r t o a v o i d e x p o n e n t i a l growth o f t h e l e n g t h o f
Z'
,
Sat,
u s i n g a t e c h n i q u e o f [ 5 ] , C h a p t e r 7, so t h a t
Sat,
,
we r e p l a c e
occurs i n
Z
In
by
Z'(Satn)
only once and t h e e q u i v a l e n c e
where
R
is a new p r e d i c a t e , i s p r o v a b l e in t h e p r e d i c a t e c a l c u l u s .
can d e f i n e by i n d u c t i o n f o r
n
>
1
Satn(x,f)
Let
of
On
On
+ On+l
whose
a l s o only l i n e a r l y .
f o r m u l a of l e n g t h
shape d o e s
,
hence a l s o
(i).
n o t depend on
We s h a l l d e s c r i b e a p r o o f
5 n+l .
n (i.e.
A
,
assume t h a t
(yI
x
is a
W
e have t o show t h a t
We can d i s t i n g u i s h t h e c a s e s :
Then
is t r u e and
On
s a t n + l ( x , f ) ++ S a t n + 2 ( x , f )
.
t h e s e proofs w i l l be
Hence t h e l e n g t h s o f t h e s e p r o o f s w i l l i n c r e a s e
Arguing i n
(iii)
x = r-ly
by
n
n
d e n o t e t h e u n i v e r s a l c l o s u r e o f (2).
i n s t a n c e s o f a p r o o f schema).
let
.
++df Z ' ( S a t n - l )
Then t h e l e n g t h o f s u c h f o r m u l a s is l i n e a r i n
h a s a p r o o f of l e n g t h l i n e a r i n
Now w e
x
5n ,
is a t o m i c , x
t h u s by
Satn(y,f)
++
.
is a n i m p l i c a t i o n e t c .
On
Satn+l(y,f)
.
E.g.
Proofs and Finitistic Consistency Statements in First Order Theories
Applying
( i t ) t o n+l and n+2 we g e t
satn+l(x,f)
,
-Satn(y,f)
++
.
S a t n + 2 ( x , f ) ++ - S a t n + l ( y , f )
The l a s t t h r e e e q u i v a l e n c e s y i e l d ( i i i ) .
proofs of
187
On
On+l
+
Since
Oo
is t r i v i a l and w e have t h e
of l i n e a r l e n g t h we g e t a proof of
On
,
i.e. of (2), of
polynomial l e n g t h .
Now we can c o n s t r u c t a p o l y n o m i a l p r o o f of ( l ) , i . e . of
h n ( x ) + ( s a t n ( x , f ) ++ Z ( S a t n ) )
.
T h i s f o l l o w s e a s i l y from ( i i ) and ( 2 ) , s i n c e a l l t h e f o r m u l a s t o which
is a p p l i e d in
Z(Satn)
a r e of l e n g t h
<
n
.
Sat
0
Lemma 5.2
Sat,
p r e s e r v e s t h e l o g i c a l axioms,
p o l y n o m i a l in
h n ( x ) & "x
(1)
Sat
in
n
A-proofs
of l e n g t h s
of
is a l o g i c a l axiom"
preserves the logical rules,
n
i.e. there a r e
+
satn(x,f)
;
i.e. t h e r e a r e p r o o f s o f l e n g t h s polynomial
of
(2)
Fmn(xr+'y)
& Sat,(x,f)
& Satn(xr+7y,f)
(3)
F m n ( r V v 7 x ) & Vf S a t n ( x , f ) + Vf Sat,('Vvf
+
Satn(y,f) ;
x;f)
.
0
P r o o f s of ( 1 ) f o r p r o p o s i t i o n a l axioms and t h e p r o o f s of ( 2 ) and ( 3 )
f o l l o w d i r e c t l y from Lemma 5 . 1 ( 1 ) .
The p r o o f o f (1) f o r q u a n t i f i e r axioms
r e q u i r e s an a d d i t i o n a l lemma, t h e r e f o r e is o m i t t e d .
P. PUDLAK
188
Lemma 5 . 3
There e x i s t s a p o l y n o m i a l
formula
4(vi
p
such t h a t f o r e v e r y
,...,
5n
V. )
of l e n g t h
m
1
d i s p l a y e d ) t h e r e e x i s t s a n A-proof
satn('41,f)
(where a l l f r e e v a r i a b l e s of
of l e n g t h
4(f[i,
++
and e v e r y
n E w
p(n)
4
are
of
I ,. . . , f [ i m I )
.
Proof:
Let
Y(+)
be t h e f o r m u l a a b o v e .
polynomial
-< p ( n )
p
a t o m i c we h a v e s u c h a p r o o f o f
Now it is s u f f i c i e n t t o show t h a t t h e r e e x i s t s a
from Lemma 5 . 1 ( 1 ) .
Y(4)
4
For
such t h a t a l l t h e f o l l o w i n g i m p l i c a t i o n s have p r o o f s o f l e n g t h s
:
Y(+)
& Y($)
Y(+)
.+
HI+)
Y(+)
.+
Y(Yv;+)
.+
Y(+
,
for
,
,
+)
.+
t h e n 'we h a v e , by Lemma 5 . 1 ( 1 1 ,
*
which, t o g e t h e r w i t h
Let
a
Y(4),
be a s e n t e n c e .
yields
Then
c o n s i d e r t h e second
a p o l y n o m i a l p r o o f of
.
P{a)(x,y)
n ;
.
E.g.
-Satn(
Y(y+)
5
;
( v v ~ +5( n
for
Satn(ry+',f)
+
5n
1-91
T h i s c a n be e a s i l y d e r i v e d from Lemma 5 . 1 ( 1 ) .
implication,
I+ $1
for
r
47 , f ) ,
c]
and
Con{a)(y)
w i l l denote t h e
a r i t h m e t i z a t i o n s of p r o v a b i l i t y and c o n s i s t e n c y where t h e a x i o m a t i z a t i o n
i s numerated by t h e f o r m u l a
{a)
.
x = a
'1
Theorem 5 . 4
Let
every
A
n E w
be s e q u e n t i a l .
Then t h e r e e x i s t s a p o l y n o m i a l
and e v e r y s e n t e n c e
A
a
1.
,
p0 a
+
5
n
.
con{a)(c)
.
p
such t h a t f o r
Proofs and Finitistic Consistency Statements in First Order Theories
189
Proof:
By t h e p r e c e d i n g lemma we have - S a t n ( T , f )
.
Hence i t is s u f f i c i e n t t o
show t h a t
a 6 P
has an
A-proof
(n,x)
-
{a1
of polynomial l e n g t h .
+
Vf S a t n ( x , f )
Denote t h i s formula by
t r i v i a l , t h e r e f o r e we need o n l y polynomial p r o o f s of
On
that
a
and
hold t r u e and
w
is a proof of
en
,
1.
x
+
.
On+l
5
.
On
n+l
Bo
is
S o assume
.
We have t o
prove
(i)
Now
Vf s a t n + l ( x , f )
w
.
is a sequence of formulas where t h e l a s t one is
of t h i s s e q u e n c e , e x c e p t of
x
, we
x
.
For every formula
have
vf Satn(y.f)
by
On
.
S a t n+l
Using ( 2 ) of Lemma 5.1 we g e t t h e same f o r
.
Now we c o n s i d e r
t h e f o l l o w i n g two c a s e s :
(a)
x
is a l o g i c a l axiom or f o l l o w s from t h e p r e c e d i n g f o r m u l a s of
some l o g i c a l r u l e .
(b)
x = ral
.
By Theorem 5 . 4 ,
A
by
Then ( i ) h a s a polynomial proof by Lemma 5.2;
a
Then we g e t a polynomial proof of ( i ) u s i n g t h e a s s u m p t i o n
and Lemma 5.3.
then i n
w
0
if
A
is a f i n i t e a x i o m a t i z a t i o n of a s e q u e n t i a l t h e o r y ,
we have p r o o f s of l e n g t h polynomial in
t h a t t h e numeration of
A
is r e a s o n a b l e ) .
n
of
Con ( n )
A -
,
(assuming
This theorem could be e a s i l y
g e n e r a l i z e d t o i n f i n i t e a x i o m a t i z a t i o n s which a r e s p a r s e , i . e .
t h e r e a r e o n l y p o l y n o m i a l l y many axioms of l e n g t h
not i n c l u d e t h e t h e o r i e s t h a t we a r e i n t e r e s t e d i n
i
n
.
for e v e r y
n
However t h i s would
(PA,ZF) , s i n c e t h e y a r e
P. PUDLAK
190
not sparse.
T h e r e f o r e we s h a l l p r o v e a d i f f e r e n t theorem.
The p r o o f of t h i s
theorem is b a s e d on t h e f a c t t h a t t h e a x i o m a t i z a t i o n s in q u e s t i o n c a n b e
r e p l a c e d by s p a r s e o n e s .
Theorem 5.5
A = {try O ( + ( y , z ) ) ( + ( y , z )
Let
formula w i t h two f r e e v a r i a b l e s
a n a x i o m a t i z a t i o n of a s e q u e n t i a l t h e o r y .
bounded i n t h e schema
so t h a t
.
@
it is p r o v a b l e in
Vy O ( + ( y , z ) ) " .
Suppose t h a t t h e v a r i a b l e
A
Suppose t h a t a n u m e r a t i o n o f
that
A
is an axiom i f f
"a
Then f o r some p o l y n o m i a l
p
and e v e r y
in
y,z)
be
y
is not
is c h o s e n
A
is of t h e form
a
n E
u)
.
Proof:
Define
Trn(x,y,z)
Let
an
Then
of
.
+
Satn(x,f))
A
is a n i n s t a n c e o f t h e schema, h e n c e
proves t h a t
a
I n f a c t t h i s p r o o f h a s l e n g t h p o l y n o m i a l in t h e l e n g t h o f
a l s o polynomial i n
(we know t h a t
n
Sat,
h a s polynomial l e n g t h ) .
be t h e c o n j u n c t i o n of f i n i t e l y many s e n t e n c e s p r o v a b l e i n
s e q u e n t i a l i t y of
Bn
.
denote
an
A
++df Vf ( f [ i ] = y 6 f [ l ] = z
h a s an
A
A-proof
f o r every polynomial
.
Let
Bn
be
an 6 6
of p o l y n o m i a l l e n g t h .
q
a n o t h e r polynomial
A
.
Since
a
A
i s a n axiom
Trn
,
thus
6
Let
which e n s u r e t h e
is a n axiom of
A
,
Using Theorem 5.4 we can c o n s t r u c t
p
p0 con{6nl(&)
such t h a t f o r e v e r y
.
n Ew
Proofs and Finitistic Consistency Statements in First Order Theories
I t s u f f i c e s t o prove now t h a t for a s u i t a h l e polynomial
q
191
we have a
polynomial proof of
in
let
F i r s t we s h a l l a r g u e in t h e m e t a t h e o r y .
A.
vo,vl
be t h e f r e e v a r i a b l e s of
0
Satn(r+l,f)
c+
u s i n g a proof of polynomial l e n g t h ,
sequential).
6
Further,
Again u s i n g t h e s e q u e n t i a l i t y of
For t h i s
x
I f we assume moreover
By Lemma 5.3 we can d e r i v e from
,
+(f[Ol,f[ll)
Bn
(which is
*
we f i n d an
6
x
such t h a t
t h e n we can d e r i v e
Vy P ( + ( y , z ) )
a
h 6)
be g i v e n and let
A
t h i s proof i n t o a proof
w'
from a s i n g l e axiom
r e p l a c e every
A
<
q()w))
'
+
l
and
+
.
<
,
Thus we have
n
is d e r i v a b l e
v i a a proof of polynomial l e n g t h .
in
lw'I
x[g]
we have t h e n
a
axiom of
is
.
+(y,z)
w
Now l e t a proof
6
of
6
shown t h a t t h e i n s t a n c e of t h e schema f o r a r b i t r a r y
from
6
a l s o implies
Trn(r+',y,z)
.
be a formula
( s i n c e t h e t h e o r y a x i o m a t i z e d by
Thus we g e t a polynomial proof from
x[l] = y
.
+
Let
.
Iw)
<
Bn
i n such a way t h a t we
n
by t h e proof of t h i s axiom from
f o r some polynomial
q
.
we can t r a n s f o r m
6"
.
Thus we have
P. PUDLAK
192
.
In order to get ( i ) we have to formalize the above argument in A
clear that this argument can be formalized in
hence we can apply Lemma 2.1.
,
,
contains Q
A
0
The usual axiomatizations of
PA
and
ZF
since instead of a single parameter (which is
arbitrarily many parameters.
+ exp
IA
It is
are not exactly of this form,
y
in try O(o(y,z)))
they allow
Since all sequential theories have a pairing
function, this is an inessential difference.
The fact that
PA
and
ZF
are
axiomat ized by such schemas is provable (for reasonable numerat ions) already in
IA
+ exp
.
Thus the polynomial upper bounds are true also for
PA
and
ZF
.
The theorem of Vaught [ I 8 1 implies that every recursively axiomatizable
sequential theory is axiomatizable by a schema. We would like to know if it
can be axiomatized by a schema of the form described in Theorem 5.5, (i.e.,
with
6.
y
free in
4).
Some problems related t o
NP = coNP?
So far we have studied only the question of the size of the shortest proof
of
in A
ConA(")
.
Rut what about the proofs of
in weaker
ConA(")
theories? The best that we can say is the following informal proposition.
Proposition 6 . 1
Let
A
be a consistent
reasonable numeration of
every
A
co-NF'
in B
axiomatization, let
Q
S
there exists.a polynomial
.
B
p
Then for a
such that for
n E w
B
p(n)
12
conA(;) .
.
Proof-sketch:
Working in
B
enumerate all sequences of length (n
of them is a proof of contradiction in
A
.
and check that none
Proofs and Finitistic Consistency Statements in First Order Theories
193
Problem 1
Is there a finite consistent axiomatization A
p
,Q
-z A
and a polynomial
such that
(2)
or
Con
(fi) ?
A+ConA(Pn)
&
A
(Mycielski)
We conjecture that the answer is "0.
We have added the finiteness
assumption in order to avoid possible pathological examples, but we do not know
any such example.
The quantifier complexity of the formulas occurring in the proofs of
Con ( n )
A-
n
that we have constructed in the preceding section increased with
A truly finitistic proof should have limited quantifier complexity.
.
Such
proofs were used in the proof-sketch of Proposition 6 . 1 , but they were
exponentially long.
Problem 2
Is there a consistent finite axiomatization A
a polynomial
in A
p
of length
such that for every
5 p(n)
n E w
,Q
C A
,
a number k
there exists a proof of
and
Con,(n)
which uses only formulas of complexity Lk ?
Again we conjecture that the answer is "0.
Rut we have the following
proposit ion.
Proposition 6.2
A negative answer to any of the two problems above would imply
(hence also P
f
NP)
.
Np
f
coNP,
P. PUDLAK
194
Proof:
We s h a l l show t h a t s u c h a n a n s w e r would imply t h e s t r o n g e r i n e q u a l i t y
NEXP
X
f
coNEXP
.
S e t s o f numbers which b e l o n g t o NEXP a r e e x a c t l y t h o s e sets
f o r which t h e r e e x i s t s an
If
is f i n i t e , t h e n
A
a l g o r i t h m which a c c e p t s
Np
Con (n)
A-
,
as a p r e d i c a t e on
w
,
2"
is i n
P r o v i n g t h e c o u n t e r p o s i t i v e i m p l i c a t i o n s assume t h a t
A
for s u f f i c i e n t l y large f i n i t e part
iff
n E X
coNMP
.
.
NMP = coNEXP.
Then
of t h e t r u e a r i t h m e t i c a l s e n t e n c e s we
have
A
f o r some
ConA(n)
ConA(x)
c*
UNP(k,2X)
k E w ; ( s e e t h e d e f i n i t i o n o f UNP i n S e c t i o n 3 ) .
in
A
it i s s u f f i c i e n t t o f i n d a c o m p u t a t i o n of t h e T u r i n g machine
w i t h t h e number
k
on t h e i n p u t
2"
and c h e c k i n
computation ( f o r t h e corresponding numeral).
in
A
A
t h a t i t is s u c h a
The l e n g t h of t h i s c o m p u t a t i o n is
p o l y n o m i a l i n t h e l e n g t h of t h e i n p u t , which is
ConA(x)
Thus t o p r o v e
12"1 = n
.
Thus t h e p r o o f of
h a s p o l y n o m i a l l e n g t h and bounded q u a n t i f i e r c o m p l e x i t y .
Acknowledgement.
M y c i e l s k i and W.N.
I would l i k e t o t h a n k J e f f P a r i s , J a n K r a j t E e k , J a n
R e i n h a r d t f o r t h e i r comments a b o u t t h i s p a p e r .
Proofs and Finitistic Consistency Statements in First Order Theories
195
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