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Kolmogorov complexity and the second incompleteness theorem
y
Makoto Kikuchi The Graduate School of Science and Technology Kobe University Rokkodai, Nada-ku, Kobe 657, Japan e-mail: [email protected] Running title: Kolmogorov complexity Abstract: We shall prove the second incompleteness theorem via Kolmogorov complexity.
0. Introduction
Kolmogorov complexity is a measure of the quantity of information in nite objects. Roughly speaking, the Kolmogorov complexity of a number n, denoted by K (n), is the size of a program which generates n, and n is called random if n K (n). Kolmogorov showed in 1960's that the set of nonrandom numbers is recursively enumerable but not recursive, and this is a version of Godel's rst incompleteness theorem (cf. Odifreddi [8]). Chaitin also gave information-theoretic formulation of the rst incompleteness theorem in terms of Kolmogorov complexity. Relations between Kolmogorov complexity and the rst incompleteness theorem have been discussed in many places (cf. Li and Vitanyi [7]). Our purpose is to show that Kolmogorov complexity also leads to the second incompleteness theorem. While Godel's proof of the rst incompleteness theorem brings the second incompleteness theorem, Kolmogorov's proof dose not yield the second incompleteness theorem by the similar manner. Nevertheless, as noted in Odifreddi [8], Kolmogorov's proof can be seen as an application of Berry's paradox (the paradox of the de nability of the least integer not nameable in fewer than nineteen syllables), and Boolos [1] and Kikuchi [4] have extracted the rst and the second incompleteness theorems from this paradox. In this note, by applying the method y
This work was partially supported by Grant-in-Aid for Cooperative Research, The Ministry
of Education, Science and Culture No. 04302009, 06302014.
Typeset by
1
AMS
-TEX
used in Kikuchi [4] to a variant of Kolmogorov's theorem, we shall prove the second incompleteness theorem. In 1, we introduce basic theorems about provability predicates and the arithmetized completeness theorem. In 2, we de ne Kolmogorov complexity and brie y review the rst incompleteness theorem in terms of Kolmogorov complexity. The Kolmogorov complexity of a number is usually de ned to be the minimal length of a program under a certain programming method which outputs the number. In this note, we adopt the de nition of Kolmogorov complexity given by Odifreddi [8], which uses the index of partial recursive functions instead of the length of programs. Then, in 3, we prove the second incompleteness theorem. x
x
x
1. Preliminaries
Let A = +; ; 0; 1; < be the rst-order language of arithmetic. A formula in A is called 10 if contains only bounded quanti ers (i.e., quanti ers of the form ( x < t) or ( x < t) with a term t that does not contain x), and 61 if is a formula of the form x for some 10 formula . It is well known that a relation R Nk is recursively enumerable if and only if there is a 61 formula (x) such that m R if and only if N = (m ) for all m Nk . Peano arithmetic, denoted by PA, is the theory in A that consists of the basic axioms of arithmetic (i.e., the axioms of discretely ordered semirings with the least positive element 1) and the axiom schema of induction. Throughout this note, T means a recursively axiomatizable extension of PA. T is said to be !-consistent when the following condition holds: for any formula (x) in A, T x (x) if T (n) for all n N. Let ProvT (x; y) be a 61 formula that denotes the relation that x is the G odel number of a sequence of formulas which forms a T -proof of a formula with the G odel number y and de ne PrT (y ) to be the formula xProvT (x; y ). Then, Con(T ) is the formula PrT ( 0 = 1 ) which means T is consistent . The formula !-Con(T ) which means T is ! -consistent is de ned similarly. For any 61 sentence in A , (i) N = PrT ( ), (ii) N = Con(T ) (PrT ( ) ), (iii) N = ! -Con(T ) (PrT ( ) ). This theorem is provable in PA. That is, For any 61 sentence in A , (i) PA PrT ( ), (ii) PA Con(T ) (PrT ( ) ), (iii) PA !-Con(T ) (PrT ( ) ). Remark that PrT (x) satis es the modus ponens and the deduction theorem in PA. Hence, PA PrT ( ) Con(T + ) L
f
1
g
L
8
9
9
2
j
2
L
L
6` 9
2
9
:
d
e
Theorem 1.1. j
!
j
L
d
e
!
j
d:
!
e
d
! :
e
!
Theorem 1.2. `
`
`
!
L
d
e
!
d:
!
e
d
! :
e
` :
!
d
e
!
2
:
:
`
for any , since PA ` :PrT (d e) ! :PrT (d: ! 0 = 1e). Refer Smorynski [10] about these theorems and remarks. Now, we are going to introduce the arithmetized version of the completeness theorem. Let C be a set of new constants and LA = LA [ C . We say a formula (x) in LA de nes a model of T in a theory S if we can prove within S that the set f : is a sentence in LA that satisfy (de)g forms an elementary diagram of a model of T with a universe from C . Theorem 1.3. (The arithmetized completeness theorem).
There exists a formula TrT (x) in LA that de nes a model of T in P A + Con(T ). Let and be structures for LA. We say is an end-extension of and
M M M M write M e M if M M and M j= a < b for any a 2 M and b 2 M n M. Note that M j= implies M j= if M e M and is a 6 formula. We say a model M of T is a de nable end-extension of a model M of PA and write M d M if M e M and they satisfy M j= PrT (d e) ) M j= ; M j= TrT (d e) , M j= 0
0
0
0
0
0
0
0
1
0
0
0
0
0
for some formula TrT (x) in LA. From Theorem 1.3, we have the following corollary. Corollary 1.4. Let M be a model of PA. Then, M satis es Con(T ) if and only if
M has a de nable end-extension which is a model of T .
Corollary 1.4 has applications to model-theoretic proofs of the incompleteness theorems (cf. Kreisel [6], Smorynski [10], and Kikuchi [4]). See Hajek and Pudlak [2] and Kaye [3] for more information about models of arithmetic, and Smorynski [10] and Kikuchi and Tanaka [5] for the proofs of Theorem 1.3 and Corollary 1.4. 2. Kolmogorov complexity
Let f'ne(x)ge N be a canonical enumeration of n-ary recursive functions. (We omit the superscript n if there is no confusion.) We write '(x) # if '(x) is de ned at x and write '(x) " otherwise. Also, we write '(x) ' ' (x) if both '(x) and ' (x) are unde ned or they are de ned and '(x) = ' (x). Note that '(x) # can be expressed by a 61 formula since the graph of '(x) can be represented by a 61 formula and '(x) # if and only if y = '(x) for some y. The Kolmogorov complexity K (x) of x is de ned by K (x) = e('e (0) ' x) where e is the least e which satis es . Remark that K is an arithmetically de nable one-to-one function from N to N, and the relation K (x) y is represented by the 61 formula 9z y('z (0) ' x). The following lemma can be easily proved by the fact that K is one-to-one. 2
0
0
0
3
Lemma 2.1. For any a, there exists b a + 1 such that a + 1 K (b) We say a number x is random if x K (x). Lemma 2.1 shows that there exists a random number b such that a + 1 K (b) for any a, hence there exists in nitely many random numbers. The set of nonrandom numbers is recursively enumerable since it is de nable by a 61 formula, and Kolmogorov showed that it is not recursive. In fact, he proved that the set is simple, that is, it is recursively enumerable, and its complement is an in nite set which does not contain any in nite recursively enumerable subset (cf. Li and Vitanyi [7] and Odifreddi [8]). Since Godel's rst incompleteness theorem is derivable from the existence of such a set, Kolmogorov's theorem is a version of the rst incompleteness theorem. The following theorem is a variant of Kolmogorov's theorem. Theorem 2.2. (The rst incompleteness theorem).
e N such that N = Con(T ) x( PrT ( e < K (x) )), (ii) N = ! -Con(T ) x (e < K (x ) PrT ( K (x) e )). That is, we cannot prove e K (a) for any a N within consistent recursively There exists
(i)
j
2
! 8
j
:
d
e
! 8
! :
d
e
2
axiomatizable theories while there exists in nitary many such a's. We shall see a proof of a formalized version of this theorem in the following section.
3. The second incompleteness theorem For any 61 formula R(x; y ), let 01 (R) and 02 (R) be formulas in LA de ned by 01 (R) , 8x8y (R(x; y ) ! y < K (x)); 02 (R) , 8x8y 8z (R(x; y ) ^ z y ! R(x; z )):
Lemma 3.1.
Let
R(x; y ) be a 61
formula. Then there exits
e 2 N such that
T + 01 (R) ^ 02 (R) ` 8x8y (R(x; y ) ! y < e): Proof.
By the selection theorem, there is a recursive function f : N ! N such that
N = xR(x; b) j
9
f (b) #; f (b) # ) N j= R(f (b); b) )
for all b 2 N. This function f is de ned by f (y ) ' (w(R ((w)0 ; y; (w)1 ; . . . ; (w)n )))0 0
where R is the 10 formula such that 0
R(x; y ) , 9z1 1 1 1 9zn R (x; y; z1 ; . . . ; zn ) 0
4
and (w)i is the i-th component of w, hence we can show T T for all b 2 N. Hence
` 9xR(x; b) ! f(b) # ` 8x(x = f(b) ! R(x; b))
(1)
T + 01 (R) ` 8x(x = f(e) ! e < K(x)):
(2)
By applying the recursion theorem to f, we have e 2 N such that 'e (0) ' f(e): In order to show the existence of this e, only we have to do is applying the S-m-n theorem just for the recursive function f which has a concrete de nition given above. So e can be computed actually from the 61 formula R. In addition, by examining the proof of the S-m-n theorem, we can prove So we have By (2) and (3), Also, by (1), Hence
T
` 8x(x = 'e(0) $ x = f(e)):
T
` 8x(x = f(e) ! K(x) e):
(3)
T + 01 (R) ` f(e) " : T + 02 (R) ` 8x8y(R(x; y) ^ e y ! f(e) #):
T + 01 (R) ^ 02 (R) ` 8x8y(R(x; y) ! y < e):
3
Lemma 3.2. (i) T ` Con(T ) ! 01 (PrT (dy < K(x)e)), (ii) T ` 02 (PrT (dy < K(x)e)). Proof. Since y < K(x) is a negation of a 61 formula, (i) is a direct consequence of Theorem 1.2 (ii). It is also easy to show (ii), because T ` 8y 8z(z y ! PrT (dz y e)) by Theorem 1.2 (i) and PrT (x) satis es the modus ponens. 3 From these two lemmas, we have a formalized version of Theorem 2.2. Its proof also depends on Theorem 1.2. Theorem 3.3. (The formalized rst incompleteness theorem). There exists e 2 N such that (i) T ` Con(T ) ! 8x(:PrT (de < K(x)e)), (ii) T ` !-Con(T ) ! 8x(e < K(x) ! :PrT (dK(x) ee)). Remark that Lemma 2.1 is also provable in T , since the pigeonhole principle is derivable from T . 5
+ 1( + 1 ( )) Now, we are ready to prove the second incompleteness theorem.
Lemma 3.4. T ` 8x9y x
x
K y
.
Theorem 3.5. (G odel's second incompleteness theorem).
( ) Proof. Suppose that is consistent and any model of satis es Con( ). Since is consistent, there is a model M0 of . By Lemma 3.4 and the least number principle in , there is 0 + 1 such that M0 = ( 0 ) 0( ( ) ) Then, by Theorem 3.3 (i), M0 = PrT ( ( 0) ) Hence M0 = Con( + ( 0 ) ). So, by Corollary 1.4, there is a de nable end-extension M1 of M0 . Again, take the least element 1 + 1 such that M1 = ( 1). Since ( ) is a 61 formula, M1 = ( ) for any 0, and M1 = ( 0) because M1 is a model of + ( 0) . Therefore, M1 = 0( ( ) ) so 1 is strictly greater than 0 . Repeating this construction + 2 times, we have a sequence of models M0 d M1 d d Me+2 of and a corresponding strictly increasing sequence of numbers 0 1 e+2 . This contradicts the choice of i's. Therefore, if is consistent, there exists a model of which does not satisfy Con( ), hence we have the second incompleteness theorem by the completeness theorem. 3 Remark. Since our proof of the second incompleteness theorem is not formalizable in the system of primitive recursive arithmetic PRA (cf. Smorynski [10], Comments 6.3), it does not directly bring the formalized version of the second incompleteness theorem, PRA Con( ) PrT ( Con( ) ) However, our proof can be carried out within a subsystem of second-order arithmetic WKL0, since the completeness theorem is provable in WKL0 (cf. Simpson [9]) and Corollary 1.4 is provable in weaker subsystem RCA0 (cf. Kikuchi and Tanaka [5]). Thus we can also obtain a new proof of the formalized second incompleteness theorem, by using a theorem of H. Friedman that any 52 theorem of WKL0 is provable in PRA (cf. Simpson [9]). Acknow ledgement. The author of this paper thanks Prof. K. Tanaka and Mr. T. Yamaguchi for stimulating discussions. If T is consistent, Con T
is not derivable from T .
T
T
T
T
T
T
a
e
j
e < K a
j
j
T
K a
:
^ 8x < a
K x
de < K a
e :
e :
e
a
j
e < K a j
K x
K a
y
j
e
T
j
a
8x a
K x
K a
e
e ;
a
e
a
111
< a
T
< 111 < a
a
T
T
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e
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6
d
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a < a
References 1. Boolos, G., A new proof of the G odel incompleteness theorem, Notices Amer. Math. Soc. 36 (1989), 388{390. 2. H ajek, P. and Pudl ak, P., Metamathematics of First-Order Arithmetic, Springer, 1993. 3. Kaye, R., Models of Peano Arithmetic, Oxford, 1991. 4. Kikuchi, M., A note on Boolos' proof of the incompleteness theorem, Math. Logic Quart. 40 (1994), 528{532. 5. Kikuchi, M. and Tanaka, K., On formalization of model-theoretic proofs of G odel's theorems, Notre Dame J. Formal Logic 35 (1994), 403{412. 6. Kreisel, G., Notes on arithmetical models for consistent formulae of the predicate calculus, Fund. Math. 37 (1950), 265{285. 7. Li, M. and Vit anyi P.M.B., Kolmogorov complexity and its applications, Handbook of Theoretical Computer Science (J. van Leeuwen, ed.), Elsevier, 1990, pp. 187{254. 8. Odifreddi, P., Classical Recursion Theory, North-Holland, 1989. 9. Simpson, S.G., Subsystems of Second-Order Arithmetic, forthcoming. 10. Smory nski, C.A., The incompleteness theorems, Handbook of Math. Logic (J. Barwise, ed.), North-Holland, 1977, pp. 821{865.
7
y
Makoto Kikuchi The Graduate School of Science and Technology Kobe University Rokkodai, Nada-ku, Kobe 657, Japan e-mail: [email protected] Running title: Kolmogorov complexity Abstract: We shall prove the second incompleteness theorem via Kolmogorov complexity.
0. Introduction
Kolmogorov complexity is a measure of the quantity of information in nite objects. Roughly speaking, the Kolmogorov complexity of a number n, denoted by K (n), is the size of a program which generates n, and n is called random if n K (n). Kolmogorov showed in 1960's that the set of nonrandom numbers is recursively enumerable but not recursive, and this is a version of Godel's rst incompleteness theorem (cf. Odifreddi [8]). Chaitin also gave information-theoretic formulation of the rst incompleteness theorem in terms of Kolmogorov complexity. Relations between Kolmogorov complexity and the rst incompleteness theorem have been discussed in many places (cf. Li and Vitanyi [7]). Our purpose is to show that Kolmogorov complexity also leads to the second incompleteness theorem. While Godel's proof of the rst incompleteness theorem brings the second incompleteness theorem, Kolmogorov's proof dose not yield the second incompleteness theorem by the similar manner. Nevertheless, as noted in Odifreddi [8], Kolmogorov's proof can be seen as an application of Berry's paradox (the paradox of the de nability of the least integer not nameable in fewer than nineteen syllables), and Boolos [1] and Kikuchi [4] have extracted the rst and the second incompleteness theorems from this paradox. In this note, by applying the method y
This work was partially supported by Grant-in-Aid for Cooperative Research, The Ministry
of Education, Science and Culture No. 04302009, 06302014.
Typeset by
1
AMS
-TEX
used in Kikuchi [4] to a variant of Kolmogorov's theorem, we shall prove the second incompleteness theorem. In 1, we introduce basic theorems about provability predicates and the arithmetized completeness theorem. In 2, we de ne Kolmogorov complexity and brie y review the rst incompleteness theorem in terms of Kolmogorov complexity. The Kolmogorov complexity of a number is usually de ned to be the minimal length of a program under a certain programming method which outputs the number. In this note, we adopt the de nition of Kolmogorov complexity given by Odifreddi [8], which uses the index of partial recursive functions instead of the length of programs. Then, in 3, we prove the second incompleteness theorem. x
x
x
1. Preliminaries
Let A = +; ; 0; 1; < be the rst-order language of arithmetic. A formula in A is called 10 if contains only bounded quanti ers (i.e., quanti ers of the form ( x < t) or ( x < t) with a term t that does not contain x), and 61 if is a formula of the form x for some 10 formula . It is well known that a relation R Nk is recursively enumerable if and only if there is a 61 formula (x) such that m R if and only if N = (m ) for all m Nk . Peano arithmetic, denoted by PA, is the theory in A that consists of the basic axioms of arithmetic (i.e., the axioms of discretely ordered semirings with the least positive element 1) and the axiom schema of induction. Throughout this note, T means a recursively axiomatizable extension of PA. T is said to be !-consistent when the following condition holds: for any formula (x) in A, T x (x) if T (n) for all n N. Let ProvT (x; y) be a 61 formula that denotes the relation that x is the G odel number of a sequence of formulas which forms a T -proof of a formula with the G odel number y and de ne PrT (y ) to be the formula xProvT (x; y ). Then, Con(T ) is the formula PrT ( 0 = 1 ) which means T is consistent . The formula !-Con(T ) which means T is ! -consistent is de ned similarly. For any 61 sentence in A , (i) N = PrT ( ), (ii) N = Con(T ) (PrT ( ) ), (iii) N = ! -Con(T ) (PrT ( ) ). This theorem is provable in PA. That is, For any 61 sentence in A , (i) PA PrT ( ), (ii) PA Con(T ) (PrT ( ) ), (iii) PA !-Con(T ) (PrT ( ) ). Remark that PrT (x) satis es the modus ponens and the deduction theorem in PA. Hence, PA PrT ( ) Con(T + ) L
f
1
g
L
8
9
9
2
j
2
L
L
6` 9
2
9
:
d
e
Theorem 1.1. j
!
j
L
d
e
!
j
d:
!
e
d
! :
e
!
Theorem 1.2. `
`
`
!
L
d
e
!
d:
!
e
d
! :
e
` :
!
d
e
!
2
:
:
`
for any , since PA ` :PrT (d e) ! :PrT (d: ! 0 = 1e). Refer Smorynski [10] about these theorems and remarks. Now, we are going to introduce the arithmetized version of the completeness theorem. Let C be a set of new constants and LA = LA [ C . We say a formula (x) in LA de nes a model of T in a theory S if we can prove within S that the set f : is a sentence in LA that satisfy (de)g forms an elementary diagram of a model of T with a universe from C . Theorem 1.3. (The arithmetized completeness theorem).
There exists a formula TrT (x) in LA that de nes a model of T in P A + Con(T ). Let and be structures for LA. We say is an end-extension of and
M M M M write M e M if M M and M j= a < b for any a 2 M and b 2 M n M. Note that M j= implies M j= if M e M and is a 6 formula. We say a model M of T is a de nable end-extension of a model M of PA and write M d M if M e M and they satisfy M j= PrT (d e) ) M j= ; M j= TrT (d e) , M j= 0
0
0
0
0
0
0
0
1
0
0
0
0
0
for some formula TrT (x) in LA. From Theorem 1.3, we have the following corollary. Corollary 1.4. Let M be a model of PA. Then, M satis es Con(T ) if and only if
M has a de nable end-extension which is a model of T .
Corollary 1.4 has applications to model-theoretic proofs of the incompleteness theorems (cf. Kreisel [6], Smorynski [10], and Kikuchi [4]). See Hajek and Pudlak [2] and Kaye [3] for more information about models of arithmetic, and Smorynski [10] and Kikuchi and Tanaka [5] for the proofs of Theorem 1.3 and Corollary 1.4. 2. Kolmogorov complexity
Let f'ne(x)ge N be a canonical enumeration of n-ary recursive functions. (We omit the superscript n if there is no confusion.) We write '(x) # if '(x) is de ned at x and write '(x) " otherwise. Also, we write '(x) ' ' (x) if both '(x) and ' (x) are unde ned or they are de ned and '(x) = ' (x). Note that '(x) # can be expressed by a 61 formula since the graph of '(x) can be represented by a 61 formula and '(x) # if and only if y = '(x) for some y. The Kolmogorov complexity K (x) of x is de ned by K (x) = e('e (0) ' x) where e is the least e which satis es . Remark that K is an arithmetically de nable one-to-one function from N to N, and the relation K (x) y is represented by the 61 formula 9z y('z (0) ' x). The following lemma can be easily proved by the fact that K is one-to-one. 2
0
0
0
3
Lemma 2.1. For any a, there exists b a + 1 such that a + 1 K (b) We say a number x is random if x K (x). Lemma 2.1 shows that there exists a random number b such that a + 1 K (b) for any a, hence there exists in nitely many random numbers. The set of nonrandom numbers is recursively enumerable since it is de nable by a 61 formula, and Kolmogorov showed that it is not recursive. In fact, he proved that the set is simple, that is, it is recursively enumerable, and its complement is an in nite set which does not contain any in nite recursively enumerable subset (cf. Li and Vitanyi [7] and Odifreddi [8]). Since Godel's rst incompleteness theorem is derivable from the existence of such a set, Kolmogorov's theorem is a version of the rst incompleteness theorem. The following theorem is a variant of Kolmogorov's theorem. Theorem 2.2. (The rst incompleteness theorem).
e N such that N = Con(T ) x( PrT ( e < K (x) )), (ii) N = ! -Con(T ) x (e < K (x ) PrT ( K (x) e )). That is, we cannot prove e K (a) for any a N within consistent recursively There exists
(i)
j
2
! 8
j
:
d
e
! 8
! :
d
e
2
axiomatizable theories while there exists in nitary many such a's. We shall see a proof of a formalized version of this theorem in the following section.
3. The second incompleteness theorem For any 61 formula R(x; y ), let 01 (R) and 02 (R) be formulas in LA de ned by 01 (R) , 8x8y (R(x; y ) ! y < K (x)); 02 (R) , 8x8y 8z (R(x; y ) ^ z y ! R(x; z )):
Lemma 3.1.
Let
R(x; y ) be a 61
formula. Then there exits
e 2 N such that
T + 01 (R) ^ 02 (R) ` 8x8y (R(x; y ) ! y < e): Proof.
By the selection theorem, there is a recursive function f : N ! N such that
N = xR(x; b) j
9
f (b) #; f (b) # ) N j= R(f (b); b) )
for all b 2 N. This function f is de ned by f (y ) ' (w(R ((w)0 ; y; (w)1 ; . . . ; (w)n )))0 0
where R is the 10 formula such that 0
R(x; y ) , 9z1 1 1 1 9zn R (x; y; z1 ; . . . ; zn ) 0
4
and (w)i is the i-th component of w, hence we can show T T for all b 2 N. Hence
` 9xR(x; b) ! f(b) # ` 8x(x = f(b) ! R(x; b))
(1)
T + 01 (R) ` 8x(x = f(e) ! e < K(x)):
(2)
By applying the recursion theorem to f, we have e 2 N such that 'e (0) ' f(e): In order to show the existence of this e, only we have to do is applying the S-m-n theorem just for the recursive function f which has a concrete de nition given above. So e can be computed actually from the 61 formula R. In addition, by examining the proof of the S-m-n theorem, we can prove So we have By (2) and (3), Also, by (1), Hence
T
` 8x(x = 'e(0) $ x = f(e)):
T
` 8x(x = f(e) ! K(x) e):
(3)
T + 01 (R) ` f(e) " : T + 02 (R) ` 8x8y(R(x; y) ^ e y ! f(e) #):
T + 01 (R) ^ 02 (R) ` 8x8y(R(x; y) ! y < e):
3
Lemma 3.2. (i) T ` Con(T ) ! 01 (PrT (dy < K(x)e)), (ii) T ` 02 (PrT (dy < K(x)e)). Proof. Since y < K(x) is a negation of a 61 formula, (i) is a direct consequence of Theorem 1.2 (ii). It is also easy to show (ii), because T ` 8y 8z(z y ! PrT (dz y e)) by Theorem 1.2 (i) and PrT (x) satis es the modus ponens. 3 From these two lemmas, we have a formalized version of Theorem 2.2. Its proof also depends on Theorem 1.2. Theorem 3.3. (The formalized rst incompleteness theorem). There exists e 2 N such that (i) T ` Con(T ) ! 8x(:PrT (de < K(x)e)), (ii) T ` !-Con(T ) ! 8x(e < K(x) ! :PrT (dK(x) ee)). Remark that Lemma 2.1 is also provable in T , since the pigeonhole principle is derivable from T . 5
+ 1( + 1 ( )) Now, we are ready to prove the second incompleteness theorem.
Lemma 3.4. T ` 8x9y x
x
K y
.
Theorem 3.5. (G odel's second incompleteness theorem).
( ) Proof. Suppose that is consistent and any model of satis es Con( ). Since is consistent, there is a model M0 of . By Lemma 3.4 and the least number principle in , there is 0 + 1 such that M0 = ( 0 ) 0( ( ) ) Then, by Theorem 3.3 (i), M0 = PrT ( ( 0) ) Hence M0 = Con( + ( 0 ) ). So, by Corollary 1.4, there is a de nable end-extension M1 of M0 . Again, take the least element 1 + 1 such that M1 = ( 1). Since ( ) is a 61 formula, M1 = ( ) for any 0, and M1 = ( 0) because M1 is a model of + ( 0) . Therefore, M1 = 0( ( ) ) so 1 is strictly greater than 0 . Repeating this construction + 2 times, we have a sequence of models M0 d M1 d d Me+2 of and a corresponding strictly increasing sequence of numbers 0 1 e+2 . This contradicts the choice of i's. Therefore, if is consistent, there exists a model of which does not satisfy Con( ), hence we have the second incompleteness theorem by the completeness theorem. 3 Remark. Since our proof of the second incompleteness theorem is not formalizable in the system of primitive recursive arithmetic PRA (cf. Smorynski [10], Comments 6.3), it does not directly bring the formalized version of the second incompleteness theorem, PRA Con( ) PrT ( Con( ) ) However, our proof can be carried out within a subsystem of second-order arithmetic WKL0, since the completeness theorem is provable in WKL0 (cf. Simpson [9]) and Corollary 1.4 is provable in weaker subsystem RCA0 (cf. Kikuchi and Tanaka [5]). Thus we can also obtain a new proof of the formalized second incompleteness theorem, by using a theorem of H. Friedman that any 52 theorem of WKL0 is provable in PRA (cf. Simpson [9]). Acknow ledgement. The author of this paper thanks Prof. K. Tanaka and Mr. T. Yamaguchi for stimulating discussions. If T is consistent, Con T
is not derivable from T .
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References 1. Boolos, G., A new proof of the G odel incompleteness theorem, Notices Amer. Math. Soc. 36 (1989), 388{390. 2. H ajek, P. and Pudl ak, P., Metamathematics of First-Order Arithmetic, Springer, 1993. 3. Kaye, R., Models of Peano Arithmetic, Oxford, 1991. 4. Kikuchi, M., A note on Boolos' proof of the incompleteness theorem, Math. Logic Quart. 40 (1994), 528{532. 5. Kikuchi, M. and Tanaka, K., On formalization of model-theoretic proofs of G odel's theorems, Notre Dame J. Formal Logic 35 (1994), 403{412. 6. Kreisel, G., Notes on arithmetical models for consistent formulae of the predicate calculus, Fund. Math. 37 (1950), 265{285. 7. Li, M. and Vit anyi P.M.B., Kolmogorov complexity and its applications, Handbook of Theoretical Computer Science (J. van Leeuwen, ed.), Elsevier, 1990, pp. 187{254. 8. Odifreddi, P., Classical Recursion Theory, North-Holland, 1989. 9. Simpson, S.G., Subsystems of Second-Order Arithmetic, forthcoming. 10. Smory nski, C.A., The incompleteness theorems, Handbook of Math. Logic (J. Barwise, ed.), North-Holland, 1977, pp. 821{865.
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