[...]... computable functions CHAPTER 8 INCOMPLETENESS 212 (a) P 2;f (n) < 1 ) all n-bit theories settle n + f (n) + O(1) cases (b) P 2;f (n) = 1 & f (n) f (n + 1) ) for in nitely many n, there is an n-bit theory that settles n + f (n) cases (c) H (theory) < n ) it settles n + O(1) cases (d) n-bit theory ) it settles n + H (n) + O(1) cases (e) g unbounded ) for in nitely many n, all n-bit theories settle n... Hence there are in nitely many N -bit theories that yield (the rst) N + log N + f (log N ) bits of Proof Take f (n) = n + f 0(n) in Theorem B Q.E.D Theorem AB First a piece of notation By log x we mean the integer part of the base-two logarithm of x I.e., if 2n x < 2n+1 , then log x = n CHAPTER 8 INCOMPLETENESS 2 08 (a) There is a c with the property that no n-bit theory ever yields more than n +... #fs : H (s) < ng2;n;k = 2;k #fs : H (s) < ng2;n 2;k : n n Thus if even one theory with H < n yields n + k bits of , for any n, we get a cover for of measure 2;k This can only be true for nitely many values of k, or would not be Martin-Lof random Q.E.D Corollary C No n-bit theory ever yields more than n + H (n) + c bits of Proof 8. 4 RANDOM REALS: H(AXIOMS) 211 This follows immediately from Theorem C... the information content of knowing the rst n bits of is n ; c (2) Now we show that the information content of knowing any n bits of (their positions and 0/1 values) is n ; c CHAPTER 8 INCOMPLETENESS 210 Lemma C X n #fs : H (s) < ng2;n 1: Proof 1 s 2;H (s) X #fs : H (s) = ng2;n = #fs : H (s) = ng2;n 2;k n n k 1 XX X = #fs : H (s) = ng2;n;k = #fs : H (s) < ng2;n : = Q.E.D X X X n k 1 n Theorem C If a theory. .. exponential diophantine equation L(n x1 : : : xm) = R(n x1 : : : xm) (8. 4) which has only nitely many solutions x1 : : : xm if the nth bit of is a 0, and which has in nitely many solutions x1 : : : xm if the nth bit of is a 1 Let us say that a formal theory \settles k cases" if it enables one to prove that the number of solutions of (8. 4) is nite or that it is in nite for k speci c values (possibly scattered)... ciently large Q.E.D Corollary C2 Let g(n) be computable and unbounded For in nitely many n, no n-bit theory yields more than n + g(n) + c bits of Proof This is an immediate consequence of Corollary C and Lemma C2 Q.E.D Note In appraising Corollaries C and C2, the trivial formal systems in which there is always an n-bit axiom that yields the rst n bits of should be kept in mind Also, compare Corollaries... And X 1 n log n(log log n)2 8. 4 RANDOM REALS: H(AXIOMS) behaves the same as X X 1 1 2n 2n n(log n)2 = n(log n)2 which converges On the other hand, behaves the same as which diverges behaves the same as which diverges And behaves the same as 209 X X1 n X n1 X 2 2n = 1 X 1 n log n X n 1 X1 2 2n n = n X 1 n log n log log n X 1 2n 2nn 1 n = n log n log which diverges Q.E.D 8. 4 Incompleteness Theorems for.. .8. 3 RANDOM REALS: jAXIOMSj 207 Recall the Cauchy condensation test Hardy (1952)]: if (n) is a nonincreasing function P n, then the series P (n) is convergent or of divergent according as 2n (2n ) is convergent... CHAPTER 8 INCOMPLETENESS 2 08 (a) There is a c with the property that no n-bit theory ever yields more than n + log n + log log n + 2 log log log n + c (scattered) bits of (b) There are in nitely many n-bit theories that yield (the rst) n + log n + log log n + log log log n bits of Proof Using the Cauchy condensation test, we shall show below that 1 (a) P n log n(log log n)2 < 1, 1 = 1 (b) P n log n... settles n + O(1) cases (d) n-bit theory ) it settles n + H (n) + O(1) cases (e) g unbounded ) for in nitely many n, all n-bit theories settle n + g(n) + O(1) cases Proof The theorem combines Theorem R8, Corollaries A and B, Theorem C, and Corollaries C and C2 Q.E.D .