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DEPARTMENT OF MATHEMATICS CORNELL UNIVERSITY ITHACA, NEW YORK c o"v..,,? 6 0ot  vec\ 7c' et  ck~&~e~ ts T~L l"^ i'c B 0 i i V? 5 Co. i^^1 > , ., e l~Sc e c l ic. e TD7 Ik~ C, Le^ J, c V7 tiA) %tLl4V(S C VC nko ce is IQ CL 9ve 5 i /W We a 7 \Awec V L~) ~ .1; \ '\L4~ Vs ~s 3;x c A ) }_ ' ^  <9? V' ~r1I e ~* ) .d~ Ale~e,'., . tI. ~ hA4~ e c e S. ' V~ urj YO [ho ~r1~ e4 O0 V,  IL r c , k K W o ~~1 \ I? r ~e r 3 Je~s DEPARTMENT OF MATHEMATICS CORNELL UNIVERSITY ITHACA, NEW YORK >^ ~~ iA.o,~? .S te (~c ve A As e Ss I e, .r4~vore47t P# i,44S V c Ie &~~~~ 7 .~t, D t C r ^ IJ: ie S t ;A l 5 >0 3I t k '. ?/\ i~ ~' ,~ '4 ) \IAA. yTe s 1e ,e c e A" c pro yac , tk V\ic\A lp v S1e ., fA  xo se? Or v o)" L,'$) Th , A^ c por A1 SIt 4 Ib SC f E cI  '9 )~ &I. Pc ) Ij )o If P0 'b c 1.e 1 ^1 \ SJ ) I a P ( E) iFo 1 fr V t ",i 0  ~\L i < o v ;1 >es k FWe exs sV e 4e I C S elue Fo s& e k. Co0~ l47 ^ R e LS e o vds Pg e/Z~ i S.c c I o A p 0 4 A. E& 0 Or Sv rK ( I vifA V~o' tV^ cvJ. 0, ic Fo \k L e pt e e5 'I ^ ys Nt \eA .v <,A\ \sh V> < v ^ Vel.^* 3 i? \ s \.k; co *\ 7 e Vko^e 'i^ l A ,t, q  lA '\ e a' cs Ah' v,k;c k h YOu Ccf 01 c<  E4lcC4 L45~ lL e e^ < ( L6 kae a l 9 J^ ve us es Hs 0' p. 2 ol : 0 FkL I LeL '3 C) . s, So yV  LI \. x e r  re,,a, rg~e~ Hl  10/5/1963 Dear Orey: . Thank you .most kindly for your letter and inclosures. Will you be so knid once tmoore to look 6ver.. my scribbling to see if you cannot specify where I did not follow. In par, cular: (1) Please indicate the mistake ,.in Th. A.5 which was not spelled odat' (2) Is the mltake in the last five lines on p. 7/4 what I thought 'it was, namely that : 'Vs>,o; 7 ) .<00 If so .is my indicated correction sufficient? As tor Th. 4.1 that is a terrible goofs but it is easy to correct. Re Lemma on p. 28, your indication may be wrong, but the following Lemma sugfices: Lemma.. If a is optional and Me] and a = a: on M., =+ on WM, M then. a 'is optional. Doob and I have a paper in which these questions will be treated, some of which is also in.Meyer's book. I.will think about your conjecture but I admit .1 have not really ""gone through Williams's papers. The MS I sent you recently is hopefully less erroneous as.I have .given a seminar on some parts with Mrs iMoy in it who 'is very. careful.. But I would really appreciate it this time.if_ you would send me corrections before it is printed. Send any on the way, if you ever lo.k at it I will refund postage. Sincerely, ' matik och matematisk statistik'3 i l 11 85 TO 1H 's VLM A,".f. w L&  'Pee M AAsrX TJANSTE STOCKHOLMS UNIVERSITET Institute f6r forsakringsmate matik och matematisk statistik Box 6701 113 85 STOCKHOLM Vt s'frAi^L ON BOREL'S HEIGHT OP A REAL NUMBER Per MartinLof Borel 1903, 1914, 1939 and 1951 repeatedly discussed what he called the height of a real number. It is defined to be an increasing function of the number of operations required to define the number starting from unity. Exactly how this is.made precise does not matter for there are only two properties of the notion that are being used. Firstly, every computable number should. have a finite height, and, secondly, there should be not more than finitely many numbers whose height is less than a given value. Let c?(h) be the minimum distance between two different numbers of height less than h. The second property of the height ensures that o is strictly positive. Suppose now that we are given two numbers a and b of height less than h and that we know P(h). Then, as soon as la bi < <(h) we can conclude that a = b, that is, to determine whether or not the numbers a and b are equal it suffices to compute them approximately with an accuracy that is.determined by the function 4. For example, choose 1 1 1 a = 1 + + and b ( eX2dx)2 0 and imagine that we did not know yet that these two numbers are equal. It seems in accordance with Borel's intuitive idea to consider the complexity K(a) in the sense of Kolmogorov 1965 as a measure of the height of the com putable real number a. Clearly, K(a) is finite for every computable number a, and there are less than 2h numbers a with K(a) < h, so that the two conditions that Borel imposes on the height are satisfied. Note that K(a) depends on the definition (GOdel number) of a and hence that two computable numbers which are extensionally equal, that is, define the same real number, need not have the same complexity. On the other hand, if two computable numbers are definitionally equal, that is, equal considered as syntactical objects, then their complexities must be the same, of course. Let f(h) be the minimum of [a bi for all computable real numbers a and b such that a A b and max(K(a),K(b)) < h. We shall show that the function 4 tends to zero so rapidly that it cannot be minorized by any strictly positive computable function. Theorem. Let 4 be a computable sequence of o putable real numbers such that 0 o '(h) 4(h) for all h. Then we can find k so big[ hat (h) 0 for all h k. Proof. By the fundamental theorem of Kolmogorov 1965 there is a constant a such that K(tV(h)) logh + 0 for all h, the logarithm being taken to the base 2. Let k be so big that log h + e,< h when h 0 k and K(O) < k where 0 denotes (some Gbdel number of) the computable real number zero. Since tr(h) o0 < (h) for all h and max(K(14(h)),K(O)) < h when h~ k, we can conclude that 1f(h) = 0 and hence (h),= 0 when h & k as desired. More generally, suppose we define the height H(a) of a computable real number a in such a way that we can effectively determine a bound on H(a) when (the GSdel number of) a is given to us, and let 4(h) be the minimum of  a bi for all computable numbers a and b such that a A b and max(H(a),H(b)) < h. Then, if is a computable sequence of computable real numbers such that 0 < (h) 4< 4(h) for all h, it is impossible that ~1(h) > 0 for arbitrarily large h, because in that case we would obtain an effective method of deciding whether a = b or a A b for arbitrary computable numbers a and b which is impossible. The problem posed by Borel, that of. determining effectively the function < is thus unsolvable as soon as the height is subjected to the very general condition of the previous paragraph. 1903 Sur I'3appr vti'. on, les uns par les *u'.3.?e, des nombref fi:nt un un &szle diniabrable C R Acad Soi Pari: 136 297 1914 Leqons sur la t ,o;ie des fonetions 2 edition Par is GauthierVil r ? 1939 Traite du calcul des nmbabilitas et dU sea applications Tome IV Pascicule TTI Paris GauthierVi liars 1951 La definition en mathk4matiques Actualites Sci Indust 1137 8999 Kolaogorov A N 1965 Tri podhoda k opredel"'.'.u ponjatija "koli .e : Problem peredaici informacii 1 3i11 

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