Correspondence, 1961-1970

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Correspondence, 1961-1970
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Mathematics -- History -- 20th century

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DEPARTMENT OF MATHEMATICS
CORNELL UNIVERSITY
ITHACA, NEW YORK


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DEPARTMENT OF MATHEMATICS
CORNELL UNIVERSITY
ITHACA, NEW YORK

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10/5/1963
Dear Orey: .

Thank you .most kindly for your letter and inclosures.
Will you be so knid- once tmoore to look 6-ver.. 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 W-M,
-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- AAsr-X


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 Martin-Lof




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 f-i: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 Gauthier-Vil r ?
1939 Traite du calcul des nmbabilitas et dU sea
applications Tome IV Pascicule TTI
Paris Gauthier-Vi liars

1951 La definition en mathk4matiques
Actualites Sci Indust 1137 89-99

Kolaogorov A N
1965 Tri podhoda k opredel"'.'.u ponjatija "koli- .e :


Problem peredaici informacii 1 3i-11




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