This note deepens a problem proposed and discussed by Kadane and O'Hagan (JASA, 1995). Kadane and O'Hagan discuss the existence of a uniform probability on the set of natural numbers (they provide a su_cient and necessary condition for the existence of such a uniform probability). I question the practical relevance of their solution. I show that a purely _nitely additive measure on the set of natural numbers cannot be constructed (its existence needs some form of the Axiom of Choice, the prototype of a nonconstructive axiom).

Lauwers, Luc

English

Research Papers in Economics

The uniform distributions puzzle
Luc Lauwers
Center for Economic Studies, K.U.Leuven
Naamsestraat 69, B-3000 Leuven, BELGIUM
Luc.Lauwers@econ.kuleuven.be
First version: July 17, 2007. This version: March 31, 2009
This note deepens a problem proposed and discussed by Kadane and O'Hagan
(JASA, 1995). Kadane and O'Hagan discuss the existence of a uniform prob-
ability on the set of natural numbers (they provide a sucient and necessary
condition for the existence of such a uniform probability). I question the prac-
tical relevance of their solution. I show that a purely nitely additive measure
on the set of natural numbers cannot be constructed (its existence needs some
form of the Axiom of Choice, the prototype of a nonconstructive axiom).
KEY WORDS: Finitely additive probabilities; Charges; Axiom of choice;
Constructivism; Natural numbers.
11 Introduction
Professors Joseph B. Kadane and Anthony O'Hagan (1995) formulate the uniform distri-
butions puzzle as follows:
Suppose that a computer scientist asks you for help with nding the probability
that a random integer has some property, ... She says that by \random" she
means something akin to a uniform distribution, in the sense that every integer,
no matter how large, will have the same probability. How can such a question
be answered in terms of modern probability theory?
This problem of a suitable denition of \pick a natural number at random" has caused
considerable diculty at the foundational level. Kadane and O'Hagan use relative frequen-
cies and residue classes to make precise notions of uniformity, and they provide a necessary
and sucient condition for the \existence" of a nitely additive probability dened on all
sets of natural numbers that agrees with these interpretations.
For a study of nitely additive measures, see Bhaskara Rao and Bhaskara Rao (1983).
They indicate that a lot of the standard theory of measure and integration can be carried
through assuming only nite additivity. For discussions on whether countable additivity
should be dispensed with, see, for example, Savage (1954) and Dubins and Savage (1976).
Wakker (1993), Stinchcombe (1997) and Kadane, Schervish, and Seidenfeld (1986) discuss
some implications of the failure of countable additivity.
More than fourty years ago, Deal et al (1963) discuss the existence of a nitely additive
probability on the set of natural numbers which attaches zero probability to all nite sets.
They use a free ultralter1 to dene a nitely additive 0-1-probability measure (sets that
belong to the ultralter obtain probability one, sets that do not belong to the ultralter
obtain probability zero). A result of Sierpi nski (1938), however, implies that each 0-1-
probability (attaching zero probability to nite sets) generates a non-measurable set of
real numbers. Since the existence of a non-measurable set hinges on the Axiom of Choice
(Solovay, 1970), such a 0-1-probability cannot be constructed.
In view of this observation and|similar to Kadane and O'Hagen|having the computer
scientist in mind, I want to deepen the uniform distributions puzzle: Is it possible to
\construct" a nitely additive measure on the set of natural numbers which attaches zero
probability to all nite sets? This note provides the answer: No!
I show that the uniform distributions puzzle cannot be solved in a constructive way.
The result of Sierpi nski extends to arbitrary nitely additive probabilities which attach
zero probability to all nite sets. More precisely and in terms of set theory: the existence
of such a measure is independent of the Zermelo-Fraenkel axioms of set theory together
with the Axiom of Dependent Choice. Although nitely additive probabilities provide the
right approach to the uniform distributions puzzle, each computer scientist is unable to
1A free ultralter on the set N of natural numbers is a collection U of subsets of N such that (i) N 2 U
and ? = 2 U, (ii) if A  B and A 2 U, then B 2 U, (iii) if A;B 2 U, then A \ B 2 U, and (iv) for each
A  N, either A or N   A belongs to U. The existence of a free ultralter follows from Zorn's Lemma.
2\implement" such a uniform distribution. Results on the \existence" of a nitely additive
probability do not provide any help.
The next section touches the notion of constructivism and recalls the Axiom of Choice
and the Axiom of Dependent choice. Section 3 states and proves the main result.
2 Constructivism
In constructive mathematics, a problem is considered as solved only if its solution can be
explicitly produced. Let me provide a typical example.
Claim. There exist two irrational real numbers a and b such that ab is rational.
Proof. The real number c =
p
2
p
2 either is rational, and then we put a = b =
p
2; or is
not rational, and then we put a = c and b =
p
2.
Classical mathematics accepts this proof. On the other hand, this proof uses the law of
the excluded middle which states the truth of `P or not-P' for each proposition P (i.c.
the real number c is rational) and does not provide an explicit way to indicate which
pair of real numbers satises the claim. Therefore, within constructivism this proof is
rejected. In constructivist logic the classical law of the excluded middle is not assumed.
Of course, constructive mathematics also rejects principles that are stronger than the law
of the excluded middle. The Axiom of Choice (AC) is such a principle.
AC postulates for each nonempty family D of nonempty sets the existence of a function
f such that f(S) 2 S for each set S in the family D. The function f is referred to as a
choice function. AC does not provide an explicit way to construct such a choice function
and provoked considerable criticism in the aftermath of Zermelo's formulation in 1904.
AC is (i) consistent: AC can be added to the Zermelo-Fraenkel axioms of set theory
(ZF) without yielding a contradiction; and (ii) independent: AC is not a theorem of ZF
(Fraenkel et al, 1973, Section II.4.2). Among the applications of AC, we mention Zorn's
Lemma, the theorem of Hahn-Banach, and the existence of free ultralters. AC also implies
a number of paradoxes such as the decomposition of a sphere into a sphere of smaller size,
and the existence of a non-measurable set of real numbers.
Constructive mathematics rejects AC. On the other hand, the Axiom of Dependent
Choice (DC) is generally accepted by constructivists (Beeson, 1988, p. 42). Let S be a
nonempty set and let R be a binary relation in S such that for each a in S there is a b in
S with (a;b) 2 R. Then, DC postulates the existence of a sequence (a1;a2;:::;an;:::) of
elements in S such that (ak;ak+1) 2 R for each k = 1;2;:::.
The nonconstructive object used in this note is known as a non-Ramsey set. Let I be
an innite set and let n be a positive integer. Let [I]n collect all the subsets of I with
exactly n elements. Ramsey (1928) shows that for each subset S of [I]n, there exists an
innite set J  I such that either [J]n  S or [J]n \ S = ?. When n is replaced by
countable innity, then Ramsey's theorem fails. There exists a subset S of [I]1 such that
for each innite subset J of I the class [J]1 intersects S and its complement [I]1   S as
3well. Such a set S is said to be non-Ramsey. Mathias (1977) showed that the existence of
non-Ramsey sets does not follow from ZF (without AC).2
3 Finitely additive probabilities
Let N be the set of natural numbers. Let F be the eld of all subsets of N. A nitely
additive probability is a map  : F ! R+ that satises (N) = 1 and the condition


A1 [ A2 [ ::: [ An

= (A1) + (A2) +  + (An);
where the sets A1;A2;:::;An are disjoint, for all nite n. We now state the main result of
this note.
Theorem. Let  be a nitely additive probability on N that attaches zero probability to
each natural number. Then,  generates a non-Ramsey set.
Proof. We start with some additional notation. For two natural numbers i > j, let [i;j[
denote the set fi;i + 1;:::;j   1g. Furthermore, to each innite set A  N we connect
a set, denoted by A0, as follows. Let A = fn0;n1;:::;nk;:::g with nk < nk+1 for each k,
then A0 = [n0;n1[ [ [n2;n3[ [ ::: [ [n2k;n2k+1[ [ ::::
Now, let  satisfy the requirements listed in the theorem. We show that
S = fA  Nj(A0) > 0:5g
is a non-Ramsey set. It is sucient to show that each innite set A = fn0;n1;:::;nk;:::g
includes an innite subset B such either A or B belongs to S (the `either-or' being exclu-
sive). We distinguish three cases.
Case 1. A = 2 S, in particular (A0) < 0:5. Let B = A fn0g. Then, [0;n0[[A0[B0 = N.
Since ([0;n0[) = 0, (A0) < 0:5, and (N) = 1; we obtain that (B0) > 0:5. Therefore,
B  A and B 2 S.
Case 2. A = 2 S, in particular (A0) = 0:5. Let B = fn0;n3;n4;n7;:::;n4k;n4k+3;:::g and
let B0 = fn0;n1;n2;n5;n6;n9;:::;n4k+2;n4k+5;:::g. Then, A0 = B0 \ B0
0. Hence, we have
(B0)  0:5 and (B0
0)  0:5. Furthermore,
[0;n0[ [ (B0 B
0
0) [ A0 = N:
Conclude that the symmetric dierence B0 B0
0 has a measure equal to 0.5. Hence, at
least one of the sets B0 or B0
0 has a measure strictly larger than 0.5. Select the subset B
or B0 of A for which the corresponding set B0 or B0
0 has the highest measure. The selected
subset of A belongs to S.
2More precisely, Solovay (1970) proposed a model in which ZF and DC are true and in which AC fails.
Mathias showed that in this Solovay-model a non-Ramsey set does not exist. Hence, the existence of a
non-Ramsey set is independent of ZF + DC.
4Case 3. A 2 S. Similar to Case 1, we put B = A   fn0g. Conclude that B  A and that
B = 2 S. 2
The next summary closes the paper. If each natural number has the same probability,
then this probability is zero and the resulting probability distribution  is nitely additive
and not countably additive (observe the inequality ([1
0 fkg) >
P1
0 (fkg)). Whether
one uses the limiting relative frequency, the residue sets, or translation invariance as an
interpretation of uniformity (Kadane and O'Hagan, 1995), the above Theorem applies:
uniformity of the natural numbers cannot be constructed.
References
Beeson, M.J. (1988), Foundations of constructive mathematics, Ergebnisse der Mathe-
matik und ihrer Grenzgebiete 3.6. Berlin: Springer-Verlag.
Bhaskara Rao, K.P.S. and Bhaskara Rao, M. (1983), Theory of charges, a study of nitely
additive measures, London: Academic Press.
Deal, R.B., Waterhouse, W.C., Alder R.L., Konheim A.G., and Blau J.G. (1963), \Prob-
ability measure for sets of positive integers," The American Mathematical Monthly
(Advanced Problems and Solutions), 70(2), 218-219.
Dubins, L.E. and Savage, L.J. (1976), Inequalities for stochastic processes, how to gamble
if you must, New York: Dover Publications.
Fraenkel, A.A., Bar-Hillel, Y., and Levy, A. (1973), Foundations of set theory, Amster-
dam: North-Holland.
Kadane, J.B. and O'Hagan, A. (1995), \Using nitely additive probability: uniform dis-
tributions on the natural numbers," Journal of the American Statistical Association,
90, 626-631.
Kadane, J.B., Schervish, M., and Seidenfeld, T. (1986), \Statistical implications of nitely
additive probability," p. 59-76 in Bayesian Inference and Decision Techniques: Essays
in Honor of Bruno de Finetti, edited by P.K. Goel and A. Zellner (eds.), Amsterdam:
North Holland.
Mathias, A.R.D. (1977), \Happy families," Annals of Pure and Applied Logic, 12, 59-111.
Ramsey, F.P. (1928), \On a problem of formal logic," Proceedings of the London Math-
ematical Society, Series 2. 30.4, 338-384.
Savage, L.J. (1954), The foundations of statistics, New York: Dover Publications, 2nd
edn, 1972.
5Sierpi nski, W. (1938), \Fonctions additives non compl etement additives et fonctions non
mesurables," Fundamenta Mathematicae, 30, 96-99.
Solovay, R. (1970), \A model of set theory where every set of reals is Lebesgue measur-
able," Annals of Mathematics, 92, 1-56.
Stinchcombe, M.B. (1997), \Countably Additive Subjective Probabilities," Review of
Economic Studies, 64(1), 125-146.
Wakker, P. (1993), \Savage's axioms usually imply violation of strict stochastic domi-
nance," Review of Economic Studies, 60, 487-493.
6

A model of set theory where every set of reals is Lebesgue measurable,&quot;

Countably Additive Subjective Probabilities,&quot;

Fonctions additives non compl etement additives et fonctions non mesurables,&quot;

Foundations of constructive mathematics,

Foundations of set theory,

Happy families,&quot;

Inequalities for stochastic processes, how to gamble if you must,

On a problem of formal logic,&quot;

Probability measure for sets of positive integers,&quot;

Savage's axioms usually imply violation of strict stochastic dominance,&quot;

Statistical implications of  additive probability,&quot;

The foundations of statistics,

Theory of charges, a study of  additive measures,

Using  additive probability: uniform distributions on the natural numbers,&quot;

The uniform distributions puzzle.

https://lirias.kuleuven.be/bitstream/123456789/223651/1/DPS0906.pdf

The uniform distributions puzzle.

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics