Drawing from a box that contains all the Natural numbers

This question is basically about the fact that there exists no uniform distribution over the set of all Natural numbers.

Imagine a situation in which we would draw from an urn that contains one ball for each positive integer. The shape and mass etc. of the balls are all identical. What would be the limiting relative frequency of the event "even number"?

Or does this question even make sense?

• Is this helpful? mathoverflow.net/questions/47134/… – present Aug 9 '18 at 21:48
• The op's question is somewhat different than @present 's (which I believe the op is acknowledging in his first sentence). For both questions, the answer exists for any finite subset N_n={1,2,...,n}. But in the op's case, I'm not seeing why the answer for the limit n-->oo wouldn't also exist. – John Forkosh Aug 10 '18 at 6:23

To make sense to OP's question, one should define what are the "events" in the probability theory one is considering. In other words, which subsets of the natural numbers are such that finding a number belonging to that set is relevant for our measurements/considerations.

Mathematically speaking, this amounts to defining a σ-algebra in the set of Natural numbers (and every element of the σ-algebra is a subset of numbers representing a possible event; in other words, it is a measurable set of numbers). The natural numbers endowed with a σ-algebra become a measurable space. In a measurable space, one can define a probability distribution, i.e. a function that assigns to each event a given probability. The probability distribution should satisfy, as well as the σ-algebra, some mathematical properties that make the probability theory sensible (in agreement with experience).

Since the natural numbers are a discrete space, the possible σ-algebras on it are rather easy to describe. Take a partition of the naturals in mutually disjoint sets. Then the collection of all the unions of an arbitrary number of elements of the partition is a σ-algebra on the naturals, and every σ-algebra on the naturals is of such form.

The simplest possibility is to take as a disjoint partition of the natural numbers each number by itself. In this way, every set of natural numbers (and in particular every number) is a measurable event. However in this case it is not possible to define a uniform probability distribution on the associated measurable space, since a probability must be σ-additive: given any countable collection of mutually disjoint measurable sets, the probability of their union should equal the sum of the probabilities of each measurable disjoint set. Hence if one takes the disjoint measurable sets to be the natural numbers (possible by our definition of σ-algebra), and the probability of picking any number to be equal and nonzero, the sum of such probabilities is never finite (and thus the probability of picking a natural number would not be one, but infinity!).

On the other hand, if one is only interested in having as events the odd numbers and the even numbers, then one could choose a different σ-algebra. In this case, the easiest choice is to partition the naturals exactly into even and odd numbers. Then the corresponding σ-algebra, i.e. the measurable events would only be the empty set, the set of odd numbers, the set of even numbers, and the set of all natural numbers. In other words, we are saying that for us the meaningful events are that a number is natural (and we want this to have probability one), that a number is even, and that a number is odd (the empty set is the event when we are picking no number at all). For the corresponding measurable space, there is a very simple uniform probability distribution: we set the probability of picking no number to be zero, the probability of the picking an odd number to be one half, and the probability of picking an even number to be one half (from which it follows that the probability of picking a natural number is one).

In this latter probability space (that in my opinion describes fairly accurately the thought experiment described by the OP, since in such experiment the distribution should be uniform, but the only relevant measurable events are being either even or odd), the limiting relative frequency of the event "even number" (to use OP's terminology) is indeed one-half.

Since the question is about the limiting frequency, the sample space is way more complex than what the answers so far suggest. Your "experiment" consists of infinitely many draws from the urn, with no replacement, but it's not clear whether the resulting sequence of numbers must contain all natural numbers. Let's say it does (I'm not going to give you a solution so it doesn't really matter). Then your sample space is the set of all permutations of N, and the first step is to define a probability measure on this space (a non-trivial thing to do). Once that is done, define a sequence {X_n} n=1,2,... of random variables on this space by the rule X_n(p) = p(n) mod 2 for each permutation p. In words, X_n(p) is one if p(n) is even, zero otherwise. Now define another random variable Z as the limit of (X_n / n). Your question is really about the distribution of Z. In particular, is it possible to define the probability measure so that P(Z = 1/2) = 1?. This clearly belongs to mathoverflow.

I think the question makes some kind of sense, or at the very least sounds like it does, but I think it is an example of an uncomputable function (https://en.m.wikipedia.org/wiki/Computable_function). It's essentially the same problem that you run into with the many world's interpretation of quantum mechanics. Finite probability has no meaning in the context of infinitely uncomputable functions.

What means "all natural numbers"? Does it mean only those natural numbers which have the characteristic property of every natural number, i.e., to be followed by infinitely many natural numbers? Then you don't get all of them in your box because always infinitely many are missing. Or do you take all natural numbers with no exception? Then you include some which are not followed by infinitely many and hence are not natural numbers because they are lacking the characteristic property of every natural number.

Modern set theorists try to save |N by storing all natural numbers, with the omission of their natural order, in a big bag. The order of the natural numbers however is their most important feature. So set theorist must not use their set for enumerating and mathematical induction. To make a long story short: |N does not exist because its elements cannot be in a set together. Therefore your box cannot contain all natural numbers.

• I cannot really follow your thought argument. I aso study maths and we deal with the set of all the natural numbers all the time. – Sebastian Aug 11 '18 at 11:53
• The set of all natural numbers is well-defined in a theory of first order logic with equality and the "belong to" symbol, together with the ZF axioms of set theory. The ordering of natural numbers is a well-ordering that is encoded in the first infinite ordinal number. (For some properties of well-ordering, but not for defining the ordering of the naturals, some form of axiom of choice may be needed) – yuggib Aug 11 '18 at 12:46
• @yuggib The set |N may be well-defined, but this definition is nonsense. You cannot use a natural number outside of the first percent. Proof: Try it, choose any larger n. Multiply it by 100. It is a natural number too. And it belongs to the first percent as you can prove by multiplying it by 100, remaining a natural number. Instead of 100 you may choose every larger number you like (from the first percent of |N). – Wilhelm Aug 18 '18 at 12:31
• @Wilhelm It may be nonsense to you, and essentially you seem to not accept/like the axiom of infinity. However you should clarify that this is your personal opinion, and not the commonly accepted mathematical praxis. On a side note, the axiom of infinity seems quite in agreement with experience: if on one hand we cannot experience infinity directly, there are many observations that support the existence of infinite sets. For example, dividing one by three using elementary school methods shows already a set with an infinite number of threes. – yuggib Aug 18 '18 at 17:28
• @yuggib Do you deny the possibility that the accepted mathematics is refutable? If not, why do you say I did not like the axiom of infinity? Can't you accept the proven fact that every natural number that can be used or quantified over belongs to the first percent? Is this argument really so hard to understand? – Wilhelm Aug 19 '18 at 12:24