The axiom of choice

A few days ago I came across the Clay Mathematics Institute’s website, and the one million dollar prizes they set for the people who will succeed in solving a few math problems. One of these problems is the famous P vs NP problem, which I came across during my studies of computer science. In the last few days I read some information about related issues – mostly mathematics and computer science issues. The main source I used was Wikipedia (in English), which is a very good source for reading information about anything.

Among the articles I read in Wikipedia are articles about the halting problem, the axiom of choice, P vs NP and other interesting issues. I decided to comment about these issues, and that’s the main reason for starting my own weblog.

The first subject I would like to comment about is the axiom of choice. The reason is because I believe this axiom is not true. Here is a short explanation why I think it is not true:

Suppose we have to select a random real number in a given range, say from 0 to 1. This can be achieved by various ways in real life, such as turning a roulette, throwing a dart, using computers random number generators etc. And suppose that each number in the range has an equal probability of being selected. Then I claim that this is already a paradox! And even if we limit ourselves only to rational numbers, it is still a paradox. It’s not possible to make such a selection. Only if we limit ourselves to rational numbers with a certain precision – that is, only if the set of possible numbers is finite, then we are able to make such a selection. Or, if we don’t require that each number will have an equal probability of being selected (this I will show later).

Why is it a paradox? Because the sum of all probabilities to select each number must be exactly 1. If there are infinite possible numbers, and all the probabilities are equal, then each probability must be exactly 0 (this is easy to prove). And if a probability is 0, then it cannot occur! Of course this still has to be proved, but I think it’s already intuitive. Things with probability 0 just don’t happen!

I want to tackle this issue from a different point of view. The issue is not selecting a number, but telling which number was selected or writing it down. Of course we can still claim that we can select a number even if we can’t tell it or write it down, but this is like claiming about anything else that we know it and can’t tell it, for example I can claim that I know whether P and NP are equal or not but I can’t tell you, or that I know whether any given algorithm halts or not but I can’t tell you, and so on. Therefore, I would assume that if we can’t tell what number we selected and we can’t write it down, then we can’t make the selection. This is true for all practical purposes.

Now, let’s consider the ways we write down numbers. There are a few ways to write numbers: The most common way is decimal (or binary) representation (either fixed point or scientific representation). We can represent any real number with binary representation (or at any integer base). The problem is: with some numbers, this representation is infinite. For rational numbers, it’s periodic, but for irrational numbers it isn’t. Therefore, if we wanted to write down a real number which is not rational in the binary form, it would take us an infinite number of digits. Which leads to the question if such numbers really exist. This is a philosophical question. But if we assume they exist, we are able to write them down in alternative ways (not binary representation) – for example: e, PI, and the square root of 2.

Lets consider all the ways to tell a number or write it down. We can decode each way in the binary form and save it to a binary file. Everything can be encoded into binary files: text, graphics, voice etc. But with n binary bits, it’s possible to write down only 2n different files. Therefore, not more than 2n different numbers can be represented with n bits. The number 2n grows exponentially, but it’s still finite. It’s not possible to represent an infinite number of different values with a finite number of bits.

How is this related to selecting a real number in a given range, or with the axiom of choice? Well, there is only a finite number of different numbers we can write down or tell with a finite number of bits (or in a finite time). Therefore, the number of numbers we can ever write down or tell is countable (I think it’s what we call definable numbers). So if a set is uncountable (for philosophical reasons, I don’t think any uncountable set really exists, but this is another story) – then it must contain numbers which we will never be able to tell! Or in other words, we can define a set of all the definable numbers (numbers we can tell) and subtract it from the set of real numbers, and the result will be a set of numbers none of which we will ever be able to tell. And therefore, we are not able to select an arbitrary number from that set.

Now, what about rational numbers? Can we randomly select a rational number in a given range? I say we can, but we can’t give all the numbers the same probability. We can give each number a positive probability (more than 0), but if we insist in giving each number the same probability – we will still not be able to tell which number we have selected. Or to be more accurate, the probability that we will be able to tell which number we have selected is 0. This is because for each number of bits n, there are only a finite numbers that can be represented with up to n bits, but an infinite numbers which require more than n bits.

On the other hand, if we don’t require all the numbers to have the same probability, then we can definitely select a number within the given range. This is trivial and very easy to prove. We can just select bits at random, and after selecting each bit select randomly whether we should stop or keep selecting more bits. We can show that each number has a positive probability to be selected. This can be shown with any countable set, not just with rational numbers.