## The Axiom Of Choice

Posted by on Thursday, October 13, 2016 Under: Mathematics

Mathematics is the pinnacle of logic and science. Most people assume that mathematics is based on solid foundations of logic, and that everything is well understood. Some sciences contain controversies and differing opinions, but surely mathematics is pure and definitive. At the very least the basics that are taught in school must be based indisputable.

And yet deep in the heart of mathematics lies a problem, known as the axiom of choice.

Suppose that you have a collection of boxes, each containing some (non-zero) number of objects. Is it possible to take one object from each of these boxes, and put them all into another box? Clearly this is true for a finite number of boxes, and it is also true for an infinite number in which we define a system of selecting the objects. But is it true for any infinite collection of boxes? Or in mathematical terms, is it true that in an infinite number of sets, it is always possible to form a new set which contains an element from each of the previous sets?

The simple fact is that we just don't know. As yet there is no way of proving that this is always possible, and so it must be assumed to be true or to be false. This is known as the axiom of choice.

At this point I am sure that most of you will be thinking that this is a trivial assumption. Clearly it must be true. But consider a simple example of the set of all subsets of the real numbers. It is an infinite collection, so we cannot simply choose elements by hand because it would take an infinite amount of time. We could try to define a system, such as always selecting the smallest number or the largest number in each set, except that sets such as {x | x > 0 and x<1} have no smallest and largest members. Suddenly it is not so obvious that the axiom of choice is true for this example.

However the problem is even worse than that, because whether we assume that the axiom is true or assume it is false, it creates some very odd mathematical results.

If the axiom of choice is true, then we can prove both the Hausdorff paradox and the Banach-Tarski paradox. I have written about both of these in previous articles, but in short the Banach-Tarski paradox proves that you can divide a ball into a finite number of pieces, apply a rigid transformation (ie you can rotate and move each piece, but you cannot change the size of each piece) and when they are re-assembled you will have two exact duplicates of the original ball. More disturbing, the Banach-Tarski paradox also proves that you can cut a pea into pieces and reassemble it as an exact, actual size replicate of the entire solar system!

Clearly something is wrong with the axiom of choice if it allows such nonsense to be proven!

And yet if it is not true, we lose most of what we know about mathematics. Without the axiom of choice, the union of countable sets may not be countable itself. In fact without the axiom of choice, it may not even be possible to define what we mean by a finite set or a countable set. There are also possibly odd geometric spaces that cannot have a coordinate system defined in them. It is possible that without the axiom of choice we might not even be able to prove that some parts of basic grade-school arithmetic is mathematically correct.

So there are the two choices facing mathematicians. If it is not true, then we lose a lot of important mathematical results. And also if it is not true, are intuitive understanding of set theory is completely wrong. But if the axiom of choice is true, then we get odd results like the Banach-Tarski paradox.

And yet because it is an axiom and cannot be proven, there is no clear answer. All we can do is keep studying, and hoping that one day there will be an answer.

And yet deep in the heart of mathematics lies a problem, known as the axiom of choice.

Suppose that you have a collection of boxes, each containing some (non-zero) number of objects. Is it possible to take one object from each of these boxes, and put them all into another box? Clearly this is true for a finite number of boxes, and it is also true for an infinite number in which we define a system of selecting the objects. But is it true for any infinite collection of boxes? Or in mathematical terms, is it true that in an infinite number of sets, it is always possible to form a new set which contains an element from each of the previous sets?

The simple fact is that we just don't know. As yet there is no way of proving that this is always possible, and so it must be assumed to be true or to be false. This is known as the axiom of choice.

At this point I am sure that most of you will be thinking that this is a trivial assumption. Clearly it must be true. But consider a simple example of the set of all subsets of the real numbers. It is an infinite collection, so we cannot simply choose elements by hand because it would take an infinite amount of time. We could try to define a system, such as always selecting the smallest number or the largest number in each set, except that sets such as {x | x > 0 and x<1} have no smallest and largest members. Suddenly it is not so obvious that the axiom of choice is true for this example.

However the problem is even worse than that, because whether we assume that the axiom is true or assume it is false, it creates some very odd mathematical results.

If the axiom of choice is true, then we can prove both the Hausdorff paradox and the Banach-Tarski paradox. I have written about both of these in previous articles, but in short the Banach-Tarski paradox proves that you can divide a ball into a finite number of pieces, apply a rigid transformation (ie you can rotate and move each piece, but you cannot change the size of each piece) and when they are re-assembled you will have two exact duplicates of the original ball. More disturbing, the Banach-Tarski paradox also proves that you can cut a pea into pieces and reassemble it as an exact, actual size replicate of the entire solar system!

Clearly something is wrong with the axiom of choice if it allows such nonsense to be proven!

And yet if it is not true, we lose most of what we know about mathematics. Without the axiom of choice, the union of countable sets may not be countable itself. In fact without the axiom of choice, it may not even be possible to define what we mean by a finite set or a countable set. There are also possibly odd geometric spaces that cannot have a coordinate system defined in them. It is possible that without the axiom of choice we might not even be able to prove that some parts of basic grade-school arithmetic is mathematically correct.

So there are the two choices facing mathematicians. If it is not true, then we lose a lot of important mathematical results. And also if it is not true, are intuitive understanding of set theory is completely wrong. But if the axiom of choice is true, then we get odd results like the Banach-Tarski paradox.

And yet because it is an axiom and cannot be proven, there is no clear answer. All we can do is keep studying, and hoping that one day there will be an answer.

In : Mathematics