# The Axiom of Choice - Part 1

One of the most surprising and astonishing axioms in mathematics, simply by the means of its assertion is the axiom of choice. Its assertion is extremely elementary, and yet this is precise fact, this nature of it portrays deep profound properties of thought, mathematical thought in particular. It has many formulations. One of them basically states that given a collection, or a set of non-empty sets, it is possible to choose one element from every one of those sets, in this way forming a set which has one element in common with every set in this set of sets. The sets in the set of sets, can be sets of anything, other sets or numbers (which are sets as well), etc. At this point the statement of the axiom seems empty, its truth being soo implicit. After all, if you have a collection of sets, which have say at least one number in them, it seems obvious that you can choose an element from the first set, then the second, and so on. Every set in this collection of sets is non-empty by assumption and so has at least one element, which we can choose. So what wrong? One inconsistency in the former logic was the use of the word ‘first’. This word, albeit invisibly, implies a great deal. Namely, it implies that one has determined an ordering on the set of things from which one chooses a first element. For otherwise, ‘first’ has no meaning. What does ‘first’ mean? That there is nothing before that first element? Then what does ‘before’ mean? To define these terms would be to define an ordering on the set, or essentially a set of rules which define a first element, and the next, etc. So that can be very simply. Lets say for instance we are talking about a set of apples. I can throw these apples on the ground, and then I can specifically define the first element to be the one closest to me in distance, the second element is closes to me in distance if the first were removed and so on. Then in this way I have an ordering and so I can take the first set and choose an element, which must be possible since its non-empty. Here I do not specify the ordinal position of the element I am choosing, simply that I am choosing an element and so the discussion of ordering does not apply here. What is wrong here? The next inconsistency arises when we speak about sets which have an infinite number of elements. More specifically, sets, the elements of which cannot be counted or basically aligned with any number. More in part two…