Math CentralQuandaries & Queries


Question from sydney, a student:

The axiom of choice asserts the existence of certain sets, but does not construct the set. What does "construct" mean here? For example, does it require showing the existence and uniqueness of some function yielding the set? In general, what does it mean to require the existence of a mathematical object be tied to a construction of it?


To construct a set would be to use an unambiguous rule to determine membership in it. For instance, I have a set pairs of pants, a set of shirts and a set of hats. Every day I dress, that is, I choose a pair of pants, a shirt and a hat. I can use a simple rule like "all blue" (provided I have exactly one blue item in each set), or I can make independent ad hoc choices, like the blue pants, the green shirt and the black hat. We have no problems making independent ad hoc choices if we have only three sets and three choices to make, but what if I had an infinity of sets, and an infinity of choices to make? I know I will not make infinitely many independent choices in practice, so I'd be happy to revert to a formal rule like "all blue". But this will not work if some sets contain more than one blue item, and others contain none. So I cannot escape the possibility of relying on infinitely many independent choices in some cases.

Here, opinions start to differ. The constructivists would say that if I cannot come up with an unambiguous rule to choose one element from each set, then I cannot assert that there is a set constructed by taking one element from each set. On the other hand, the axiom of choice asserts that such a set exists, even though its construction would require infinitely many independent choices.

There are nontrivial consequences in accepting the axiom of choice, even with problems that are easy to state. How many colours are required to colour all the points in the plane in such a way that pairs of points at distance 1 always have different colours? This is the "Hadwiger-Nelson problem'' discussed in the Wikipedia article
The problem dates from 1950 and the exact answer is still not known. There is a set of seven points requiring four colours, showing that the answer is at least four, and there is a coloured hexagonal tiling of the plane with seven colours showing that the answer is at most seven. The coloured hexagonal tiling is an "unambiguous rule to colour the plane with seven colours'', but ultimately, the problem allows for infinitely many independent choices, that is, one choice of colour for every point in the plane. If we accept the axiom of choice, this is fine, and we know that the true answer is between four and seven, though we don't know the exact value yet. What if we reject the axiom of choice? Then we can instead adopt the axiom of Solovay stating that every set is "Lebesgue measurable'', and it is known that if every colour class is Lebesgue measurable, then at least five colours are needed. So in this case we reject the value four as a possible answer.



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