Is it possible that you really don't have choice?
Miguel Moreno is a PhD student at the Department of Mathematics and Statistics at the University of Helsinki. He works on set theory and model theory, and studies the complexity of mathematical questions.
Everyone has answered a yes or no question at least once. Maybe in an elementary school exam, a referendum or even on a lazy Sunday playing games with the family. We have assumed that we can answer every question yes or no, although we might sometimes get it wrong.
Some mathematicians study the ability to choose an answer. They have found out that actually we cannot always give an answer.
Mathematicians obtain these paradoxical results from something they call the axiom of choice, “for every collection of nonempty boxes, it is possible to choose an element of each box.” In 1960s mathematicians found a system in which we cannot always choose, this was made by constructing a mathematical universe in which the statement “given a set of elements, the elements can be ordered such that for every subset of it, we can choose the smallest element” is false.
“Classic results in calculus or algebra are false in the system in which we don't have a choice.”
Before that all the science was done assuming that we always have a choice. Classic results in calculus or algebra are false in the system in which we don't have a choice. Even the simple task of putting things in order from smaller to bigger requires choices to be made.
Some recent results even appear to contradict one of the best known facts of science: that matter cannot be created or destructed, it can be only transformed. But what if we can transform one tennis ball into two tennis balls identical to the first one, are we creating matter?
The axiom of choice allows us to construct many interesting mathematical objects. It seems that it is possible to construct objects that cannot be measured. These objects are basically the union of small pieces, each one chosen by the axiom of choice with certain property. These objects are given more by a mathematical formula than by a physical example. These objects with no measure have interesting applications. The best known one is the Banach-Tarski paradox, a way to duplicate objects. By dividing a ball in six pieces that have no measure, and gluing them back in a certain way, we indeed obtain two balls identical to the original one.
“Some recent results even appear to contradict one of the best known facts of science: that matter cannot be created or destructed, it can be only transformed.”
Some scientists look at these results in an optimistic way. In the same way as physics gives an explanation to the paradox of duplicating balls, there should be other physical notions that explain other paradoxes given by the axiom of choice.
One way or another, the axiom of choice gives clear limits for what we can and what we cannot do. Either we always have a choice, but there are results that cannot be put in practice (like duplicating tennis balls), or we don't always have a choice and many results in science need to be reconsidered.
Most everyone accepts the axiom of choice as true, because it is natural to think that we can always choose. It is extremely useful, and there is no reason to deny it. On the other hand, a small group of people are trying to understand the axiom of choice by studying its negation: What would happen if there are situations in which we cannot make a choice?
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