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The axiom of choice has been one of the most hotly contested principles of 20th-century mathematics. Once the concept of the infinite had begun to be investigated by Georg Cantor (1845 - 1918), the axiom of choice was something it seemed obvious to formulate; it is (loosely) something that makes infinite sets behave far more like finite ones. Several different formulations of the axiom of choice have been made, among others by , Ernst Zermelo (1871 - 1953) and , Adolf Abraham Fraenkel (1891 - 1965), the inventors of modern axiomatic set theory.
Some of these equivalent formulations are as follows: (1) The Classic Axiom of Choice For every set A of non-empty sets, there is a set B (known as the choice set) which has one and only one member in common with each member of A. (2) The Well-Ordering Principle Every set can be well-ordered. (A set is well-ordered if for every element a,b,c we have a < b or b < a but not both; if a < b and b < c then a < c and any non-empty subset has a minimal element.) (3) Every Set has a Cardinality This means that the notion of size is well-defined for infinite sets. There are several other, more complicated formulations of the same principle, one of the most commonly used being known as Zorn\'s Lemma.
Even once these definitions have been disentangled, there at first seems to be no problem. Some of the consequences of the axiom are much less desirable, being completely at odds with common sense. For example, it is possible (given the axiom of choice) to take a spherical object (such as an orange), cut it into a small finite number of pieces, which can then be reassembled to produce two spheres of the same size as the original one. This contradicts our understanding of how space works, but that in itself is not a sufficient objection to make mathematicians discard the axiom: it is not contradictory, but merely strange. Besides, the axiom of choice is vastly useful. A second objection is that it is often easy to use the axiom of choice to show the existence of some kind of mathematical object, but difficult to see what such an object looks like. An example of this is the well-ordering of the real numbers predicted to exist by the well-ordering principle (see below). This gives mathematicians a problem, for they like to be able actually to look at the objects they produce, and so try to produce them in a ‘constructive’ way.
Pure mathematicians collectively heaved a sigh of relief in 1939, when Gödel (1906 - 1978) proved that the axiom of choice was consistent with the other axioms of set theory. He did not prove that it was necessarily true, merely that its use would never produce a result which contradicted the other axioms, which were all widely accepted. However, in 1963 Paul Cohen showed that the negation, or opposite, to the axiom of choice was also consistent with the rest of set theory. He demonstrated that there was a kind of set theory in which the set of real numbers was not well-orderable (this means that even if the axiom of choice is accepted, it is not possible explicitly to find a way of well-ordering the real numbers).
This demonstration left the axiom of choice in a kind of limbo. Today, mathematicians will use it if they can find no other way to prove a theorem, but its use still carries a stigma from the time when some thought it false, and it is always flagged when used. SMcL
Further reading W.V. Quine, Set Theory and Its Logic. |
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