stoutfellow (![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png)
stoutfellow) wrote2009-06-24 01:52 pm
stoutfellow) wrote2009-06-24 01:52 pm
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Ramble, Part 73: You Pays Your Money...
The last axiom in Zermelo's list requires a post of its own. It is rather complicated to describe, and - unusually - it is quite controversial. Details, as usual, under the cut.
Suppose you have a collection of sets in mind, and you'd like to pick an element from each set. Obviously, if one of them is empty, you can't, so let's assume that all of the sets actually have elements. If the collection is finite, then, in principle, at least (remember, "finite" doesn't mean "not very big"!) making your choice is straightforward: just work your way down the list, picking an element each time. If the collection is infinite, though, you might have some qualms; making an infinite number of choices is chancy. Maybe, if the sets are special, you can lay down a selection rule, but most of the time that won't be possible. How can you be sure that it's possible to make such a choice?
The last of Zermelo's axioms is called the Axiom of Choice, and it simply declares that it's always possible to make that choice.
For example, if the Axiom of Choice is true, then the following is also true. Take a solid ball of radius 1. It is possible to disassemble the ball into a small number of pieces, which can then be reassembled into two solid balls of radius 1. (We are, of course, ignoring such inconsequences as the existence of atoms and the uncertainty principle....) This is the Banach-Tarski Paradox. Note that it is different in kind from Russell's Paradox. Russell's Paradox points out that certain assumptions lead to a logical impossibility; Banach-Tarski, that certain assumptions lead to something bizarre and unexpected.
One objection that might be raised would be the question of volume; surely B-T is impossible, because the two balls have greater volume, taken together, than the one original ball. The only way around this objection is to accept that there are certain sets of points - like the pieces into which we disassemble the sphere - which do not have a volume. I do not mean that they have volume zero - 0 is, after all, a number - but that there is no number which can be identified as the volume of one of these pieces. In technical language, they are "not measurable", and their behavior under disassembly and reassembly cannot be predicted in terms of volume.
Some mathematicians find the Banach-Tarski Paradox and the existence of non-measurable sets distressing - enough so that they reject the Axiom of Choice, for leading to such an absurd result. For others, the Axiom of Choice has so many useful consequences that these mathematicians are willing to bite the bullet and accept a few oddities. (One might compare the willingness of so many mathematicians to ignore the logical flaws of Newtonian calculus, although here both sides are equally committed to mathematical rigor.) It has been shown that both positions are reasonable. That is: if the Zermelo axioms (or, rather, a modified version called the Zermelo-Frankel axioms) without the Axiom of Choice are consistent, then they remain consistent after adding it. On the other hand, they also remain consistent if one adds, not the Axiom of Choice, but its negation! If consistency is the only criterion, then, it is equally sensible to affirm as to deny the Axiom of Choice; all that matters beyond that is the amount of strangeness you are willing to accept.
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Suppose you have a collection of sets in mind, and you'd like to pick an element from each set. Obviously, if one of them is empty, you can't, so let's assume that all of the sets actually have elements. If the collection is finite, then, in principle, at least (remember, "finite" doesn't mean "not very big"!) making your choice is straightforward: just work your way down the list, picking an element each time. If the collection is infinite, though, you might have some qualms; making an infinite number of choices is chancy. Maybe, if the sets are special, you can lay down a selection rule, but most of the time that won't be possible. How can you be sure that it's possible to make such a choice?
The last of Zermelo's axioms is called the Axiom of Choice, and it simply declares that it's always possible to make that choice.
Technically, what it says is this: given any collection of nonempty sets, the Cartesian product of the sets in the collection is nonempty. Why is this equivalent to being able to choose one element from each? Think of, for instance, a Cartesian product of three sets, A, B, C. The elements of the Cartesian product are ordered triples (a,b,c) where a is an element of A, b an element of B, and c an element of C. The reason this works is that our three sets are indexed by the set {1,2,3}; A is the first set, B the second, and C the third. The first element of the triple comes from the first set, the second from the second, and the third from the third. Thus, we can think of the elements of AxBxC as functions from the index set {1,2,3} to the union of the sets A, B, C, with the proviso that the function must take each index to a member of the set with that index. The general case is no different; our collection of sets is {Ai}, where i ranges over an index set I (which may be infinite, uncountable or even larger); the Cartesian product is the set of functions from I to the union of the sets Ai, subject to the same condition - the image of i must be an element of Ai. Viewed this way, it is, perhaps, a bit clearer that elements of the Cartesian product amount to choices of elements from the various sets.It seems like a reasonable assumption, but, as frequently when infinite things come into play, it has some distinctly odd consequences.
For example, if the Axiom of Choice is true, then the following is also true. Take a solid ball of radius 1. It is possible to disassemble the ball into a small number of pieces, which can then be reassembled into two solid balls of radius 1. (We are, of course, ignoring such inconsequences as the existence of atoms and the uncertainty principle....) This is the Banach-Tarski Paradox. Note that it is different in kind from Russell's Paradox. Russell's Paradox points out that certain assumptions lead to a logical impossibility; Banach-Tarski, that certain assumptions lead to something bizarre and unexpected.
One objection that might be raised would be the question of volume; surely B-T is impossible, because the two balls have greater volume, taken together, than the one original ball. The only way around this objection is to accept that there are certain sets of points - like the pieces into which we disassemble the sphere - which do not have a volume. I do not mean that they have volume zero - 0 is, after all, a number - but that there is no number which can be identified as the volume of one of these pieces. In technical language, they are "not measurable", and their behavior under disassembly and reassembly cannot be predicted in terms of volume.
Some mathematicians find the Banach-Tarski Paradox and the existence of non-measurable sets distressing - enough so that they reject the Axiom of Choice, for leading to such an absurd result. For others, the Axiom of Choice has so many useful consequences that these mathematicians are willing to bite the bullet and accept a few oddities. (One might compare the willingness of so many mathematicians to ignore the logical flaws of Newtonian calculus, although here both sides are equally committed to mathematical rigor.) It has been shown that both positions are reasonable. That is: if the Zermelo axioms (or, rather, a modified version called the Zermelo-Frankel axioms) without the Axiom of Choice are consistent, then they remain consistent after adding it. On the other hand, they also remain consistent if one adds, not the Axiom of Choice, but its negation! If consistency is the only criterion, then, it is equally sensible to affirm as to deny the Axiom of Choice; all that matters beyond that is the amount of strangeness you are willing to accept.
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