1. The Axiom of Choice "The axiom gets its name not because mathematicians prefer it to other axioms." — A.K. Dewdney Roger Sheu May 4, 2011
2. Another Quote… "The Axiom of Choice is necessary to select a set from an infinite number of socks, but not an infinite number of shoes." — Bertrand Russell
3. So what did Mr. Russell mean? Finite number of shoes/socks. How do you pick a set of shoes or socks? Take an infinite number of shoes/socks. How do you pick in this case? (Socks are indistinguishable, but there are left and right shoes)
4. The choice function The choice function is a function that picks exactly one element out of any set in a whole collection of sets. Example: the set S={1,2,5} could have the choice function F={1}, F={2}, or F={5} Apply the choice function to socks and shoes
5. The axiom of choice and the choice function Axiom of choice assumes the existence of a choice function Fixes the infinite sock problem
6. Where is Axiom of Choice from, fundamentally speaking? Zermelo-Fraenkel Set Theory (ZF) 10 Axioms The axiom of extension Emptyset/Pairset Axiom Union axiom Powerset Axiom Separation Axiom Replacement Axiom Infinity Axiom Regularity Axiom Axiom of Choice Ernst Zermelo (1871-1953)
7. Why is there controversy over the Axiom of Choice? Kurt Gödel (1940): not having Axiom of Choice does not contradict any of the other axioms or results from ZF theory (not necessarily false) Paul Cohen(1963): ZF theory cannot prove the Axiom of Choice (not necessarily true). Axiom Of Choice Independent Kurt Gödel Paul Cohen
8. Well-ordering Theorem (Not to be confused with the well-ordering property) States that any set can be ordered in such a way that there is guaranteed to be a smallest element (a well-ordered set) Equivalent to the Axiom of Choice
9. Proof of Axiom of Choice from the Well-Ordering Theorem Take the set and use the ordering as described by the well-ordering theorem Every set will have an order such that there’s a smallest element Take smallest element from each set, and this is your choice function
10. So does the Axiom of Choice make sense? "The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn’s lemma?" — Jerry Bona All three of these are equivalent.
11. Zorn’s Lemma – An Introduction As noted above, Zorn’s Lemma is one of the equivalent statements to the Axiom of Choice Reminder: A well ordered set is one that is guaranteed to have a smallest element Partially ordered set: A set with order between any two elements, but not necessarily full order Totally ordered set: a set with an order between all elements
12. Zorn’s Lemma (and some other results of the axiom of choice) Given a partially ordered set, if every totally ordered subset has an upper bound, there is a maximal element. Other results… Trichotomy Tukey’s Lemma Maximal principle Every vector space has a basis Tychonoff’s Theorem
13. Banach-Tarski Paradox However, the Axiom of Choice does not always produce intuitively obvious results. If we take one sphere, then we can get two identical spheres back with some disjoint cuts and pastes. How is this possible? Where does the Axiom of Choice come in?
17. A Sketchy Explanation of the Banach-Tarski Paradox Take a sphere and define its four quadrants as sets. Imagine you take one of those sets and you have a rotation in mind that will guarantee that your location on the sphere has changed. The only rotation that will lead to a problem is the reverse rotation back to the original position. With two axes of rotation and a forward/backward, there will be four different rotations. Pick one rotation type and for your next rotation, you have three choices because one of those rotations will simply cancel out with the original rotation. However, if we take one pre-rotation, then this one inverse rotation type accounts for the possibilities of the three other rotations, because it is equivalent to the next one rotation out of the three. Credit: http://www.irregularwebcomic.net/2339.html
18. Banach-Tarski explanation continued There are two different rotations, so there are now two equations that add up to give that the new volume of the sets is two times the original volume. So where does the axiom of choice come in? Well, let there be a set of points S. This set contains all the points possible such that they are all not reachable by rotating other points in the set S. We can do this because of the choice function provided by the axiom of choice.
19. Hausdorff Paradox If we have a sphere, then we can remove a part of the sphere and define sets A, B, C such that A, B, C, and B U C are congruent, essentially implying that A is 1/3 and ½ of the sphere at the same time.
20. What if the Axiom of Choice were false? There exists an infinite set of real numbers with no countable subset. The set of all real numbers is a union of countably many countable sets.
21. In conclusion… The Axiom of Choice is a powerful axiom from Zermelo-Fraenkel Set Theory that results in important, sometimes unintuitive, results spanning various fields of mathematics. What do you think? Is the Axiom of Choice true?
Two sets equal iff they have same elementsExistence of null set…given sets x and y, there is a set with elements of x and yGiven set x, union of all elements in x is a setThere is a set with all the subsets of xf(x) is a setA set x with null set such that if y in x, then y U {y} is in x.Given set x, there is a y in x so that y (intersection) x = empty set.
Non-measurable sets Measure theory becomes complex at higher dimensions, like 3-D as the sphere.Non-measurable sets is a result of the Axiom of Choice.Non-measurable sets are collections of points that require an infinite number of arbitrary points to identify.
