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# Axiom of choice presentation

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• 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.
• ### Axiom of choice presentation

1. 1. The Axiom of Choice<br />"The axiom gets its name not because mathematicians prefer it to other axioms." — A.K. Dewdney<br />Roger Sheu<br />May 4, 2011<br />
2. 2. Another Quote…<br />"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<br />
3. 3. So what did Mr. Russell mean?<br />Finite number of shoes/socks. How do you pick a set of shoes or socks?<br />Take an infinite number of shoes/socks. How do you pick in this case?<br />(Socks are indistinguishable, but there are left and right shoes)<br />
4. 4. The choice function<br />The choice function is a function that picks exactly one element out of any set in a whole collection of sets.<br />Example: the set S={1,2,5} could have the choice function F={1}, F={2}, or F={5}<br />Apply the choice function to socks and shoes<br />
5. 5. The axiom of choice and the choice function<br />Axiom of choice assumes the existence of a choice function<br />Fixes the infinite sock problem<br />
6. 6. Where is Axiom of Choice from, fundamentally speaking?<br />Zermelo-Fraenkel Set Theory (ZF)<br />10 Axioms<br />The axiom of extension<br />Emptyset/Pairset Axiom<br />Union axiom<br />Powerset Axiom<br />Separation Axiom<br />Replacement Axiom<br />Infinity Axiom<br />Regularity Axiom<br />Axiom of Choice<br />Ernst Zermelo (1871-1953)<br />
7. 7. Why is there controversy over the Axiom of Choice?<br />Kurt Gödel (1940): not having Axiom of Choice does not contradict any of the other axioms or results from ZF theory (not necessarily false)<br />Paul Cohen(1963): ZF theory cannot prove the Axiom of Choice (not necessarily true).<br />Axiom<br /> Of Choice<br /> Independent<br />Kurt Gödel<br />Paul Cohen<br />
8. 8. Well-ordering Theorem<br />(Not to be confused with the well-ordering property)<br />States that any set can be ordered in such a way that there is guaranteed to be a smallest element (a well-ordered set)<br />Equivalent to the Axiom of Choice<br />
9. 9. Proof of Axiom of Choice from the Well-Ordering Theorem<br />Take the set and use the ordering as described by the well-ordering theorem<br />Every set will have an order such that there’s a smallest element<br />Take smallest element from each set, and this is your choice function<br />
10. 10. So does the Axiom of Choice make sense?<br />"The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn’s lemma?" — Jerry Bona<br />All three of these are equivalent.<br />
11. 11. Zorn’s Lemma – An Introduction<br />As noted above, Zorn’s Lemma is one of the equivalent statements to the Axiom of Choice<br />Reminder: A well ordered set is one that is guaranteed to have a smallest element<br />Partially ordered set: A set with order between any two elements, but not necessarily full order<br />Totally ordered set: a set with an order between all elements<br />
12. 12. Zorn’s Lemma (and some other results of the axiom of choice)<br />Given a partially ordered set, if every totally ordered subset has an upper bound, there is a maximal element.<br />Other results…<br />Trichotomy<br />Tukey’s Lemma<br />Maximal principle<br />Every vector space has a basis<br />Tychonoff’s Theorem<br />
13. 13. Banach-Tarski Paradox<br />However, the Axiom of Choice does not always produce intuitively obvious results.<br />If we take one sphere, then we can get two identical spheres back with some disjoint cuts and pastes.<br />How is this possible? Where does the Axiom of Choice come in?<br />