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cs2102: Discrete Mathematics Class 3: Well-Ordering David Evans University of Virginia

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Plan Well-Ordering Principle Well-Ordered Sets (and Comparators) Well-Ordered Theorem? Well-Ordered Data Types Well-Ordering Principle Proofs The Well-Ordering Principle is the crux of mathematical induction, which is the basis for many of the most important proofs in CS.

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There are lots of office hours! Don’t wait to get help.

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Well Ordering Principle

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Well-Ordering Principle Every nonempty set of non-negative integers has a smallest element.

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Ordering Sets Definition. A set is well-ordered with respect to an ordering function (e.g., <), if all of its non-empty subsets has a minimum element. Why does this mean the whole set is well-ordered?

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Definition. A set is well-ordered with respect to an ordering function (e.g., <), if all of its non-empty subsets has a minimum element.

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Sensible Ordering Functions Definition. A set is well-ordered with respect to an ordering function (e.g., <), if all of its non-empty subsets has a minimum element.

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Java Transgression

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Java Transgression

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Well-Ordered Sets? Set: the integers Comparator: a < b

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Well-Ordered Sets? Set: the integers Comparator: |a| < |b|

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Well-Ordered Sets? Set: the integers Comparator: if | a | = | b |: a < b else: |a| < |b|

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Well-Ordered Sets? Set: the non-negative rationals Comparator: < Definition. A number is rational if it can be written as a ratio of two integers.

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Set: the non-negative rationals Comparator: ? Is there some comparator that makes the non-negative rationals well-ordered?

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Well-Ordered Theorem For every set, there is some comparator (that is transitive and trichotomic) that makes the set well-ordered. Is this true?

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Slack break… For every set, there is some comparator that is transitive and trichotomic that makes the set well-ordered.

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Georg Cantor Well-ordered theorem is a fundamental law of thought. (1883)

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Gyula Kőnig There is no comparator that results in a well- ordering of the real numbers.

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Gyula Kőnig Claimed counter-proof: There is no comparator that results in a well- ordering of the real numbers. Proof withdrawn as incorrect

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Georg Cantor Well-ordered theorem is a fundamental law of thought. (1883) What should we (mathematics) do when there is something that seems like it should be true, but no one can prove?

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Axiom of Choice For any collection of non-empty sets, there is a choice function that picks exactly one element from each set.

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Bertrand Russell (1872-1970) To choose one sock from each of infinitely many pairs of socks requires the Axiom of Choice, but for shoes the Axiom is not needed.

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For every set, there is some comparator that is transitive and trichotomic that makes the set well-ordered. For any collection of non-empty sets, there is a choice function that picks exactly one element from each set. implies Well-Ordered Theorem Axiom of Choice

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For every set, there is some comparator that is transitive and trichotomic that makes the set well-ordered. For any collection of non-empty sets, there is a choice function that picks exactly one element from each set. implies Well-Ordered Theorem Axiom of Choice

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Still Controversial… ZF Axioms: Zermelo–Fraenkel Set Theory ZFC Axioms: ZF + Axiom of Choice

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Well-Ordering in Programming Is a + 1 > a always true?

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“glitch in control unit causes generators to shut down if left powered on for 248 days”

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Can I bet any dollar amount using just $2 and $5 chips?

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Betable Numbers A number is betable if it can be produced using some combination of $2 and $5 chips. Are all integers greater than $3 betable?

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Betable Numbers A number is betable if it can be produced using some combination of $2 and $5 chips.

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Charge • Maintain Well Orderliness! • If you need to enroll in the class, bring me a course action form to sign (now, or later) • Due Friday (6:29pm): PS1 • Read MCS Ch 2, 3 (at least through 3.5) before Thursday’s class