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CS120: Lecture 15

CS120: Lecture 15. MP Johnson Hunter mpjohnson@gmail.com. Claims. Some infinities are larger than others. Some problems are unsolvable. Infinite sets. Equinumerous sets N, Z, squares, primes, Q / cross-product, graph paper R, [0,1] Self-reference Smullyan’s brother Gmail Barber

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CS120: Lecture 15

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  1. CS120: Lecture 15 MP Johnson Hunter mpjohnson@gmail.com

  2. Claims • Some infinities are larger than others. • Some problems are unsolvable.

  3. Infinite sets • Equinumerous sets • N, Z, squares, primes, Q / cross-product, graph paper • R, [0,1] • Self-reference • Smullyan’s brother • Gmail • Barber • Set of all sets • “non-self-describing” • Diagonalization • R / P(N)

  4. Undecidability • Decision problems • Primality • 3-colorability • Has TSP <k • Pgms either halt or loop • Von Neuman: pgms are data • Given H, can define: D(x) = { down, if x(x) up { up, if x(x) down • Assume H is decidable • Then D(D) = {down, if D(D) up {up, if D(D) down 

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