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Nathan Brunelle Department of Computer Science University of Virginia Theory of Computation CS3102 – Spring 2014 A tale of computers, math, problem solving, life, love and tragic death

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Today: Infinities and Paradoxes Themes: Dovetailing Diagonalization Contradiction Cardinality Describability

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Historical Perspectives Georg Cantor ( ) Created modern set theory Invented trans-finite arithmetic (highly controvertial at the time) Invented diagonalization argument First to use 1-to-1 correspondences with sets Proved some infinities “bigger” than others Showed an infinite hierarchy of infinities Formulated continuum hypothesis Cantor’s theorem, “Cantor set”, Cantor dust, Cantor cube, Cantor space, Cantor’s paradox Laid foundation for computer science theory Influenced Hilbert, Godel, Church, Turing

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Problem: How can a new guest be accommodated in a full infinite hotel? ƒ(n) = n+1

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Problem: How can an infinity of new guests be accommodated in a full infinite hotel? … ƒ(n) = 2n

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… one-to-one correspondence Problem: How can an infinity of infinities of new guests be accommodated in a full infinite hotel?

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Problem: Are there more integers than natural #’s? ℕ ℤℕ ℤ ℕ ℤℕ ℤ So | ℕ |<| ℤ | ? Rearrangement: Establishes 1-1 correspondence ƒ: ℕ ℤ |ℕ|=|ℤ||ℕ|=|ℤ| ℤ ℕ ℤ

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Problem: Are there more rationals than natural #’s? … … … … … … … … ℕ ℚℕ ℚ ℕ ℚℕ ℚ So | ℕ |<| ℚ | ? Dovetailing: Establishes 1-1 correspondence ƒ: ℕ ℚ |ℕ|=|ℚ||ℕ|=|ℚ|

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Problem: Are there more rationals than natural #’s? … … … … … … … … ℕ ℚ ℕ ℚ So | ℕ |<| ℚ | ? Dovetailing: Establishes 1-1 correspondence ƒ: ℕ ℚ | ℕ |=| ℚ | Avoiding duplicates!

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Problem: Are there more rationals than natural #’s? … … … … … … … … ℕ ℚ ℕ ℚ So | ℕ |<| ℚ | ? Dovetailing: Establishes 1-1 correspondence ƒ: ℕ ℚ | ℕ |=| ℚ |

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Problem: Why doesn’t this “dovetailing” work? … … … … … … … … There’s no “last” element on the first line! So the 2 nd line is never reached! 1-1 function is not defined!

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Dovetailing Reloaded Dovetailing: ƒ: ℕ ℤ … … To show | ℕ |=| ℚ | we can construct ƒ: ℕ ℚ by sorting x/y by increasing key max(|x|,|y|), while avoiding duplicates: max(|x|,|y|) = 0 : {} max(|x|,|y|) = 1 : 0/1, 1/1 max(|x|,|y|) = 2 : 1/2, 2/1 max(|x|,|y|) = 3 : 1/3, 2/3, 3/1, 3/2...{finite new set at each step} Dovetailing can have many disguises! So can diagonalization! ℕ ℤ Dovetailing!

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Theorem: Some numbers have no description. Proof: Dovetailing!

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Problem 1: Why not just insert X into the table? Problem 2: What if X=0.999… but 1.000… is already in table? ƒ(1) = … ƒ(2) = … ƒ(3) = … ƒ(4) = … ƒ(5) = … X = 0. ℝ Table with X inserted will have X’ still missing! Inserting X (or any number of X’s) will not help! To enforce unique table values, we can avoid using 9’s and 0’s in X. ℕℝ Non-existence proof! Diagonalization

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Corollary: Some real numbers do not have finite descriptions! ƒ(1) = … ƒ(2) = … ƒ(3) = … ƒ(4) = … ƒ(5) = … X = 0. ℝ Table with X inserted will have X’ still missing! Inserting X (or any number of X’s) will not help! To enforce unique table values, we can avoid using 9’s and 0’s in X. ℕℝ Non-existence proof! Diagonalization

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Non-Existence Proofs Must cover all possible (usually infinite) scenarios! Examples / counter-examples are not convincing! Not “symmetric” to existence proofs! Ex: proof that you are a millionaire: “Proof” that you are not a millionaire ? Existence proofs can be easy! Non-existence proofs are often hard! P NP

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Historical Perspectives Bertrand Russell ( ) Philosopher, logician, mathematician, historian, social reformist, and pacifist Co-authored “Principia Mathematica” (1910) Axiomatized mathematics and set theory Co-founded analytic philosophy Originated Russell’s Paradox Activist: humanitarianism, pacifism, education, free trade, nuclear disarmament, birth control gender & racial equality, gay rights Profoundly transformed math & philosophy, mentored Wittgenstein, influenced Godel Laid foundation for computer science theory Won Nobel Prize in literature (1950)

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Russell’s paradox was invented by Russell in 1901 to show that naïve set theory is self-contradictory: Define: set of all sets that do not contain themselves S = { T | T T } Q: does S contain itself as an element? S S S S contradiction! Similar paradoxes: “A barber who shaves exactly those who do not shave themselves.” “This sentence is false.” “I am lying.” “Is the answer to this question ‘no’?” “The smallest positive integer not describable in twenty words or less.”

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Star Trek, 1967, “I, Mudd” episode Captain James Kirk and Harry Mudd use a logical paradox to cause hostile android “Norman” to crash Diagonalization!

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Historical Perspectives Kurt Gödel ( ) Logician, mathematician, and philosopher Proved completeness of predicate logic and Gödel’s incompleteness theorem Proved consistency of axiom of choice and the continuum hypothesis Invented “Gödel numbering” and “Gödel fuzzy logic” Developed “Gödel metric” and paradoxical relativity solutions: “Gödel spacetime / universe” Made enormous impact on logic, mathematics, and science

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Gödel’s Incompleteness Theorem Frege & Russell: Mechanically verifying proofs Automatic theorem proving A set of axioms is: Sound: iff only true statements can be proved Complete: iff any statement or its negation can be proved Consistent: iff no statement and its negation can be proved Hilbert’s program: find an axiom set for all of mathematics i.e., find a axiom set that is consistent and complete Gödel: any consistent axiomatic system is incomplete! (as long as it subsumes elementary arithmetic) i.e., any consistent axiomatic system must contain true but unprovable statements Mathematical surprise: truth and provability are not the same!

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Gödel’s Incompleteness Theorem That some axiomatic systems are incomplete is not surprising, since an important axiom may be missing (e.g., Euclidean geometry without the parallel postulate) However, that every consistent axiomatic system must be incomplete was an unexpected shock to mathematics! This undermined not only a particular system (e.g., logic), but axiomatic reasoning and human thinking itself! Truth Provability Justice Legality

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Gödel’s Incompleteness Theorem Gödel: consistency or completeness - pick one! Which is more important? Incomplete: not all true statements can be proved. But if useful theorems arise, the system is still useful. Inconsistent: some false statement can be proved. This can be catastrophic to the theory: E.g., supposed in an axiomatic system we proved that “1=2”. Then we can use this to prove that, e.g., all things are equal! Consider the set:{Bush, Pope} | {Bush, Pope} | = 2 | {Bush, Pope} | = 1 (since 1=2) Bush = PopeQED All things become true: system is “complete” but useless!

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Gödel’s Incompleteness Theorem Moral: it is better to be consistent than complete, If you can not be both. “It is better to be feared than loved, if you cannot be both.” -Niccolo Machiavelli ( ), “The Prince” “You can have it good, cheap, or fast – pick any two.” - Popular business adage

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Gödel’s Incompleteness Theorem Thm: any consistent axiomatic system is incomplete! Proof idea: Every formula is encoded uniquely as an integer Extend “Gödel numbering” to formula sequences (proofs) Construct a “proof checking” formula P(n,m) such that P(n,m) iff n encodes a proof of the formula encoded by m Construct a self-referential formula that asserts its own non-provability: “I am not provable” Show this formula is neither provable nor disprovable George Boolos (1989) gave shorter proof based on formalizing Berry’s paradox The set of true statements is not R.E.! Dovetailing! Algorithm! Diagonalization!

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Gödel’s Incompleteness Theorem Systems known to be complete and consistent: Propositional logic (Boolean algebra) Predicate calculus (first-order logic) [Gödel, 1930] Sentential calculus [Bernays,1918; Post, 1921] Presburger arithmetic (also decidable) Systems known to be either inconsistent or incomplete: Peano arithmetic Primitive recursive arithmetic Zermelo–Frankel set theory Second-order logic Q: Is our mathematics both consistent and complete? A: No [Gödel, 1931] Q: Is our mathematics at least consistent? A: We don’t know! But we sure hope so.

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Gödel’s “Ontological Proof” that God exists! Formalized Saint Anselm's ontological argument using modal logic: For more details, see:

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Continuum Hypothesis

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Axiom of Choice Given any set of sets, it is possible to construct a new set by picking exactly one item from each set. Obvious for case where the set is finite, tricky for infinite Non-constructive! Statement of possibility, bot procedure Is it true? Mathematics has no answer!

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Banach-Tarski Paradox Non-intuitive side-effect of the Axiom of Choice Any solid sphere can be broken into a finite number of pieces and reassembled into 2 spheres of the same size as the original

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