Theory of Computation CS3102 Spring 2014

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1 Theory of Computation CS0 Spring 0 A tale of computers, math, problem solving, life, love and tragic death Nathan Brunelle Department of Computer Science University of Virginia

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

others Showed an infinite hierarchy of infinities Formulated continuum hypothesis Cantor s theorem, Cantor set, Cantor dust,

3 Historical Perspectives Georg Cantor (-9) Created modern set theory Invented trans-finite arithmetic (highly controvertial at the time) Invented diagonalization argument First to use -to- 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

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

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

6 Problem: How can an infinity of infinities of new guests be accommodated in a full infinite hotel? 0 9 one-to-one correspondence

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

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

10 Problem: Are there more rationals than natural # s? N Q N Q So N < Q? Dovetailing: Establishes - correspondence ƒ: N Q N = Q 9 0 9

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

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

13 Dovetailing: ƒ:n Z Dovetailing Reloaded 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 ) = : 0/, / max( x, y ) = : /, / max( x, y ) = : /, /, /, /... {finite new set at each step} Dovetailing can have many disguises! So can diagonalization! Z N

14 Theorem: Some numbers have no description. Proof: Sort descriptions of numbers by their length: The integer after 9 The ratio of the circumference of a circle to its diameter Conclusion: Only countably many descriptions of numbers! Some numbers are not on this list of descriptions! Some numbers are not finitely describable!

15 Problem : Why not just insert X into the table? Problem : What if X=0.999 but.000 is already in table? N R ƒ() =. 9 ƒ() = ƒ() =. ƒ() =. ƒ() = X = 0. 9 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.

16 Corollary: Some real numbers do not have finite descriptions! N R ƒ() =. 9 ƒ() = ƒ() =. ƒ() =. ƒ() = X = 0. 9 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.

17 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? P NP

Historical Perspectives Bertrand Russell (-90) Philosopher, logician, mathematician, historian, social reformist, and pacifist Co-authored Principia Mathematica (90) Axiomatized mathematics and set

18 Historical Perspectives Bertrand Russell (-90) Philosopher, logician, mathematician, historian, social reformist, and pacifist Co-authored Principia Mathematica (90) 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 (90)

S S S S contradiction! Similar paradoxes: A barber who shaves exactly those who do not shave themselves.

20 Russell s paradox was invented by Russell in 90 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.

21 Star Trek, 9, I, Mudd episode Captain James Kirk and Harry Mudd use a logical paradox to cause hostile android Norman to crash

continuum hypothesis Invented Gödel numbering and Gödel fuzzy logic Developed Gödel metric and

23 Historical Perspectives Kurt Gödel (90-9) 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

Gödel s Incompleteness Theorem Frege & Russell: Mechanically verifying proofs Automatic theorem proving A set of axioms is: Sound: iff only true statements can

set for all of mathematics i.e., find a axiom set that is consistent and complete Gödel: any consistent axiomatic system is incomplete!

26 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!

27 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

28 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 =. Then we can use this to prove that, e.g., all things are equal! Consider the set: {Bush, Pope} {Bush, Pope} = {Bush, Pope} = (since =) Bush = Pope QED All things become true: system is complete but useless!

29 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 (9-), The Prince You can have it good, cheap, or fast pick any two. - Popular business adage

30 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 (99) gave shorter proof based on formalizing Berry s paradox The set of true statements is not R.E.!

32 Gödel s Incompleteness Theorem Systems known to be complete and consistent: Propositional logic (Boolean algebra) Predicate calculus (first-order logic) [Gödel, 90] Sentential calculus [Bernays,9; Post, 9] 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, 9] Q: Is our mathematics at least consistent? A: We don t know! But we sure hope so.

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

34 Continuum Hypothesis Posed by Cantor in Are there any infinities between the naturals and the reals? i s. t. ℵ 0 < ℵ i < ℵ? Consumed Cantor s life Possibly drove him mad Independent of the standard set theory axioms Equivalent to the Axiom of Choice

35 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!

36 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 spheres of the same size as the original

37 Problem: Show that is not rational Assume FSORC that is rational. a, b Z s. t. = a b We will say, without loss of generality, that a is b in simplest form, that is GCD a, b =. = a = a, thus b b a. Since a, it must be that a. However a a, so for = a b to hold, it must be that b b. This means that GCD a, b. Contradiction!

Lecture 14 Rosser s Theorem, the length of proofs, Robinson s Arithmetic, and Church s theorem. Michael Beeson

Lecture 14 Rosser s Theorem, the length of proofs, Robinson s Arithmetic, and Church s theorem Michael Beeson The hypotheses needed to prove incompleteness The question immediate arises whether the incompleteness