Class 34: Proving Unprovability

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Axiomatic Systems
Proving 1+1=2
Russell's Paradox

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Class 34: Proving Unprovability

  1. 1. cs1120 Fall 2011 Class 34:David Evans Proving11 - 11 - 11 Unprovability
  2. 2. PlanExplanation of PS8 OptionsUnprovability Any questions about the interpreter? Comments about Ivan’s lecture Wednesday 2
  3. 3. 3
  4. 4. Problem Set 8 Option J: Option C: Option W: Aazda (Charme) Conveying Web Interpreter in Computing ApplicationJava + static type checking Only Option J automatically satisfies the prerequisite for taking cs2110 this Spring. If you indicated interest in Computer Science major on your PS0 survey you are expected to do Option J. If you prefer to cs2110: do a different option, must provide aSoftware Development Methods convincing reason why. 4
  5. 5. Meta-Circularity?Much of the course so far: Getting comfortable with recursive definitions Learning to write programs to do (almost) anythingStarting today and next week: Getting un-comfortable with recursive definitions Things no program can do!
  6. 6. Computer Science/Mathematics Computer Science (Imperative Knowledge) Monday Are there (well-defined) problems that cannot be solved by any procedure? Mathematics (Declarative Knowledge)Today Are there true conjectures that cannot be the shown using any proof?
  7. 7. Mechanical ReasoningAristotle (~350BC): Organon Codify logical deduction with rules of inference (syllogisms) Every A is a P Premises X is an A X is a P Conclusion Every human is mortal. Gödel is human. Gödel is mortal.
  8. 8. Euclid (~300BC): Elements Newton (1687): Reduce geometry to a few Philosophiæ Naturalisaxioms and derive the rest by Principia Mathematica following rules Reduce the motion of objects (including planets) to following axioms (laws) mechanically
  9. 9. Mechanical Reasoning1800s – mathematicians work on codifying“laws of reasoning” Augustus De Morgan (1806-1871)George Boole (1815-1864) De Morgan’s laws Laws of Thought proof by induction
  10. 10. Bertrand Russell (1872-1970) 1910-1913: Principia Mathematica (with Alfred Whitehead) 1918: Imprisoned for pacifism 1950: Nobel Prize in Literature 1955: Russell-Einstein Manifesto 1967: War Crimes in Vietnam Note: this is the same Russell who wrote In Praise of Idleness!
  11. 11. When Einsteinsaid, “Great spiritshave alwaysencountered violentopposition frommediocre minds.”he was talking aboutBertrand Russell.
  12. 12. All true statements about numbers
  13. 13. Perfect Axiomatic System Derives all true statements, and no false statements starting from a finite number of axioms and following mechanical inference rules.
  14. 14. Incomplete Axiomatic System incomplete Derives some, but not all true statements, and no false statements starting from a finite number of axioms and following mechanical inference rules.
  15. 15. Inconsistent Axiomatic System Derives all true statements, and some false statements starting from a finite number of axioms and following mechanical inference rules. some false statements
  16. 16. Principia Mathematica [1910] 2000 pages Attempted to axiomatizeAlfred Whitehead Bertrand Russell mathematical (1861-1947) (1872-1970) reasoning Claimed to be complete and consistent: All true theorems could be derived No falsehoods could be derived
  17. 17. Proving 1+1 = 2 17
  18. 18. More Understandable ProofDefine the natural numbers 18
  19. 19. More Understandable ProofDefine the natural numbers Peano’ s Postulates: N is the smallest set satisfying these postulates: P1. 1 is in N . P2. If x is in N , then its "successor" (succ x) is in N . P3. There is no x such that (succ x) = 1. P4. If x is not 1, then there is a y in N such that (succ y) = x. P5. If S is a subset of N , 1 is in S, and the implication (x in S=> (succ x) in S) holds, then S=N. 19
  20. 20. Proving 1+1 = 2 N is the smallest set satisfyingDefine +: N × N  N these postulates: 1. 1 is in N . 2. If x is in N , then its "successor" (succ x) is in N . 3. There is no x such that (succ x) = 1. 4. If x is not 1, then there is a y in N such that (succ y) = x. 5. If S is a subset of N , 1 is in S, and the implication (x in S=> (succ x) in S) holds, then S=N. 20
  21. 21. Proving 1+1 = 2 N is the smallest set satisfyingDefine +: N × N  N these postulates: 1. 1 is in N . 2. If x is in N , then its "successor" Call the inputs a and b. (succ x) is in N . 3. There is no x such that (succ x) = 1. If b is equal to 1: 4. If x is not 1, then there is a y in (+ a b) = (succ a) N such that (succ y) = x. 5. If S is a subset of N , 1 is in Otherwise: S, and the implication (x in S=> by P4, there exists (succ x) in S) holds, then S=N. c such that b = (succ c) (+ a b) = (succ (+ a c)) 21
  22. 22. Now the Proof! Definition of (+ a b): If b is equal to 1: (+ a b) = (succ a) Otherwise: by P4, there exists c such that b = (succ c) (+ a b) = (succ (+ a c)) 22
  23. 23. Now the Proof!“2” = (succ 1) Definition of (+ a b): If b is equal to 1:1 + 1 = (succ 1) (+ a b) = (succ a)By definition of +, Otherwise: by P4, there exists 1 + 1 = (succ 1) = “2” c such that b = (succ c) (+ a b) = (succ (+ a c))QED! 23
  24. 24. Bertrand Russell, My Philosophical Development, 1959 24
  25. 25. Russell’s ParadoxSome sets are not members of themselves e.g., set of all JeffersoniansSome sets are members of themselves e.g., set of all things that are non-JeffersonianS = the set of all sets that are not members of themselves Is S a member of itself?
  26. 26. Russell’s ParadoxS = set of all sets that are not members of themselvesIs S a member of itself?
  27. 27. Russell’s ParadoxS = set of all sets that are not members of themselvesIs S a member of itself? If S is an element of S, then S is a member of itself and should not be in S. If S is not an element of S, then S is not a member of itself, and should be in S.
  28. 28. Ban Self-Reference?Principia Mathematica attempted to resolve this paragraph by banning self-referenceEvery set has a type The lowest type of set can contain only “objects”, not “sets” The next type of set can contain objects and sets of objects, but not sets of sets
  29. 29. Russell’s Resolution (?)Set ::= SetnSet0 ::= { x | x is an Object }Setn ::= { x | x is an Object or a Setn - 1 }S: SetnIs S a member of itself? No, it is a Setn so, it can’t be a member of a Setn
  30. 30. Epimenides ParadoxEpidenides (a Cretan): “All Cretans are liars.”Equivalently: “This statement is false.” Russell’s types can help with the set paradox, but not with these.
  31. 31. Gödel’s “Solution”All consistent axiomatic formulations ofnumber theory include undecidablepropositions. undecidable: cannot be proven either true or false inside the system.
  32. 32. The Information, Chapter 6 Kurt GödelBorn 1906 in Brno (now Czech Republic, then Austria-Hungary)1931: publishes Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme (On Formally Undecidable Propositions of Principia Mathematica and Related Systems)
  33. 33. 1939: flees ViennaInstitute for Advanced Study, PrincetonDied in 1978 – convinced everything was poisoned and refused to eat
  34. 34. ChargeToday: Incompleteness: there are theorems that cannot be provenMonday Uncomputability: there are problems that cannot be solved by any algorithmWednesday: PS7 Due

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