Victoria Gitman and Thomas Johnstone. New York City College of Technology, CUNY

Transcription

1 Gödel s Proof Victoria Gitman and Thomas Johnstone New York City College of Technology, CUNY vgitman@nylogic.org tjohnstone@citytech.cuny.edu March 17, 2009 Victoria Gitman and Thomas Johnstone (CUNY) Gödel s Proof March 17, / 21

2 Mathematical Logic Mathematical Logic Logic: Study of reasoning Mathematical logic: study of type of reasoning done by mathematicians examines the methods used by mathematicians Mathematics, as opposed to other sciences, uses proofs instead of observations. impossible to prove all mathematical laws certain first laws, axioms, are accepted without proof the remaining laws, theorems, are to be proved from axioms How do we accept certain axioms? How do we choose reasonable axioms? Non-contradictory axioms? Powerful axioms? What constitutes a proof from a given set of axioms? Which rules do we have do follow at each step? Victoria Gitman and Thomas Johnstone (CUNY) Gödel s Proof March 17, / 21

3 Mathematical Logic Gottlob Frege ( ) In his Begriffsschrift (1879), Frege introduces symbolism for predicate logic invents quantified variables: for all,and there exists makes iterations of and understandable Every boy loves some girl vs. Some girl is loved by all boys invents axiomatic predicate logic In his Grundgesetze der Arithmetik(1893, 1903), Frege attempts to axiomatize the theory of sets. Victoria Gitman and Thomas Johnstone (CUNY) Gödel s Proof March 17, / 21

4 Mathematical Logic Frege s Set Building Axiom For any formal criterion, there exists a set whose members are those objects (and only those objects) that satisfy the criterion. Frege s axioms allows us to build various sets: the set N of all natural numbers the set R = {x : x is a real number} the set I = {x : x is an infinite set} the set of all sets, V = {x : x = x} Some sets are members of themselves, while others are not! Consider the set B of all objects that are not members of themselves, i.e. B = {x : x / x} Question: What s the problem with B? Victoria Gitman and Thomas Johnstone (CUNY) Gödel s Proof March 17, / 21

5 Mathematical Logic Russell s Paradox (1901) Bertrand Russell ( ) discovers that Frege s axioms lead to a contradiction. The key ideas: The set B = {x : x / x} cannot exist. Self-reference: x is not an element of itself Similar to the Liar Paradox (Epimenides, 400 BC): This sentence is false! Russell fixed Frege s system using type theory. This led to the Comprehension Axiom and Zermelo-Fraenkel set theory. Victoria Gitman and Thomas Johnstone (CUNY) Gödel s Proof March 17, / 21

6 The 19th century work of Frege, Russell, Hilbert, Peano, Cantor, etc. leads to development of formal systems: A formal system consists of A formal language Axioms Rules of inference (how to conclude theorems from axioms) Victoria Gitman and Thomas Johnstone (CUNY) Gödel s Proof March 17, / 21

7 Hilbert s Program (1921) David Hilbert ( ) aimed to provide a secure foundation for mathematics. Two Key Questions Consistency: How do we know that contradictory consequences cannot be proved from the axioms? Completeness: What if there are statements that cannot be decided by the axioms? No one shall expel us from the paradise that Cantor has created for us....hilbert Victoria Gitman and Thomas Johnstone (CUNY) Gödel s Proof March 17, / 21

8 Hilbert s Program (continued...) Translate all mathematics into a formal language and demonstrate by finitary means that Peano Axioms (PA) for Number Theory, Zermelo-Fraenkel (ZF) Axioms for Set Theory, Euclidian Axioms for Geometry, Principia Mathematica (PM) Axioms, are consistent and complete! What are Hilbert s finitary means? Problem: natural numbers, the universe of sets, the Euclidian space are not finite! Solution: Strip away the meaning of mathematical assertions! Victoria Gitman and Thomas Johnstone (CUNY) Gödel s Proof March 17, / 21

9 Hilbert s Program (continued...) A mathematical assertion can be viewed in two fundamentally different ways: as a sentence, namely a sequence of letters and symbols, or as the meaning of the sentence. Advantages of the syntactical view: Mathematical concepts are very abstract A sentence, studied as a syntactical object with no meaning, is very concrete. Proofs are finite sequences of sentences that follow a few simple rules. Provability can be studied without any understanding of the underlying subject. Mathematics is a game played according to certain simple rules with meaningless marks on paper....hilbert Victoria Gitman and Thomas Johnstone (CUNY) Gödel s Proof March 17, / 21

10 Crash Course in 1) Logical symbols: Equality: = Boolean connectives:,,, Quantifiers:, 2) Functions, relations, and constants symbols: specific to subject Number Theory: +,, <, 0, 1 Set Theory: Group Theory:, 1, e 3) Variables: x 1, x 2, x 3, x 4,... infinitely many! 4) Punctuation symbols: (, ) For notational convenience, we will write x, y, z instead of x 1, x 2, x 3, respectively. Victoria Gitman and Thomas Johnstone (CUNY) Gödel s Proof March 17, / 21

11 Writing : Formulas Examples of formulas in Number Theory x is even: x divides y: even(x) := y y + y = x x y := z z x = y x is prime: prime(x) := ( y (y x (y = 1 y = x)) x = 1) 3 x = y: suggestions? (problem is that definition is recursive) x y = z: (same problem) 3 is even: x (x = (1 + 1) + 1 even(x)) There are infinitely many primes: x y (y > x prime(y)) Every even number > 2 is the sum of two primes: x ((x > even(x)) y z ((prime(y) prime(z)) x = y + x))) Victoria Gitman and Thomas Johnstone (CUNY) Gödel s Proof March 17, / 21

12 Formulas (continued...) So what is a formula? Slogan: A formula, should if translated to English, correspond to a complete sentence. Really, after stripping away any meaning: A formula is a string of symbols built according to a finite set of simple rules. This string is a formula: (why?) z(z > 0 x + y = z) This string is not a formula: (why not?) x(y z z > 0) What are the rules? Victoria Gitman and Thomas Johnstone (CUNY) Gödel s Proof March 17, / 21

13 Formula Witnessing Sequences Recursive rules for building formulas: Equality statements are formulas: x = y, Less than statements are formulas: x + 1 < z x + y = z z Boolean combinations of formulas are formulas: if ϕ and ψ are formulas, then so are (ϕ ψ), (ϕ ψ), ϕ, (ϕ ψ). A formula with a quantifier-variable pair attached in front is a formula: if i is any natural number and ϕ is a formula and, then so are x i ϕ, x i ϕ. Nothing else is a formula This recursive definition gives rise to formula witnessing sequences. Victoria Gitman and Thomas Johnstone (CUNY) Gödel s Proof March 17, / 21

14 The Peano Axioms Axiomatization of Number Theory proposed by Giuseppe Peano ( ). Peano Axioms Addition and Multiplication x y z (x + y) + z = x + (y + z) x y x + y = y + x x y z (x y) z = x (y z) x y x y = y x x y z x (y + z) = x y + x z. x (x + 0 = x x 1 = x) (associativity of addition) (commutativity of addition) (associativity of multiplication) (commutativity of multiplication) (distributive law) (additive and multiplicative identity) Victoria Gitman and Thomas Johnstone (CUNY) Gödel s Proof March 17, / 21

15 Peano Axioms (continued) Order x y z ((x < y y < z) x < z) x x < x x y ((x < y x = y) y < x) x y z (x < y x + z < y + z) x y z ((0 < z x < y) x z < x z) x y (x < y z (z > 0 x + z = y)) x (x 0 (x > 0 x 1)) Induction Scheme For every formula ϕ(x) we have (ϕ(0) x (ϕ(x) ϕ(x + 1))) xϕ(x) (the order is transitive) (the order is anti-reflexive) (any two elements are comparable) (order respects addition) (order respects multiplication) (the order is discrete) Hilbert s question: Are the Peano Axioms consistent? Are they complete? Victoria Gitman and Thomas Johnstone (CUNY) Gödel s Proof March 17, / 21

16 The Group Theory Axioms: An Easy Example Language:, 1, e Group Theory Axioms x y z x (y z) = (x y) z x (e x = x x e = x) x x x 1 = e (associativity) (e is the identity) ( 1 is the inverse) Question Are the group theory axioms consistent? Victoria Gitman and Thomas Johnstone (CUNY) Gödel s Proof March 17, / 21

17 Group Theory Axioms: (continued...) Hilbert would like: Z 4 e a b c e e a b c a a b c e b b c e a c c e a b Question Are the group theory axioms complete? Victoria Gitman and Thomas Johnstone (CUNY) Gödel s Proof March 17, / 21

18 Propositional Logic Language: Variables: A, B, C,... Boolean connectives:,,, Punctuation symbols: (, ) Think of these variables as standing in for any sentence: A = Bush is a great public speaker B = It is going to rain tomorrow C = This talk is boring Rules for building formulas: A variable is a formula. Boolean combinations of formulas are formulas: (A B), (A B) The Rule of Inference is Modus Ponens: From A and A B, infer B. Propositional Logic Axioms (A A) A A (A B) (A B) (B A) (A B) ((C A) (C B)) Victoria Gitman and Thomas Johnstone (CUNY) Gödel s Proof March 17, / 21

19 Propositional Logic is Consistent The following formulas are provable from the axioms: A A (A A) (A A) B The axioms are inconsistent if for some formula A, both A and A are provable. It suffices to find a single formula that is not provable! Hilbert s Strategy: Find a structural, syntactical property that every derivable formula has. Show that not every possible formula has that property. What is the desired property? The formula is a tautology, i.e. true in all possible worlds Theorem (Hilbert, 1905) Propositional logic is consistent. Victoria Gitman and Thomas Johnstone (CUNY) Gödel s Proof March 17, / 21

20 Presburger Arithmetic Arithmetic without multiplication: Presburger Axioms Addition x 0 = x + 1 x y (x + 1 = y + 1 x = y) x x + 0 = x x y (x + y) + 1 = x + (y + 1) Induction Scheme For every formula ϕ(x) we have (ϕ(0) x (ϕ(x) ϕ(x + 1))) xϕ(x) Mojzesz Presburger ( ) showed in 1929 using finitary arguments that Presburger Arithmetic is consistent and complete! Victoria Gitman and Thomas Johnstone (CUNY) Gödel s Proof March 17, / 21

21 Gödel ends Hilbert s Program Theorem (Gödel, 1931) The Peano Axioms are not complete. In fact, any reasonable collection of axioms for Number Theory or Set Theory is necessarily incomplete. Theorem (Gödel, 1931) No proof of the consistency of the Peano Axioms can be given by finitary means. Victoria Gitman and Thomas Johnstone (CUNY) Gödel s Proof March 17, / 21