G ö del’s Theorem

1. Gödel’s Theorem Carmen Serrano

2. Overview • Kurt Gödel • 20th century mathematical logician • Incompleteness Theorem • Prove incompleteness of formal systems

3. Life • Born in 1906 in Austria-Hungary • Early interest in mathematics • Studies in University of Vienna • Publishes famous theorems at 25 • Nazi Germany annexation of Austria • Institute for Advanced Study, Princeton

4. History • Formalized system for mathematics • Based on axioms • Works by Russell, Whitehead, Hilbert • Widely believed to be possible

5. The theorem • “In any sound, consistent, formal system containing arithmetic there are true statements that cannot be proved…” • …solely with axioms from that system

6. Gödel’s Method • Three steps: • 1. Set up axioms and rule of inference for predicate calculus • 2. Set up axioms for arithmetic in predicate calculus • 3. Establish numbering system for formulas

7. Predicate Calculus • Ax - “For all values of x…” • Ex – “There exists an x…” • P, Q, … – formulas • x – variable •  - implication; “if… then…” • ┐ - “not” operator • & - “and” operator

8. Standard Arithmetic • Natural numbers: 0, 1, 2, 3, … • Addition and multiplication • Successor function, s: • sx = x+1

9. The axioms • Six axioms and a rule of inference for predicate calculus • If F and F  G, then G. • Nine axioms and rule of induction for arithmetic • (P(0) & Ax(P(x)P(sx)))  AxP(x)

10. Symbol 0 + = x Code Number 1 3 5 9 Gödel Numbering

11. Computing a Gödel Number • x + 0 = x • 2^9 * 3^3 * 5^1 * 7^5 * 11^9 • Gödel number: 512*27*5*16807*2357947691 • A very large number! • Expression can be extracted from number

12. Proof (x,y,z) • Representation from textbook • Proof X: Proof(x,y,z) • x – Gödel number of proof X • y – Gödel number of formula Y • Y – formula being proven • z – integer substituted in formula Y

13. Proof(x,y,z) • Actual expression is more complicated • Textbook version is concise: • “x is the Gödel number of a proof X of a formula Y (with one free variable and Gödel number y) which has the integer z substituted into it.” • Making sense?

14. Proof (x,y,z) • Review: • Proof X: Proof(x,y,z) • x – Gödel number of proof X • y – Gödel number of formula Y • Y – formula being proven • z – integer substituted in formula Y

15. Gödel’s Theorem • Gödel’s Theorem: • ┐ExProof(x,g,g) is true but not provable in the formal arithmetic system. • Proof(x,g,g): “x is the Gödel number of a proof of the formula obtained by substituting its own Gödel number g for its one free variable”

16. The Proof • Suppose ┐ExProof(x,g,g) can be proven • Using axioms and rules of system • Call proof P, with Gödel number p • Proof P: Proof(p,g,g) • P is true since we supposed it was provable • But Proof(p,g,g) contradicts ┐ExProof(x,g,g) • Therefore P can’t exist • Original claim is true – doesn’t have a proof • Proof with Universal Truth Machine

17. Gödel’s Theorem • Consistent – true statements are not contradicted • Complete – true statements can be proven • “Within any sound, consistent, formal system containing arithmetic there are true statements that cannot be proved [within that system].” • Consistent vs. Complete

18. Implications • Major discovery in mathematical theory • Limits of machine intelligence • Also contributed to theory of recursive functions • Cognitive science theory

19. Sources • The New Turing Omnibus – A.K. Dewdney • http://www-history.mcs.st-andrews.ac.uk/history/Biographies/Gödel.html • http://www.time.com/time/time100/scientist/profile/Gödel.html • http://plato.stanford.edu/entries/goedel/ • http://www.britannica.com/eb/article-9037162/Kurt-Gödel • http://diglib.princeton.edu/ead/eadGetDoc.xq?id=/ead/mss/C0282.EAD.xml&query=kw%3A(Gödel)

20. Homework • 1. Calculate the Gödel number for: • 1 + 0 = 1 • Hint: See pg. 32 of textbook. • 2. Briefly explain Gödel’s theorem or a component of the theorem as best as you can.