G

1
Gödels 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
Gödels 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 x1

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
Gödel Numbering

• Symbol
• 0
• x

• Code Number
• 1
• 3
• 5
• 9

11
Computing a Gödel Number

• x 0 x
• 29 33 51 75 119
• Gödel number 512275168072357947691
• 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

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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
Gödels Theorem

• Gödels 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

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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 cant exist
• Original claim is true doesnt have a proof
• Proof with Universal Truth Machine

17
Gödels 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

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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/Bi
  ographies/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/e
  ad/mss/C0282.EAD.xmlquerykw3A(Gödel)

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Homework

• 1. Calculate the Gödel number for
• 1 0 1
• Hint See pg. 32 of textbook.
• 2. Briefly explain Gödels theorem or a component
  of the theorem as best as you can.