Godel's Incompleteness Theorems

Godel's Incompleteness Theorems are among the most famous results in modern mathematics. They are often informally stated as follows:

1. Any formal system that can express elementary arithmetic contains statements that are "true but not provable".

2. Any consistent formal system that can express elementary arithmetic cannot prove its own consistency.

These informal statements, however, often raise more questions than they answer. What exactly is a "formal system"? What does it mean for a statement to be "true but not provable"? What does it mean to "express arithmetic", and why does arithmetic have anything to do with proof or consistency?

This talk will introduce the audience to the main ideas behind the incompleteness theorems. We'll discuss the historical context in which the incompleteness theorems were discovered, develop the concepts necessary to properly understand what the theorems actually say, and present a sketch of the theorems' proof.

Transcript

1. G¨ odel’s Incompleteness Theorems Scott Sanderson (@scottbsanderson) March 8, 2017 Papers We Love - Boston
2. The Stage
3. Hilbert’s Program David Hilbert In 1900, published a list of the 23 most important unsolved problems in mathematics for the 20th century. 1
4. Hilbert’s Program David Hilbert In 1900, published a list of the 23 most important unsolved problems in mathematics for the 20th century. Problem 2 is “prove the consistency of aritmetic”. 1
5. Principia Mathematica and Russell’s Logicism Alfred North Whitehead Bertrand Russell 2
6. The Paper
7. The Paper 3
8. Informal Statement of the Incompleteness Theorems Theorem (G¨ odel’s First Incompleteness Theorem) Let F be a consistent formal system that can express arithmetic. There are statements well-formed statements in the language of F which are neither provable nor disprovable in F. 4
9. Informal Statement of the Incompleteness Theorems Theorem (G¨ odel’s First Incompleteness Theorem) Let F be a consistent formal system that can express arithmetic. There are statements well-formed statements in the language of F which are neither provable nor disprovable in F. Theorem (G¨ odel’s Informal Statement of the Second Incompleteness Theorem) Let F be a consistent formal system that can express arithmetic. F cannot prove its own consistency. 4
10. lolwut? What do we mean by “formal system”? 5
11. lolwut? What do we mean by “formal system”? What does it mean for F to be “consistent”? 5
12. lolwut? What do we mean by “formal system”? What does it mean for F to be “consistent”? How can a statement be neither provable nor disprovable? 5
13. lolwut? What do we mean by “formal system”? What does it mean for F to be “consistent”? How can a statement be neither provable nor disprovable? What would it mean for F to prove its own consistency? 5
14. What is a Formal System? A formal system F is: • An alphabet of symbols (e.g. 0, , ¬, ∃, ∀, →). • A grammar, describing the strings of symbols that are well-formed sentences. • A set of axioms containing initial theorems of F. • A set of rules of inference describing how to transform existing theorems into new theorems. 6
15. Robinson Arithmetic Robinson Arithmetic (Q) is a simple axiomatization of arithmetic on the natural numbers. 7
16. Robinson Arithmetic Robinson Arithmetic (Q) is a simple axiomatization of arithmetic on the natural numbers. Defines + and × in terms of successor . 7
17. Robinson Arithmetic Robinson Arithmetic (Q) is a simple axiomatization of arithmetic on the natural numbers. Defines + and × in terms of successor . 0 is the only constant. Other numeric constants written as 0 ... . 7
18. Robinson Arithmetic Symbols 0, , ¬, =, +, ×, ∧, ∨, ∀, ∃, →, (, ) a, b, c, . . . , x, y, z, . . . (as many as needed) 8
19. Robinson Arithmetic Well-Formed Sentences 0 + 0 = 0 0 + 0 = 0 ∀x(x + 0 = x) ∀x∃y(y + y = x) Invalid Sentences 0 = +xy ∀∃ 9
20. Robinson Arithmetic Axioms 1. ∀x(¬(0 = x )) 10
21. Robinson Arithmetic Axioms 1. ∀x(¬(0 = x )) 2. ∀x∀y(x = y → x = y) 10
22. Robinson Arithmetic Axioms 1. ∀x(¬(0 = x )) 2. ∀x∀y(x = y → x = y) 3. ∀x(¬(x = 0) → ∃y(x = y )) 10
23. Robinson Arithmetic Axioms 1. ∀x(¬(0 = x )) 2. ∀x∀y(x = y → x = y) 3. ∀x(¬(x = 0) → ∃y(x = y )) 4. ∀x(x + 0 = 0) 10
24. Robinson Arithmetic Axioms 1. ∀x(¬(0 = x )) 2. ∀x∀y(x = y → x = y) 3. ∀x(¬(x = 0) → ∃y(x = y )) 4. ∀x(x + 0 = 0) 5. ∀x((x + y ) = (x + y) ) 10
25. Robinson Arithmetic Axioms 1. ∀x(¬(0 = x )) 2. ∀x∀y(x = y → x = y) 3. ∀x(¬(x = 0) → ∃y(x = y )) 4. ∀x(x + 0 = 0) 5. ∀x((x + y ) = (x + y) ) 6. ∀x(x × 0 = 0) 10
26. Robinson Arithmetic Axioms 1. ∀x(¬(0 = x )) 2. ∀x∀y(x = y → x = y) 3. ∀x(¬(x = 0) → ∃y(x = y )) 4. ∀x(x + 0 = 0) 5. ∀x((x + y ) = (x + y) ) 6. ∀x(x × 0 = 0) 7. ∀x∀y(x × y = (x × y) + x) 10
27. Robinson Arithmetic Rules of Inference p, (p → q) q ¬¬p → p p ∧ q p (p → q) (¬q → ¬p) . . . 11
28. Consistency A formal system F is inconsistent if, for some proposition p: F p and F ¬p 12
29. Consistency A formal system F is inconsistent if, for some proposition p: F p and F ¬p A formal system is consistent if it is not inconsistent. 12
30. The First Theorem
31. The First Incompleteness Theorem Theorem (First Incompleteness Theorem) Let F be a consistent formal system that contains Q. Then we can construct a sentence GF such that: 1. F GF . 2. F ¬GF . 13
32. Liar Paradox Consider the sentence “This sentence is false.”. 14
33. Liar Paradox Consider the sentence “This sentence is false.”. If it’s true, then by its own assertion it must be false. 14
34. Liar Paradox Consider the sentence “This sentence is false.”. If it’s true, then by its own assertion it must be false. If it’s false, then, by its own assertion, it must be true. 14
35. Liar Paradox Consider the sentence “This sentence is false.”. If it’s true, then by its own assertion it must be false. If it’s false, then, by its own assertion, it must be true. Many answers have been proposed for how to understand the logical content of this sentence. 14
36. Proof Paradox Consider the sentence “This sentence is not provable”. 15
37. Proof Paradox Consider the sentence “This sentence is not provable”. If it’s provable, then we have a proof that the sentence isn’t provable. 15
38. Proof Paradox Consider the sentence “This sentence is not provable”. If it’s provable, then we have a proof that the sentence isn’t provable. If it’s not provable, then we don’t have such a proof. 15
39. Proof Paradox Consider the sentence “This sentence is not provable”. If it’s provable, then we have a proof that the sentence isn’t provable. If it’s not provable, then we don’t have such a proof. If our deductive system is consistent, then the sentence is true, which means we can’t prove that it’s true 15
40. Proof Idea for First Incompleteness Theorem Proof Idea Construct a sentence in Q that asserts its own unprovability. 16
41. Representability Definition (Weak Representability) A set S of natural numbers is weakly representable in F if there exists a formula A(x) in the language of F such that: n ∈ S ⇐⇒ F A(n) 17
42. Representability Definition (Strong Representability) A set S of natural numbers is strongly representable in F if there exists a formula A(x) in the language of F such that: n ∈ S =⇒ F A(n) n / ∈ S =⇒ F ¬A(n) 18
43. Representability (in plain English) Weak representability means that we can write down a formula A such that F proves A(x) whenever x ∈ S. 19
44. Representability (in plain English) Weak representability means that we can write down a formula A such that F proves A(x) whenever x ∈ S. Strong representability means that can write down a formula A such that F proves A(x) whenever x ∈ S and F proves ¬A(x) whenever x / ∈ S. 19
45. Representability (in plain English) Weak representability means that we can write down a formula A such that F proves A(x) whenever x ∈ S. Strong representability means that can write down a formula A such that F proves A(x) whenever x ∈ S and F proves ¬A(x) whenever x / ∈ S. We can define representability for relations analogously using formulae with more than one free variable. 19
46. Examples of Representability Even(x) ≡ ∃y(x = y × 0 ) Odd(x) ≡ ∃y(x = y ∧ Even(y)) LessThan(x, y) ≡ ∃n(x + n = y) Divides(x, y) ≡ ∃n(n × y = x) Prime(x) ≡ ∀y((Divides(x, y) → (y = 0 ∨ y = x))) 20
47. G¨ odel Numbering Problem How to make assertions about provability of Q-sentences in Q? 21
48. G¨ odel Numbering Problem How to make assertions about provability of Q-sentences in Q? Idea Encode assertions about proofs as assertions about numbers. 21
49. G¨ odel Numbering Problem How to make assertions about provability of Q-sentences in Q? Idea Encode assertions about proofs as assertions about numbers. Requires that we can produce a correspondence between numbers and sentences. 21
50. G¨ odel Numbering Assign a positive integer to each symbol in the alphabet of the language of F. C(0) = 1 C( ) = 2 C(+) = 3 C(×) = 4 C(=) = 5 C(() = 6 C()) = 7 C(→) = 8 C(¬) = 9 C(∀) = 10 C(∧) = 11 C(∨) = 12 C(a) = 13 C(b) = 14 . . . C(xi ) = 12 + i 22
51. G¨ odel Numbering Encode sequences of symbols in the exponents of the powers of prime numbers. 0 + 0 = 0 = 2C(0)3C(+)5C07C(=)11C(0) = 21335175111 = 4537890 23
52. G¨ odel Numbering Encode sequences of symbols in the exponents of the powers of prime numbers. 0 + 0 = 0 = 2C(0)3C(+)5C07C(=)11C(0) = 21335175111 = 4537890 We can use similar techniques to encode sequences of sequences. 23
53. Encoding Proof Theorem (Valid Proofs are Strongly Representable in Q) There is a formula of F, ProofF (x, y) that strongly represents the relation: x is the G¨ odel number of a proof of a formula with G¨ odel number y 24
54. Encoding Provability Corollary (Provability is Weakly Representable in Q) There is a formula of F, ProvableF (x), that weakly represents the relation: x is the G¨ odel number of a provable sentence. 25
55. Encoding Provability Corollary (Provability is Weakly Representable in Q) There is a formula of F, ProvableF (x), that weakly represents the relation: x is the G¨ odel number of a provable sentence. ProvableF (x) ≡ ∃y(ProofF (x, y)) 25
56. Encoding Self-Reference Theorem (Diagonalization Lemma) Let A(x) be a formula in the language of F with one free variable. We can construct a sentence DA such that: F (DA ↔ A( DA )) We refer to DA as the diagonalization of A. 26
57. Proof of the First Incompleteness Theorem Let GF be the diagonalization of ¬ProvableF (x). 27
58. Proof of the First Incompleteness Theorem Let GF be the diagonalization of ¬ProvableF (x). By the Diagonalization Lemma: F (GF ↔ ¬ProvableF ( GF )) (1) 27
59. Proof of the First Incompleteness Theorem By the Diagonalization Lemma: F (GF ↔ ¬ProvableF ( GF )) (1) Suppose F GF : By weak representability of ProvableF : F ProvableF ( GF ) 27
60. Proof of the First Incompleteness Theorem By the Diagonalization Lemma: F (GF ↔ ¬ProvableF ( GF )) (1) Suppose F GF : By weak representability of ProvableF : F ProvableF ( GF ) but by (1), F ¬ProvableF ( GF ) 27
61. Proof of the First Incompleteness Theorem By the Diagonalization Lemma: F (GF ↔ ¬ProvableF ( GF )) (1) Suppose F GF : By weak representability of ProvableF : F ProvableF ( GF ) but by (1), F ¬ProvableF ( GF ) So, if F is consistent, F GF . 27
62. Proof of the First Incompleteness Theorem Suppose F ¬GF : No n is the G¨ odel number of a proof of GF , so by strong representability of ProofF , for every numeral n: F ¬ProofF (n, GF ) 27
63. Proof of the First Incompleteness Theorem Suppose F ¬GF : No n is the G¨ odel number of a proof of GF , so by strong representability of ProofF , for every numeral n: F ¬ProofF (n, GF ) We’d like to conclude that: F ¬∃n(Proof (n, GF )) ∴ F ¬Provable( GF ) 27
64. Proof of the First Incompleteness Theorem Suppose F ¬GF : No n is the G¨ odel number of a proof of GF , so by strong representability of ProofF , for every numeral n: F ¬ProofF (n, GF ) We’d like to conclude that: F ¬∃n(Proof (n, GF )) ∴ F ¬Provable( GF ) 27
65. Proof of the First Incompleteness Theorem Suppose F ¬GF : No n is the G¨ odel number of a proof of GF , so by strong representability of ProofF , for every numeral n: F ¬ProofF (n, GF ) We’d like to conclude that: F ¬∃n(Proof (n, GF )) ∴ F ¬Provable( GF ) but this actually requires a slightly stronger assumption ω-consistency. 27
66. Consistency vs. ω-Consistency A formal theory is consistent if it is not the case that for some A: F A F ¬A 28
67. Consistency vs. ω-Consistency A formal theory is consistent if it is not the case that for some A: F A F ¬A A formal theory is ω-consistent if is is not the case that for some A(x): F A(n) for every numeral n. F ∃n(¬A(n)) 28
68. Consistency vs. ω-Consistency A formal theory is consistent if it is not the case that for some A: F A F ¬A A formal theory is ω-consistent if is is not the case that for some A(x): F A(n) for every numeral n. F ∃n(¬A(n)) ω-consistency is a stronger condition, and is what G¨ odel used in his original proof. 28
69. Final Statement of the First Incompleteness Theorem Theorem (G¨ odel’s First Incompleteness Theorem) Let F be a formal system that contains Q. Then we can construct a sentence GF such that: 1. If F is consistent, F GF . 2. If F is ω-consistent, F ¬GF . 29
70. Review of First Incompleteness Theorem Summary: • We can encode statements about statements in arithmetic by encoding statements as numbers. 30
71. Review of First Incompleteness Theorem Summary: • We can encode statements about statements in arithmetic by encoding statements as numbers. • We can express “x is a proof of y” (suitably encoded) in arithmetic. 30
72. Review of First Incompleteness Theorem Summary: • We can encode statements about statements in arithmetic by encoding statements as numbers. • We can express “x is a proof of y” (suitably encoded) in arithmetic. • We can express “x is provable” as “∃y s.t. y is a proof of x”. 30
73. Review of First Incompleteness Theorem Summary: • We can encode statements about statements in arithmetic by encoding statements as numbers. • We can express “x is a proof of y” (suitably encoded) in arithmetic. • We can express “x is provable” as “∃y s.t. y is a proof of x”. • Diagonalization transforms a predicate into a statement that is provable in iff its G¨ odel numbers satisfies the predicate. 30
74. Review of First Incompleteness Theorem Summary: • We can encode statements about statements in arithmetic by encoding statements as numbers. • We can express “x is a proof of y” (suitably encoded) in arithmetic. • We can express “x is provable” as “∃y s.t. y is a proof of x”. • Diagonalization transforms a predicate into a statement that is provable in iff its G¨ odel numbers satisfies the predicate. • Applying diagonalization to “x is not provable” yields a statement that’s provable only if it’s not provable. 30
75. The Second Theorem
76. The Second Incompleteness Theorem Theorem (G¨ odel’s Second Incompleteness Theorem) Let F be a consistent formal system that contains Peano Arithmetic, and define: ConsF ≡ ¬ProvableF (0 = 0 ) then F ConsF 31
77. The Second Incompleteness Theorem Theorem (G¨ odel’s Second Incompleteness Theorem) Let F be a consistent formal system that contains Peano Arithmetic, and define: ConsF ≡ ¬ProvableF (0 = 0 ) then F ConsF In other words, if F is consistent, then F can’t prove that it doesn’t prove contradictions. 31
78. Proof Sketch Main Idea: Let GF be the G¨ odel sentence constructed in the proof of the first theorem. Prove F (ConsF ↔ GF ) By the first theorem, if F is consistent, then F GF . Therefore, F ConsF . 32
79. Proof Sketch How do we show that F proves the equivalence of ConsF and GF ? 33
80. Proof Sketch How do we show that F proves the equivalence of ConsF and GF ? Essentially, the idea is to translate the argument from theorem 1 into the language of F. 33
81. Proof Sketch How do we show that F proves the equivalence of ConsF and GF ? Essentially, the idea is to translate the argument from theorem 1 into the language of F. We show that F (¬GF → ⊥ → ¬ConsF ), which implies that F ConsF → GF . 33
82. Challenges Derivation requires L¨ ob’s Derivability Conditions (or equivalent) to be shown for the provability predicate ProvableF (abbreviated here as ProvF ): F A =⇒ ProvF ( A ) (1) F ProvF ( A ) → ProvF ( ProvF ( A ) ) (2) F ProvF ( A ) ∧ ProvF ( A → B ) → ProvF ( B ) (3) 34
83. Epilogue
84. Implications for Hilbert’s Program David Hilbert Second Theorem rules out proving consistency of infinitistic mathematics using finitistic means. 35
85. Implications for Hilbert’s Program David Hilbert Second Theorem rules out proving consistency of infinitistic mathematics using finitistic means. Still possible to prove arithmetic consistent via external means. 35
86. Implications for Hilbert’s Program David Hilbert Second Theorem rules out proving consistency of infinitistic mathematics using finitistic means. Still possible to prove arithmetic consistent via external means. Proven consistent by Gentzen in 1936 using transfinite induction. 35
87. Implications for Hilbert’s Program David Hilbert It is likely that all mathematicians ultimately would have accepted Hilbert’s approach had he been able to carry it out successfully. The first steps were inspiring and promising. But then G¨ odel dealt it a terrific blow (1931), from which it has not yet recovered. — Hermann Weyl (1946) 35
88. Implications for Logicism Alfred North Whitehead Bertrand Russell • Logicism went into decline after G¨ odel’s proofs were announced. 36
89. Thanks! Questions? 37