Background Regarding The Incompleteness Theorems
Modern logic began in the mid 19th century with the work of Boole, DeMorgan, and Jevons. Their goal was to develop a “mathematics of logic,” rather than thinking of logic as conceptually prior to mathematics. At the time, this offered little insight into logic or mathematics, instead leading to mathematical problems regarding, for example, Boolean algebras. 2
Perceived Need for a “Foundation” for Mathematics Causes for concern about the reliability of mathematical results—late 19th century: • Persistent false theorems: For 50 years, a function was differentiable wher- ever continuous. Reaction: Formulate definitions and theorems precisely in a stylistically formal language. • Inability to make sense of uses of infinity in Calculus—infinitesimals and mysticism. Reaction: Weierstrass; Hilbert and Formalism. 3
Formalists sought to reduce all uses of the infinite in mathematics to the finite, using that proofs are finite witnesses that mathematical statements, even ones about infinite objects, are true. This approach puts a premium on (1) precise rules for what counts as a proof, (2) the consistency of formal systems, and (3) their strength in settling mathematical questions, that is, their completeness . 4
Further perceived causes for concern: • Counterintuitive theorems in analysis. Native intuition is an unreliable guide. • Consistency of non-Euclidean geometry. Truth in the physical world cannot cannot be relied upon to correct errant mathematics. • Inadvertent use of the Axiom of Choice At the time the strength and consistency of the Axiom of Choice were unknown. Some viewed it as an unjustified existence assertion. 5
Frege’s Begriffschrift (1879) was a breakthrough—the first full fledged account of propositional and quantifier logic. Frege was obscure and his ideas were revolutionary. It took 20 years for the importance of his work to be appreciated. Simultaneously, many worked to reduce mathematics to logic: • Peano (1889): Arithmeticus Principia . Development of arithmetic more or less from a sort of class theory. In modern terms: logical and mathematical notions are confounded. • Dedekind (1890): Principles of Arithmetic . “Construction” of the natural numbers, rationals and reals . 6
• Hilbert: Axiomatization of Geometry (1899); reduction of geometry to an axiomatization of the reals (1900) • Norbert Wiener (1914): reduction of relations to classes • Papers of Hilbert: 1917, 1922, 1925, 1927, 1928, 1931 on Foundations Russell, et al. : Logicism Hilbert, et al. : Formalism Brouwer, et al. : Intuitionism 7
• Culmination of Logicism: Russell (1908): Theory of Types Russell & Whitehead (1910): Principia Mathematica 8
From the start, problems were evident: • Burali-Forti paradox (1897) • Russell’s paradox (1902) • Richard’s paradox (1905) • K¨ onig’s paradox (1905) Cantor, Russell, Zermelo, Hilbert, and others proposed solutions. 9
Hilbert’s Program sought to reduce all of mathematics to Arithmetic, then to prove the consistency of a formal theory of Arithmetic within that same formal theory. Good progress was made towards reducing all of mathematics to arithmetic. Gentzen proved the consistency of arithmetic—but not from just arithmetic. 100 years later, it seems unlikely that all mathematics is actually finite, but this is still controversial. Nevertheless, proofs of mathematical statements are finite. In 1931 G¨ odel’s Incompleteness Theorems showed that Hilbert’s plan to prove the foundational consistency of arithmetic within arithmetic would have to be dramatically changed in order to succeed. 10
Informal Effective Computability A set X of finite objects is effectively decidable when there is an algorithm that, on input x , answers in finitely many steps whether x ∈ X or x / ∈ X . A set X of finite objects is effectively enumerable if there is an algorithm that lists the elements of X (given an infinite amount of time). A set X of finite objects is semi-decidable if there exists an algorithm that, on input x recognizes that x ∈ X , if it is, but spins on forever if x / ∈ X . Effectively enumerable = semi-decidable. Both X and its complement are semi-decidable ⇔ X is effectively decidable. 11
Algorithms Instructions are provided to an abstract computing agent and • instructions are finite; • execution of these instructions yields certain results (presumably precluding appeal to random events or divine oracles); and • the computing agent is unhindered by limitations of space and time in performing computations. Nevertheless, any completed computation must be finite. 12
Precise Versions of Effective Decidability • Recursive functions (Kleene) • Finite equation systems (Herbrand, Kleene) • Arithmetical representability (G¨ odel, Tarski, Robinson) • Turing machines (Turing, Post) • Flowchart computability (Goldstine, Von Neumann) • Structured programming languages • λ -calculus (Kleene, Church) 13
Church’s Thesis All attempts to make precise the informal notion of effective decidability turned out to be equivalent. Church’s Thesis A set of natural numbers is effectively decidable iff it is recursive. 14
Evidence: (1) Every known “effectively computable” function is known to be recursive (and the proof is never deep). (2) Every proposed precise version of “effectively computable function” is equiv- alent to recursive function . (3) The various precise formulations of “effectively computable” are not only equivalent, but “algorithmically equivalent”, providing evidence that this is not a coincidence. 15
Incompleteness Language of Arithmetic: L = { + , ·, 0 , � , S} Let N be the standard model of arithmetic. Associate a number � φ � to each formula φ of L in a systematic way. Say that a theory T of L is recursively axiomatized when { � φ � : φ ∈ T } is recursive. First Incompleteness Theorem (sub-optimal version) There exists a finite set Q of axioms such that � N Q and if T ⊇ Q is recursively axiomatized and consistent, then T is incomplete. 16
It follows that if T ⊇ Q is a recursively axiomatized theory that is true in N , then T is not complete. Church’s Theorem If T ⊇ Q is consistent, then T HM T = { � φ � : T ⊢ φ } is not recursive. Corollary In L , { � φ � : � φ } is not recursive. Proof: � � Q → φ iff Q ⊢ φ 17
Landmarks in the Proof Arithmetization of Syntax If T is a recursively axiomatized theory of L , then P R T = { ( � φ � , b ) : b codes a proof of φ from T } is recursive and T HM T = { � φ � : T ⊢ φ } is recursively enumerable. 18
If a ∈ � , let a be S a 0. Say that A ⊆ � k is representable in T if there exists a formula φ ( x 1 , . . . , x k ) such that ( a 1 , . . . , a k ) ∈ A ⇒ T ⊢ φ ( a 1 , . . . , a k ) ( a 1 , . . . , a k ) / ∈ A ⇒ T ⊢ ¬ φ ( a 1 , . . . , a k ) Representability Theorem For A ⊆ � k , A is recursive iff A is representable in Q . 19
Second Incompleteness Theorem Let T be recursively axiomatized theory in L . Let φ ( x, y ) represent in Q the recursive set of pairs P R T = { ( � φ � , b ) : b codes a proof of φ from T } . Let Con T = ¬∃ yφ ( � 0 � = 0 � , y ) Then � N Con T iff T is consistent. 2nd Incompleteness Theorem If T is a recursively axiomatized consistent extension of Peano Arithmetic, then T �⊢ Con T . 20
Proof Sketches Church’s Theorem If T ⊇ Q is consistent, then T HM T = { � φ � : T ⊢ φ } is not recursive. Let D IAG = { � φ � : T �⊢ φ ( � φ � ) } . Assume for a contradiction that T HM T is recursive. Then D IAG is recursive. Let δ ( x ) represent D IAG in Q : a ∈ D IAG ⇒ Q ⊢ δ ( a ) a / ∈ D IAG ⇒ Q ⊢ ¬ δ ( a ) 21
For any a : a ∈ D IAG ⇔ Q ⊢ δ ( a ) ⇔ T ⊢ δ ( a ) T is consistent ⇔ � δ ( a ) � ∈ T HM T Recall D IAG = { � φ � : T �⊢ φ ( � φ � ) } . For a = � δ � : a ∈ D IAG ⇔ T �⊢ δ ( a ) ⇔ � δ ( a ) � / ∈ T HM T � 22
First Incompleteness Theorem There exists a finite set Q of axioms such that � N Q and if T ⊇ Q is recursively axiomatized and consistent, then T is incomplete. For a contradiction, suppose that T ⊇ Q is complete, consistent, and recursively axiomatized. Since T is recursive, T HM T is recursively enumerable. Since T is complete, � φ � / ∈ T HM T iff � ¬ φ � ∈ T HM T . So � \ T HM T is recursively enumerable. So T HM T is recursive, contradicting Church’s Theorem. 23
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