Incompleteness – A very rich dessert

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Knights and KnavesA tribute to Raymond Smullyan

Raymond Merill Smullyan (born 1919). American logician, ma-thematician, concert pianist, Taoist philosopher, and magician.Many books on logic puzzles, among them: What is the Nameof This Book? (1978), Forever Undecided (1987).First-Order Logic (1968), Set Theory and the ContinuumProblem (1996), Gödel’s Incompleteness Theorems (1992), . . .

Suppose you are in Smullyan-land, where knights say always thetruth and knaves always lie. You meet someone who tells you“I am not a knight”.What kind of person is she?

As you will (hopefully) see soon:This puzzle contains the essence of Gödel’s (first) incompleteness theorem!

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Another Gödelian Puzzle (with nods to Tarski)Imagine a computing machine that can print strings based on the alphabet:

¬,P,N, (, )A string is called printable, if the machine can print it.The machine is programmed to print all printable eventually.Def. 1: The norm of string w is the string w(w).Def. 2: A sentence is a string of form P(w),PN(w),¬P, or ¬PN(w).Def. 3: P(w) is called true iff w is printable.

¬P(w) is called true iff w is not printable.PN(w) is called true iff the norm of w is printable.¬PN(w) is called true iff the norm of w is not printable.

Presuming that the machine never prints non-true sentences,can it print all true sentences?Note:

Defs. 1 and 2 are purely syntactic; Def. 3 concerns semantics.“true” is a precisely defined, technical term here:check yourself by replacing “true” by “grmph”! 4

Enter the heroes . . .Kurt Gödel (1906-1978). Austrian-American logician, ma-thematician and philosopher. Established the completeness offirst-order logic in his PhD thesis (1929). Proved his two fa-mous incompleteness results in 1931. Many other importantcontributions to logic and philosophy, e.g.: consistency of thecontinuum hypothesis in ZF set theory.

Alfred Tarski (1901-1983). Polish-American mathematicianand logician. Best known for his work on model theory, me-tamathematics, and algebraic logic. Important contributionsalso to abstract algebra, topology, geometry, measure theory,set theory, and analytic philosophy.

Note: The modern conception of logic, in particular as used in computerscience today, is largely to due to Tarski and Gödel (– who built on Hilbert,Frege, Skolem, Gentzen, Boole, Bolzano, and many other “heroes”).

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Semi-formal statements of Gödel’s and Tarski’s TheoremsFirst Incompleteness Theorem (Gödel [Rosser]):Every [ω-]consistent and reasonably expressive system of arithmeticcontains sentences that are neither provable nor refutable.

First Incompleteness Theorem (with shades of Tarski):Every correct and reasonably expressive system of arithmeticcontains true, but unprovable sentences.

Second Incompleteness Theorem (Gödel):No [ω-]consistent and sufficiently strong system of arithmeticcan prove its own consistency.

Undefinability Theorem (Tarski):No correct and reasonably expressive system of arithmeticcan define the set of (arithmetically) true sentences.

Important: To be handled with care due to remaining informality! 6

An abstract form of Gödel’s and Tarski’s TheoremsDefinition.A system Σ is a set containing the following components:E . . . expressions of Σ,S . . . sentences of Σ (S ⊆ E),T . . . true sentences of Σ (T ⊆ S),P . . . provable sentences of Σ (P ⊆ S),R . . . refutable sentences of Σ (R ⊆ S),H . . . predicates of Σ (H ⊆ E),a function Φ : E × N 7→ E : If E ∈ H then Φ(E , n) = E (n) ∈ S.

By a number-set A we mean any set of natural numbers (A ⊆ N ).Its complement (w.r.t. N ) will be denoted by A.Definition.A predicate H ∈ H expresses a number-set A if for every n ∈ N :

n ∈ A ⇐⇒ H(n) ∈ TI.e., A is expressible if A = {n | H(n) ∈ T } for some predicate H. 7

Expressibility, correctness, completenessNote: Expressibility only concerns “truth”, but not “provability”.This should be contrasted with the related notion of representability of Aby H within a (proof) system, defined by n ∈ A ⇐⇒ H(n) ∈ P.We will work with countable, even finite alphabets and consequently thelanguage/system is always countable. Therefore it is clear that not allnumber sets can be expressed. (Remember Cantor’s diagonal argumentthat demonstrates the uncountability of R!)

Definition (Correctness [aka. soundness]).A system is correct iff all provable sentences are true (P ⊆ T ).Definition (Completeness).A system is complete iff all true sentences are provable (T ⊆ P).Note:It is trivial to present sound systems or complete systems for, e.g.,arithmetic. However Gödel (in Tarski’s interpretation) proved:No (sufficiently strong) system for arithmetic is sound and complete. 8

Gödel numbering and diagonalizationNote (on self-reference):Strings of a language can always be coded as natural numbers.Therefore every concrete arithmetic system (also) “talks about itself”.Definition (Gödel number).A 1-1-function p·q : E 7→ N is called Gödel numbering.For any expression E ∈ E , pEq is called its Gödel number.We write En if n = pEq. (Thus, pEnq = n.)Definition (Diagonalization).By the diagonalization of an expression En we mean En(n)[= E (pEq)].The function d : N 7→ N that maps any n into pEn(n)q is calleddiagonal function of the system.For any number-set A we denote d−1(A) = {n | d(n) ∈ A} by A∗.– Remember: if En is a predicate, then En(n) is a sentence.– A∗ collects the Gödel numbers of exactly those expressions, who’sdiagonalizations are named (by their Gödel number) in A.

Notation: We will use P to denote {pSq | S ∈ P}. 9

An abstract form of Gödel’s (first) incompleteness theorem

Theorem GT (After Gödel with a shade of Tarski)If Σ is correct and P∗ is expressible in it, then Σ is incomplete.

Proof. Suppose H expresses P∗; let h = pHq.Let G be the diagonalization of H, i.e. G is the sentence H(h).We show that G is true but not provable in Σ.For all n ∈ N : H(n) is true (i.e., ∈ T ) iff n ∈ P∗. In particular(by “diagonalizing”): G = H(h) is true iff h ∈ P∗. Now observe:

h ∈ P∗ ⇐⇒ d(h) ∈ P ⇐⇒ d(h) 6∈ P.

But d(h) = pH(h)q = pGq. Therefore d(h) ∈ P means (via Gödelization)“G is provable in Σ” and d(h) 6∈ P means “G is not provable in Σ”.Summing up:

G is true ⇐⇒ G is not provable in Σ.

Since Σ is correct, G cannot be non-true and provable.Therefore we have obtained a sentence that witnesses the incompletenessof Σ: G is true but unprovable in Σ. QED. 10

Towards concrete systems: expressiveness conditionsAccording to Theorem GT, we can establish the incompleteness of acorrect system Σ by verifying the hypothesis that P∗ is expressible in Σ.This can be broken down to verifying three conditions:G1: For any A: A is expressible in Σ =⇒ A is expressible in Σ.G2: For any A: A is expressible in Σ =⇒ A∗ is expressible in Σ.G3: P is expressible in Σ.Remark:

Proving G1 will turn out to be trivial.Proving G2 is relatively straightforward.Proving G3 is extremely laborious.(But, with hindsight, we get a “free ride” from computability theory!)

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Gödel sentences, Diagonal Lemma

Definition (Gödel sentences).S is a Gödel sentence for a number-set A if: S ∈ T ⇐⇒ pSq ∈ A.

Lemma D (A Diagonal Lemma)If A∗ is expressible in Σ, then there exists a Gödel sentence for A in Σ.

Proof. Suppose H expresses A∗; let h = pHq. Thus d(h) = pH(h)q.H(n) ∈ T ⇐⇒ n ∈ A∗, in particular H(h) ∈ T ⇐⇒ h ∈ A∗.Since h ∈ A∗ ⇐⇒ d(h) ∈ A, H(h) is a Gödel sentence for A. QED.Note:

Condition G2 implies that Lemma D can be applied.Lemma D thus straightforwardly implies Theorem GT.P and R are irrelevant for Lemma D.A Gödel sentence of P can be read as “I am unprovable”.

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An abstract form of Tarski’s Undefinability TheoremNotation: We use T to denote {pSq | S ∈ T }.

Theorem T (After Tarski)(1) T ∗ is not expressible in Σ.(2) If G2 holds, then T is not expressible in Σ.(3) If G1 and G2 hold, then T is not expressible in Σ.

Proof. First note: There cannot be a Gödel sentence GT for T sinceGT were true ⇐⇒ pGTq is not the Gödel number of a true sentence,which is clearly is absurd.(1) If T ∗ were expressible in Σ, then by Lemma D, there would be a

Gödel sentence GT for T in Σ.(2) If G2 holds, the expressibility of T would imply that of T ∗.(3) If also G1 holds, then the expressibility of T would imply that of T .Thus we have reduced (3) to (2), (2) to (1), and (1) to the initialobservation about the non-existence of a Gödel sentence for T . QED. 13

Back to Gödel: a (more) syntactic form of incompletenessNote: So far R (set of refutable sentences) played no role.Definition (Undecidable sentences, Gödel-incompleteness)Relative to a system Σ, a sentence S is called (formally) undecidable ifS 6∈ P and S 6∈ R.Σ is called Gödel-incomplete if it contains formally undecidable sentences.

Theorem GI (Gödel-Incompleteness)If Σ is correct and P∗ is expressible, then Σ is Gödel-incomplete.

Definition (Consistency)Σ is consistent if there is no sentence that is both provable and refutable.(I.e., P ∩R = ∅.)Note:

Correctness implies consistency, but not vice versa.To establish purely syntactic incompleteness — consistency =⇒Gödel-incompleteness — more specific properties are needed (Rosser).

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Formal ArithmeticWe formulate a standard theory (= set of formulas) in first-order logic.(If needed re-visit syntax and semantics of first-order logic!)Syntax

signature:– predicate symbols: <, =– function symbols: s (unary, prefix), +, · (binary, infix)– constant: 0formulas: defined inductively as usual, but officially using only thefollowing additional symbols: ( ) , ¬ ∨ ∃ v ′

(∀, ¬, ∧, and ⊃ are used as ‘macros’ only)

Note: By representing variable v1 as v ′ (two symbols), v2 as v ′′(three symbols) etc., all expressions remain strings over a finite alphabet.SemanticsDefined as usual for first-order formulas. But we care only about oneparticular model, called standard model N with domain N andthe obvious signature interpretation. (s is interpreted as λx(x + 1)). 15

True Arithmetic TARemember: A sentence is a closed formula (no free variables).Definition:TA is the set of sentences that are satisfied (true) in model N.Our current aim is to show Tarski’s Undefinability Theorem.Remember the following abstract version of Tarski’s theorem.

Theorem T (After Tarski)(1) T ∗ is not expressible in Σ.(2) If G2 holds, then T is not expressible in Σ.(3) If G1 and G2 hold, then T is not expressible in Σ.

It remains to show:Formal arithmetic constitutes a system Σ.There is a concrete Gödel numbering p...q (modulo which T can beidentified with the set of Gödel numbers of sentences in TA).Conditions G1 and G2 are satisfied w.r.t. to p...q. 16

Formal Arithmetic as a systemRemember the definition of a system:A system Σ is a set containing the following components:

E . . . expressions of Σ – here: all strings over our 14 symbolsS . . . sentences of Σ (S ⊆ E) – here: closed formulasT . . . true sentences of Σ (T ⊆ S) – here: theory TAP . . . provable sentences of Σ (P ⊆ S) – currently irrelevantR . . . refutable sentences of Σ (R ⊆ S) – currently irrelevantH . . . predicates of Σ (H ⊆ E) – here: formulas with one free variablea function Φ : E × N 7→ E : If E ∈ H then Φ(E , n) = E (n) ∈ Shere:for every number n ∈ N there is a numeral n̂ =

n times︷ ︸︸ ︷s . . . s 0,

for any predicate (= formula) F (x), Φ(F (x), n) = F (n̂).

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A Gödelization of formal arithmeticWe assign digits of base-14 numbers to our symbols:symbol ( ) , ¬ ∨ ∃ v ′ < = s + · 0code 0 1 2 3 4 5 6 7 8 9 α β γ δ

Note:This induces a Gödel numbering p...q for all expressions.E.g. p∃v ′ s0 < v ′q = (567αδ867)14 = 576433739.Remarks:

(Syntactic) operations on strings, like concatenation and variablesubstitution, turn into functions over N .Gödel used a different form of Gödelization, based on prime numbers.He found an ingenious method to represent all relevant syntacticmanipulations of formulas, in particular also provability in a concretearithmetic proof system, within that system.

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Arithmetic definabilityDefinitionA formula F (v) (arithmetically) defines (expresses) a number-set X if

n ∈ X ⇐⇒ N |= F (n̂)

Read N |= F (n̂) as “F (n̂) is true in N”. (Equivalently: F (n̂) ∈ TA.)A formula F (v1, . . . , vk) defines a k-ary relation R (⊆ N k) if

R(n1, . . . , nk) ⇐⇒ N |= F (n̂1, . . . , n̂k)

A formula F (v1, . . . , vk , vk+1) defines a k-ary function f : N k → N iff (n1, . . . , nk) = m ⇐⇒ N |= F (n̂1, . . . , n̂k , m̂)

Examples: [A rich source of exercises!]The set of odd numbers is defined by (∃v ′ v = (s0 + (v ′ + v ′))).The relation ≥ is defined by (v ′ = v ′′ ∨ v ′′ < v ′).The function λx .3x + 2 is defined by ((sss0 · v ′) + ss0) = v ′′.

Note:It is a not easy to define exponentialization, which is needed to define thearithmetic counterparts (modulo p...q) of non-trivial syntactic operations. 19

Definability of recursive functions

Theorem ADRFAll recursive functions and predicates are arithmetically definable.

Proof.Remember that recursive functions are built up from projection functions,constant-0 functions, the successor function using composition, primitiverecursion, and minimization.We thus have to proceed inductively.Base cases:

projni is defined by vn+1 = vi (1 ≤ i ≤ n)zeron is defined by vn+1 = 0 (n ≥ 0)succ is defined by sv1 = v2

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(Proof cont.)Inductive cases

Composition: f ◦ [g1, . . . , gn], f : N n → N , g1, . . . , gn : Nm → NRemember that f ◦ [g1, . . . , gn] is of type Nm → NSuppose F (v1, . . . , vn, vn+1), Gi (v1, . . . vm, vm+1) for 1 ≤ i ≤ nare the formulas defining f , g1, . . . , gn, respectively.Then the formula

∃vm+2 . . . ∃vm+n+1(G1(v1, . . . vm, vm+2) ∧ . . . ∧ Gn(v1, . . . vm, vm+n+1)∧ F (vm+2, . . . , vm+n+1, vm+1))

defines the function f ◦ [g1, . . . , gn].Primitive recursion: Pr [g , h] where g : N k → N , h : N k+2 → NRemember that f = Pr [g , h] : N k+1 → N is defined as follows::f (0, x1, . . . , xk) = g(x1, . . . , xk)f (y + 1, x1, . . . , xk) = h(y , f (x1, . . . , xk), x1, . . . , xk)

Warning: The defining formula Pr [g , h] for is tricky and lengthy,since it requires coding of sequences. See, e.g., Smullyan: Gödel’sIncompleteness Theorems for a detailed discussion. 21

(Proof cont.)

Minimalization: µf (like before, we only consider f : N 2 → N )(µf )(x , y) = least y s.t. f (x , y) = 0 and f (x , z)↓ for all z < ySuppose f is defined by F (v1, v2, v3).Then the formulaF (v1, v2, 0) ∧ ∀v3(v3 < v2 ⊃ ∃v4(F (v1, v3, v4) ∧ ¬(v4 = 0)))

defines µf .For predicates consider their characteristic functions. QED.Note: Theorem ADRF invites us to appeal to the Church-Turing Thesis:all (informally) computable functions are arithmetically definable!Actually, an ever stronger fact holds for standard arithmetic proof systems:Definition: f is representable in system Π if all true instances of itsdefining formula F are provable in Π. (Analogously for number-sets.)

Theorem ARRF (schemtic)All recursive functions are representable in usual arithmetic proof systems.

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Tarski’s Undefinability TheoremIt remains to establish conditions G1 and G2.

G1: If F (v) defines A then ¬F (v) defines A.G2: According to Theorem ADRF and the Church-Turing thesis

the following functions are arithmetically definable:I subst : N 2 → N , defined by subst(pF (v)q, n) = pF (n̂)q for all

formulas F (v) and numbers n,I consequently also the diagonal function d : N → N , defined by

d(m) = subst(m,m) = pF (m̂)q where m = pF (v)q.We thus obtain the followingLemma (G2 for Formal Arithmetic)If A is definable then A∗ is definable as well.

By instantiating the (abstract) Theorem T, we obtain:

Theorem (Tarski’s Undefinability Theorem)The set T = {pSq | N |= S} is not definable in Formal Arithmetic.

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Gödel(-Tarski) incompleteness for Formal ArithmeticRemember:

Theorem GT (After Gödel with a shade of Tarski)If Σ is correct and P∗ is expressible in it, then Σ is incomplete.

To obtain the incompleteness of Formal Arithmetic we need the following:G3: The set provable sentences is arithmetically definable.Note: By again appealing to Theorem ADRF and to the Church-Turingthesis, we don’t even have to specify a concrete proof system. It suffices toobserve that — by definition of a (formal) proof system — we caneffectively decide whether a given sequence of strings of symbols is aproof. Therefore G3 must hold.

Gödel-Tarski Incompleteness TheoremEvery correct proof system for Formal Arithmetic is incomplete.I.e., no proof system can prove exactly the arithmetically true sentences.

Corollary: TA is not recursive and thus not recursively axiomatizable. 24

g

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Unprovability of ConsistencyTo establish Gödel’s second incompleteness theorem for a proof system Πits provability predicate (i.e., defining formula) B(v) has to satisfy thefollowing provability conditions:P1: if `Π S then `Π B(pSq),P2: if `Π B(pS ⊃ Tq) ⊃ (B(pSq) ⊃ B(pTq)),P3: if `Π B(pSq) ⊃ B(pB(pSq)q).

Second Incompleteness TheoremIf the provability predicate for a proof system Π is representable in Π andsatisfies the provability conditions, then ¬B(p0 < 0q) is only provable if Πis inconsistent.

Note:All usual arithmetic proof systems satisfy the mentioned conditions andtherefore cannot prove their own consistency.Of course, this does not mean that the consistency of given proof systemscannot be shown at all! 25

Further topics[to be discussed in class, if time permits]

incompleteness (Gödel, 1931) and undecidabillity (Turing, 1936)a standard system: Peano Arithmetic PAconsequence of incompleteness (discuss, e.g., the ω-rule)models of PA (Tennenbaum’s Theorem, etc.)compare: nonstandard models of TA due to compactnessincompleteness via Kolmogorov complexity:“randomness is hardly ever provable”provability logic (Boolos, Gödel-Löb, etc.)consequences for software verificationconsequences for mathematics in general (?)(E.g., the “Gödel-Friedman Program”)consequences beyond mathematics (??)

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Further ReadingRaymond Smullyan:Gödel’s Incompleteness Theorems.Oxford Logic guides, 1992.Torkel Franzen:Gödel’s Theorem: An Incomplete Guide to Its Use and Abuse.A.K. Peters, 2005.A translation of Gödel’s original paper can be found at:http://people.cs.umass.edu/∼immerman/

cs601/goedelIncompletenessTh.pdf

A short introduction (with also historical remarks) on Gödel’s paperby Richard Zach is online at:citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.3.8423

Sol Feferman:The impact of the incompleteness theorems on mathematicsmath.stanford.edu/∼feferman/impact.pdf

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