Incompleteness Theorems MT 2010

8/6/2019 Incompleteness Theorems MT 2010

1/125

OXFORD UNIVERSITY

Part C Mathematics, Mathematics and Philosophy, Mathematics and ComputationM.Sc. in Mathematics and Foundations of Computer Science

B.Phil. and M.St. in Philosophy

16 lectures on

Godels Incompleteness TheoremsMichaelmas Term 2010

Daniel IsaacsonFaculty of Philosophy

Oxford University

1st December 2010

Copyright c 2010 by Daniel IsaacsonAll rights reserved. No part of this publication may be reproducedwithout prior permission by anyone other than for their own use in

studying this subject. Enquiries and corrections [email protected]


2/125

Contents

0 Background: first-order logic and formal systems 2

0.1 First-order formal languages with identity . . . . . . . . . . . . . . . 20.2 Interpretations of first-order formal languages with identity; truth of

a first-order formula in an interpretation; logical validity and logicalconsequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

0.3 A system of natural deduction for first-order logic with identity . . . 40.4 Prenex normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

0.4.1 Model theory and proof theory . . . . . . . . . . . . . . . . . 110.5 Completeness of a system of natural deduction with respect to first-

order logical consequence . . . . . . . . . . . . . . . . . . . . . . . . . 110.5.1 Lindenbaums Lemma . . . . . . . . . . . . . . . . . . . . . . 11

0.5.2 The Completeness Theorem . . . . . . . . . . . . . . . . . . . 110.5.3 The Compactness Theorem for first-order logic . . . . . . . . . 110.5.4 The existence of non-standard models of the truths of arithmetic 110.5.5 Completeness of other systems of derivation with respect to

first-order logical consequence . . . . . . . . . . . . . . . . . . 11

1 Introduction: a weak form of Godels First Incompleteness Theo-rem; the symbols and expressions of a language for arithmetic LE;Godel numbering of the expressions of LE 121.1 Introduction: a weak form of Godels First Incompleteness Theorem 121.2 The symbols and expressions of a language for arithmetic LE . . . . . 16

1.3 Godel numbering of the expressions of LE . . . . . . . . . . . . . . . 17

2 Terms and formulas of the language LE; expressibility of diagonalsubstitution in the language LE 202.1 Terms and formulas of the language LE . . . . . . . . . . . . . . . . . 20

2.1.1 Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.1.2 Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.1.3 Free and bound variables; open formulas and sentences . . . . 22

1


3/125

CONTENTS 2

2.2 Designation by terms in LE, truth of sentences of LE, and express-

ibility of sets and relations of natural numbers by formulas of LE . . 232.2.1 Designation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2.2 Truth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.2.3 Expressibility . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3 Concatenation of numbers in a given base notation is Arithmetical. . 242.4 Substitution and diagonal functions, and their arithmetization . . . . 25

3 The Diagonal Lemma; expressibility of properties of sequence num-bers 283.1 The Diagonal Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 Expressibility of properties of sequence numbers in the language LE 293.2.1 Properties of sequences of digits . . . . . . . . . . . . . . . . . 293.2.2 Sequence numbers . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3 Coding of finite sequences of Godel numbers . . . . . . . . . . . . . . 31

4 A formal system PAE for arithmetic; an Arithmetical proof predi-cate for PAE; a weak version of Godels first incompleteness theo-rem for PAE 344.1 A formal system PAE for arithmetic . . . . . . . . . . . . . . . . . . . 344.2 An Arithmetical proof predicate for PAE . . . . . . . . . . . . . . . . 374.3 An inefficient and a weak version of Godels First Incompleteness

Theorem for PAE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5 The system PA with zero, successor, addition, multiplication, and as primitive; 0- and 1-formulas; a 0-coding of finite sets ofordered pairs; the relation xy = z is 1-expressible in the languageof PA 425.1 The system PA with zero, successor, addition, multiplication, and

as primitive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.2 0-formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.3 1- and 1-formulas; 1- 1- and 1-relations . . . . . . . . . . . . . 44

5.4 Arithmetization of syntax in the language of PA . . . . . . . . . . . . 465.5 A 0-coding of finite sets of ordered pairs of numbers . . . . . . . . . 475.6 The relation xy = z is 1-expressible in the language of PA . . . . . . 49

6 Every -formula is provably equivalent to a 1-formula; the arith-metized proof predicate for PA is 1; the arithmetical hierarchy 506.1 -formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506.2 The arithmetized proof predicate for PA is 1 . . . . . . . . . . . . . 526.3 The arithmetical hierarchy . . . . . . . . . . . . . . . . . . . . . . . . 53


4/125

CONTENTS 3

7 0-completeness and 1-completeness; weak systems of arithmetic

Q and R (without induction); 0-completeness of systems R, Q, andPA; 0-soundness and 1-soundness 557.1 0-completeness and 1-completeness . . . . . . . . . . . . . . . . . . 557.2 Weak systems of arithmetic Q and R (without induction) . . . . . . . 577.3 0-completeness of systems R, Q, and PA . . . . . . . . . . . . . . . 597.4 0-soundness and 1-soundness . . . . . . . . . . . . . . . . . . . . . 62

8 The notions of consistency, -consistency and 1-consistency; incom-pleteness from the assumption of 1-consistency; truth of the Godelsentence; -incompleteness. 63

8.1 The notions of consistency, -consistency and 1-consistency. . . . . . 638.2 Incompleteness of PA from the assumption of 1-consistency . . . . . . 678.3 Truth of the Godel sentence . . . . . . . . . . . . . . . . . . . . . . . 688.4 PA is -incomplete . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

9 Enumerability and the Separation Lemma; incompleteness of PAfrom the assumption of consistency (Rossers Theorem); weak andstrong definability of a function in a system; formal provability ofthe Diagonal Lemma 719.1 Enumerability and the Separation Lemma . . . . . . . . . . . . . . . 729.2 Incompleteness of PA from the assumption of consistency (Rossers

Theorem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739.3 Weak and strong definability of a function in a system . . . . . . . . 749.4 Formal provability of the Diagonal Lemma . . . . . . . . . . . . . . . 76

10 Arithemization of consistency; provability predicates; Godels Sec-ond Incompleteness Theorem; Lobs Theorem; analyzing and strength-ening the First Incompleteness Theorem 7910.1 Arithmetization of the statement that a system S is consistent . . . . 7910.2 Provability predicates. . . . . . . . . . . . . . . . . . . . . . . . . . . 8010.3 Godels Second Incompleteness Theorem . . . . . . . . . . . . . . . . 82

10.4 Lobs Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8310.5 Analyzing and strengthening the First Incompleteness Theorem . . . 84

10.5.1 S GS cannot be proved from the consistency of S . . . . . 8510.5.2 Strengthened second half of the First Incompleteness Theorem 8510.5.3 Consistency ofS{ConS} is strictly weaker than 1-consistency

of S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

11 Provable 1-completeness 88


5/125

CONTENTS 4

12 The -rule and uniform reflection; PA proves that PA proves every

instance of the Godel sentence; 1-uniform reflection and consis-tency; PA is 1-conservative over PA2 {ConP A} 9812.1 The -rule and uniform reflection . . . . . . . . . . . . . . . . . . . . 9812.2 PA proves that PA proves every instance of the Godel sentence . . . . 10012.3 Equivalence of 1-Uniform Reflection and consistency . . . . . . . . . 10112.4 PA is 1-conservative over PA2 {ConP A} . . . . . . . . . . . . . . 102

13 Provability logic: the system GL 10413.1 The language of GL . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10413.2 The axioms and inference rules of GL . . . . . . . . . . . . . . . . . . 105

13.3 Some derivations in GL . . . . . . . . . . . . . . . . . . . . . . . . . . 10713.4 Closure of GL under substitution by provably equivalent formulas . . 10813.5 Closure of GL under substitution of provably equivalent formulas is

provable in GL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11013.6 Strengthened proof that the closure of GL under substitution of prov-

ably equivalent formulas is provable in GL . . . . . . . . . . . . . . . 111

14 The fixed-point theorem for GL 11414.1 The notion of a sentence letter modalized in a sentence, and arithme-

tized substitution for modalized sentences . . . . . . . . . . . . . . . 11414.2 The fixed point theorem for GL . . . . . . . . . . . . . . . . . . . . . 117


6/125

Lecture 0

Background: first-order logic and

formal systems

This unspoken lecture reviews essential results on first-order logic and formal systemsas background to the development of Godels incompleteness theorems.

The notion of logical consequence is the starting point. A rough characterizationof this notion is the following. A sentence is a logical consequence of a set ofsentences , symbolized as , if and only if is true in every interpretation ofthe language of {} in which all the sentences of are true. If is empty,

reduces to the condition that is true in every interpretation of the language of ,which is to say, is logically valid, symbolized by .

Accordingly, we need to make precise the notion of a formal language and thenotion of interpretation of a formal langauge.

0.1 First-order formal languages with identity

The notion of a formal language begins from the specification of a finite alphabetof symbols which are strung together (concatenated) to produce the expressions ofthe language. The symbols of the alphabet of a formal language have no intrinsicmeaning (they are purely formal symbols) but are chosen with the intention thatthey should be interpretable in certain ways (which in general does not rule outtheir being interpreted in other ways). Some but not all of the expressions of thelanguage constitute well-formed expressions. Which expressions are well-formed isa matter of stipulation, according to our intended use of the formal language. Thesymbols of a first-order language are of three sorts, logical, non-logical, and syntactic.The logical symbols include two sorts: propositional, and quantificational, and mayinclude a third, identity. The non-logical are of two sorts, function symbols andpredicate or relation symbols. Functions and relations are of a particular arity

5


7/125

LECTURE 0 6

(unary, binary, ternary, and so on). A zero-ary function symbol is a constant term,

a zero-ary relation symbol represents a sentence. The shape of a symbol is completelyarbitrary, though there are some conventional choices, e.g. the universal quantifieris usually written , but used often to be written ( ) (the brackets enclosingthe variable of quantification), and is sometimes written

(called a California

quantifier). But the symbols can be anything and in particular, as we shall exploit,they can be digits also used to generate numerals.

We shall in this course always have full first-order classical (as opposed to intu-itionistic) predicate logic with identity as our background logic. Quite apart fromthe shapes of the symbols, there are choices to be made as to which are our prim-itive propositional functions and quantifiers. In our background logic we shall take

as our primitives of propositional logic negation, conjunction, disjunction, and im-plication, for which the symbols will be , , , and , and both the universaland existential quantifiers as primitive, written as and . As you will know froma previous logic course, we could take as primitive just negation with any one ofconjunction, disjunction, and implication and either the universal or the existentialquantifier, and in our official formal language we shall take as primitive , , and .We shall operate with the langauge as if it also contained symbols for conjunction,disjunction, and the existential quantifiers, but strictly (A B) will be an abbrevi-ation for ( A B), etc. There is also the connective if and only if, which wewill symbolize as . Even in our background logic we will take (A B) to be an

abbreviation for ((A B) (B A)).Stipulating that our background logic is first-order classical logic with identity,means that we have a primitive symbol for identity, which we will take as thecommon symbol for identity = (though others are sometimes used, e.g. , ,

.=),

but having a symbol and first-order axioms for identity is not sufficient to make alogical system first-order logic with identity. Identity is a two place relation whichevery object bears to itself and to no other object. The first property is easilyexpressed: v1v1 = v1, but the second property is not easily expressed, in particularnot expressed by v1v2(v1 = v2 v1 = v2) (if v1 is a different object from v2 thenv1 = v2).

terms in the languageformulas in the language


8/125

LECTURE 0 7

0.2 Interpretations of first-order formal languages

with identity; truth of a first-order formula inan interpretation; logical validity and logicalconsequence

0.3 A system of natural deduction for first-orderlogic with identity

This system is based on natural deduction as developed by Gerhard Gentzen. In

Gentzen-style systems of natural deduction, deductions consist of branching trees.For ease of formulation on the page, I give here a variant form of natural deduction inwhich deductions are linear1. I shall call this system LND, standing either for LinearNatural Deduction (but in that case be aware that this usage has nothing to dowith Girards usage in what he calls Linear Logic), or Lemmon Natural Deduction.[Strictly I should call this system something like CLND, for Classical, as opposedto Intuitionistic, Linear Natural Deduction, but since in this course our backgroundlogic is always classical and not intuitionistic, I wont mark the distinction.]

Deductions are generated by using the Rule of Assumption and fourteen Rules ofInference. There are no axioms. The rules of inference come in pairs, an Introduction

and an Elimination rule for each one of the seven logical constants , , , , , ,and =. We take (F G) to be defined as ((F G) (G F)). The first formulaof a Deduction is always an Assumption; later formulas can also be Assumptions.An Introduction Rule for a given logical constant results in a formula that has thatlogical constant as its main logical constant and is deduced from formulas that thatdo not have that logical constant as their main constant (though that constant mayoccur in a subformula). An Elimination Rule for a given logical constant deduces aformula that does not have that logical constant as its main logical constant from aformula that has that logical constant as its main logical constant.

A deduction consists of a numbered sequence of formulas, with each of which

is associated on the left a finite set of numbers (possibly empty) which are thenumbers of the formulas that constitute the assumptions on which the formula inthat line depends, and on the right notation indicating how and from what otherformulas that formula was derived, if its not an Assumption. Viewed verticallyrather than horizontally, a deduction consists of four columns. The middle twocolumns are a numbered sequence of formulas; the third column is a sequence offormulas and the second column is an enumeration of those formulas. To the left of

1A linear formulation of natural deduction was given by E.J. Lemmon in Beginning Logic,

Nelson, London, 1965; the system here is, in essentials, as in that book.


9/125

LECTURE 0 8

each numbered formula are the numbers of the Assumptions, if any, on which that

formula depends. The entry in the right-hand column gives the basis on which theformula in the third column at that line is introduced into the deduction, i.e. eitheras an Assumption or by one of the Rules of Inference. An Assumption depends onitself, so in an application of the rule of Assumption there is one number in the firstcolumn which is the same number as in the second column. If the formula in thethird column of a given line is introduced by a Rule of Inference, the entry in thefourth column for that line says what Rule of Inference has been used and numbersof the formulas to which that Rule of Inference has been applied. The Assumptionson which those formulas depend, as given by the numbers in the first column atthe lines for those formulas, are gathered together as the Assumptions on which

the formula that results from application of that Rule of Inference depends. Fourof the Rules of Inference, -Introduction, -Introduction, -Elimination, and -Elimination, discharge an Assumption, so it is possible to arrive at a formula whichdepends on no assumptions. Such formulas are, by the Soundness of the Rules ofInference, logically valid.

-Introduction

assumptions numbering formulas justificationsA (a) F [whatever]B (b) G [whatever]A B (c) (F G) (a) (b) -Introduction

-Elimination1

assumptions numbering formulas justificationsA (a) (F G) [whatever]A (b) F (a) -Elimination1

-Elimination2

assumptions numbering formulas justificationsA (a) (F G) [whatever]A (b) G (a) -Elimination1

-Introduction1


10/125

LECTURE 0 9

assumptions numbering formulas justifications

A (a) F [whatever]A (b) (F G) (a) -Introduction1

-Introduction2

assumptions numbering formulas justificationsA (a) G [whatever]A (b) (F G) (a) -Introduction1

-Elimination

assumptions numbering formulas justificationsA (a) (F G) [whatever]{b} (b) F AssumptionB {b} (c) H [whatever]{d} (d) G AssumptionC {d} (e) H [whatever]A B C (f) H (a)(c)(e) -Elimination

-Introduction

assumptions numbering formulas justifications{a} (a) F AssumptionA {a} (b) G [whatever]A (c) (F G) (a)(b) -Introduction

-Elimination


A (a) (F G) [whatever]B (b) F [whatever]A B (c) G (a)(b) -Elimination

The Introduction and Elimination rules for negation in this system of rules arenot as natural as those for the other logical connectives. A more natural formulationof logic in natural deduction is to take negation to be defined in terms of a primitivefalse sentence, sometimes symbolized as , by the equivalence . How-ever, in terms of formalizing informal maths talk it is more natural to take negation


11/125

LECTURE 0 10

as a primitive symbol, which we do here. It is subject to the following Introduction

and Elimination rules.

-Introduction

assumptions numbering formulas justifications{a } (a) F AssumptionA {a} (b) (G G) [whatever]A (c) F (a)(b) -Introduction

-Elimination

assumptions numbering formulas justificationsA (a) F [whatever]A (b) F (a) -Elimination

-Introduction

assumptions numbering formulas justificationsA (a) F(vi) [whatever]A (b) viF(vi) (a) -Introduction, ifvi does not occur free

in any formula enumerated by A

-Elimination

assumptions numbering formulas justificationsA (a) viF(vi) [whatever]A (b) F(t) (a) -Elimination, for t any term free for vi in F(vi)

-Introduction

assumptions numbering formulas justificationsA (a) F(t) [whatever] for t any term free for vi in F(vi)A (b) viF(vi) (a) -Introduction

-Elimination


12/125

LECTURE 0 11


A (a) viF(vi) [whatever]{b} (b) F(vi) AssumptionB {b} (c) G [whatever]A B (d) G (a)(b)(c) -Elimination, if vi does not occur free

in any formula enumerated by B

General restriction on terms t in -elimination and -introduction: The term tmust not contain any free variable which in (x) is quantified by a quantifier whosescope in (x) includes an occurrence of the variable x, i.e. the substitution of t

in (x) must not result in (t) having a different quantifier structure from that of(x).

=-Introduction

assumptions numbering formulas justifications(a) t = t =-Introduction, for t any term

=-Elimination


A (a) t1 = t2 [whatever] for t1, t2 any termsB (b) F(t1) [whatever]A B (c) F(t2) =-Elimination

Definition 1 For F a formula and a set of formulas, we say that F is (logically)derivable from , notated LDN F, or just F, if there is deduction from theRule of Assumption and the 14 Rules of Inferences of LDN in which the last linehas the numbered formula F and the assumptions on which F in that line depends

on are exactly the formulas of .

LDN (F F)

(1) (1) (F F) Assumption(2) (2) F Assumption(2) (3) (F F) (2) -Introduction(1)(2) (4) ((F F) (F F)) (1)(3) -Introduction(1) (5) F (2)(4) -Introduction


13/125

LECTURE 0 12

(1) (6) (F F) (5) -Introduction

(1) (7) ((F F) (F F)) (1)(6) -Introduction(8) (F F) (1)(7) -Introduction(9) (F F) (8) -Elimination

A modification to make LND more readily usable: Any truth functionaltautology can be written down as a line of a derivation, resting on no Assumption,with Tautology as the Justification, e.g.

(1) (F F) Tautology

0.4 Prenex normal form

Lemma 1 (change of quantified variable) For any formula in a first-order lan-gauge, any variable of quantification can be changed to any one of infinitely manyother variables while preserving the free variables of the formula, if there are any, ina way that results in a formula that is logically equivalent to the original formula.

Remark. There are restrictions on logically equivalent changes of bound vari-

able, e.g. v1v2v1 = v2 is not logically equivalent to v2v2v2 = v2 (the latter islogically valid while the former is not).

Lemma 2 (prenex equivalences) The following formulas are logically valid.(ia) ( viF(vi) vi F(vi))(ib) ( viF(vi) vi F(vi))In the following the variable vi does not occur free in the formula G.(iia) ((viF(vi) G) vi(F(vi) G))(iia) ((G viF(vi)) vi(G F(vi)))(iib) ((viF(vi) G) vi(F(vi) G))(iib) ((G v

iF(v

i)) v

i(G F(v

i)))

(iiia) ((viF(vi) G) vi(F(vi) G))(iiia) ((G viF(vi)) vi(G F(vi)))(iiib) ((viF(vi) G) vi(F(vi) G))(iiib) ((G viF(vi)) vi(G F(vi)))(iva) ((viF(vi) G) vi(F(vi) G))(iva) ((G viF(vi)) vi(G F(vi)))(ivb) ((viF(vi) G) vi(F(vi) G))(ivb) ((G viF(vi)) vi(G F(vi)))


14/125

LECTURE 0 13

Proof.

Theorem 3 (Prenex Normal Form) Each formula F in a first-order languagewith and is logically equivalent to a formula F whose quantifiers all occur as astring at the beginning of the formula, and such that F contains exactly the samefree variables as F does.

Proof The proof is by induction on the number of quantifiers in F.Base case: F is atomic. Then F has no quantifiers so, vacuously, all its quanti-

fiers, namely none, occur in a string at the beginning of the formula.Induction steps: (i) F is of the form G. By induction hypothesis G has a

prenex normal form G

. IfG

has no quantifiers, we are done, as in the base case. IfG has quantifiers, they occur in a string at the beginning of the formula. Then bylogical equivalences (ia) and (ib) from Lemma 2, the negation sign can be pushedpast all the quantifiers, changing each to the other quantifier in the process.

(ii) F is of the form (G H). By Induction Hypothesis G and H have logicallyequivalent prenex normal forms G and H. Then F is logically equivalent to (G H). By Lemma 1 and (iva), (ivb), and (iva), (iv) of Lemma 2, the quantifiers ofG and H can be pulled out into prenx normal form, those from G changing tothe other quantifier.

(iii) F is of the form (G H), (iv) F is of the form (G H). These two casesare as (ii) except simpler, with no changes of quantifiers as the quantifiers are putinto prenex form. (v) F is of the form viG. Then by induction hypothesis G hasprenex normal form G which has the same free variables as G has, particular vi(in the case when the quantification vi of F is not vacuous. Then F is logicallyequivalent to viG

, which is in prenex normal form. (vi) F is of the form viG.Same argument as for (v).

Note that the Prenex Normal Form Theorem holds only on the assumption thatall domains of interpretation are non-empty. Otherwise we have, for example, thatfor v1 not free in G, and G true in the empty domain (e.g. G = v1v1 = v1, orequally G = v1 v1 = v1),

(v1F(v1) G) is true in the empty domain, since G is true, but v1(F(v1) G))is false, since every existentially quantified statement is false in the empty domain.

Note that prenex normal forms are not in general unique, e.g.(v1F(v1) v2G(v2) has as prenex normal form both v1v2(F(v1) G(v2)) andv2v1(F(v1) G(v2)).


15/125

LECTURE 0 14

0.4.1 Model theory and proof theory

0.5 Completeness of a system of natural deduc-tion with respect to first-order logical conse-quence

0.5.1 Lindenbaums Lemma

0.5.2 The Completeness Theorem

The completeness theorem for a given formal system of first-order logic means thata model theoretic argument for a logical consequence (as above) establishes theexistence of a formal derivation of that logical consequence in the system whosecompleteness has been proved without actually finding a derivation.

While we talk about the completeness theorem for first order logic, there are ac-tually many completeness theorems, one for each different complete formal systemfor first-order logical consequence. On the other hand there is an intrinsic com-pleteness theorem, namely completeness of a system which consists of exactly thoseaxioms and rules of inference needed for the proof of the completeness theorem.

Comparison between the completeness theorem for first-order logical consequenceand the incompleteness theorem for truth in arithmetic. Both these results are

due to Godel, in 1930 and 1931. (The first was his Ph.D. thesis, the second hisHabilitation thesis. They were written under the nominal supervision of the verynotable mathematician Hans Hahn, but Godel was essentially working on his own.)

0.5.3 The Compactness Theorem for first-order logic

0.5.4 The existence of non-standard models of the truths ofarithmetic

0.5.5 Completeness of other systems of derivation with re-

spect to first-order logical consequence

Definition 2 LetS1 and S2 be formal systems such that the language L1 of S1 is asub-language of the language L2 of S2. We say that S1 is a subsystem of S2 if forevery formula in L1, if S1 , then S2

Lemma 4 If S1 is a complete system for first-order logical consequence and S1 is asubsystem of S2, then S2 is complete for first-order logical consequence.


16/125

Lecture 1

Introduction: a weak form of

Godels First IncompletenessTheorem; the symbols andexpressions of a language forarithmetic LE; Godel numberingof the expressions of LE

(Tuesday, 12 October 2010)

1.1 Introduction: a weak form of Godels FirstIncompleteness Theorem

The context of Godels discovery of the phenomenon of formal incompleteness isDavid Hilberts programme for giving mathematics a secure foundation by estab-lishing the consistency of systems formalizing it. In 1918 he declared that

we must make the concept of specifically mathematical proof itself intoan object of investigation (Hilbert [2]).

Hilbert formulated the distinction between finitary and infintary mathematics.The paradigm of finitary mathematics is arithmetical calculation. Finitary mathe-matics is mathematical bedrock, corresponding to observation statements in science.A calculation such as 27 = 128 is finitary, but the claim that exponentiation to the

15


17/125

LECTURE 1 16

power 2 always yields a value, i.e. xy(2x = y) is infinitary, and more generally,

quantification over the infinite domain of natural numbers is infinitary. However,quantification over a bounded, i.e. initial segment of the natural numbers, which isfinite, belongs to finitary mathematics.

Hilberts deep insight was to recognize that the formal manipulation of all sym-bols, not just the symbols for numbers, i.e. numerals and terms built up fromnumerals and symbols for arithmetical operations, belongs to finitary mathematics.In particular,

a formalized proof, like a numeral, is a concrete and surveyable object.([3], p. 383 and also in [4], p. 471.)

Hilbert recognized two sorts of finitary statements, general and particular (thoughhe did not introduce terminology for this distinction). Particular finitary statementsare decided by computations, e.g. 7 5 = 35, and 210 = 1024 and truth functionalcombinations of them (the truth values of such combinations being computable fromthe truth values of the component statements). General finitary statements containfree variables, and can be thought of as a template for particular finitary state-ments that result by substitution of numerals for the free variables, for examplex + y = y + x, and (n > 2 xn + yn = zn). On the other hand, xy x + y = y + xand nxyz(n > 2 xn + yn = zn) are infinitary.

For F(v1) a general finitary statement with free variable v1, bounded quantifica-tion on the variable v1, which is finitary, is expressible using (apparently) unboundedquantification by, in the case of universal quantification, v1(v1 t F(v1)), fort a term in the language of arithmetic, which we abbreviate as (v1 t)F(v1),and in the case of existential quantification, v1(v1 t F(v1)), which we abbre-viate (v1 v2)F(v1). For t a numerical term (a numeral or a composition ofarithmetical functions applied to numerals), (v1 t)F(v1) and (v1 t)F(v1) areparticular finitary statements if v1 is the only free variable in F(v1). If t is a freevariable or a composition of arithmetical functions applied to one or more variables,(v1 t)F(v1) and (v1 t)F(v1) are general finitary statements.

Hilbert noted that general finitary statements are not closed under negation, i.e.

the negation of a general finitary statement cannot be expressed as a general finitarystatement. For example, Fermats Last Theorem is expressible as a general finitarystatement, (n > 2 xn + yn = zn), but to say (falsely) that Fermats Last Theoremis false requires existential quantification, nxyz(n > 2 xn + yn = zn). Onthe other hand, the statement that a specific quadruple of numbers a,b,c,d is acounterexample, i.e. (a > 2 an + bn = cn), is a particular finitary statement.

Statements about particular formal proofs are, as Hilbert recognized, finitarystatements, e.g. that a particular formal proof is or is not a proof of a particularstatement.


18/125

LECTURE 1 17

Hilbert missed something about his insight which Godel realized, namely that

formal proofs can be literally identified with natural numbers, i.e. they could betaken to be numerical expressions, rather than merely like them. As Godel put thispoint in (1931),

Of course, for metamathematical considerations it does not matter whatobjects are chosen as primitive signs, and we shall assign natural numbersto this use, that is, we map the primitive signs one-to-one onto somenatural numbers.

Numbers assigned to formulas of a formal language in this way are called Godelnumbers.

Definition 3 We denote the Godel number of a formula F byF.

To carry out the arithmetization of syntax, the system must be able to talkabout numbers, i.e. there must be for each natural number a formal numeral in thelanguage of the system that denotes that numbers.

Definition 4 For formal languages that have a numeral for each natural number,we denote by n the numeral for the natural number n.

Definition 5 Whenever in these notes I speak of a sentence in the specified language

of arithmetic as being true, I mean true in the usual (intended) structure consistingof the domain of natural numbers with the usual arithmetical functions and relationson the natural numbers (also known as the standard model).

What Godel showed is that the property of being (the G odel number of) aprovable formula is expressible within any system which can express basic arithmetic,i.e. there is a formula P r(v1) with one free variable, in the language of a formalsystem for arithmetic, S, such that for every formula X in the language ofS, S Xif and only ifP r(X) is true. We shall establish the existence of such a formula fora particular formal system of arithmetic in Lecture 4. Godel also showed that for anyformula with one free variable (in particular a formula that expresses the propertyof being the Godel number of an unprovable formula) there is a diagonal sentence,i.e for formula F(v1) there is a sentence D such that the equivalence (D F(D))is true. We shall establish this result in Lecture 3.

From these results and on the assumption that everything provable in a givensystem S is true (about the natural numbers) (a strong assumption, much strongerthan is needed to establish incompleteness, but it is illuminating to consider thissimple case), it is easy to see that for G such that (G P r(G)), S G, and alsothereby that G is true, which implies, from the assumption that everything provablein a system S is true, that S G.


19/125

LECTURE 1 18

Theorem 5 (weak form of Godels first incompleteness theorem) LetS be

a theory such that for each natural number n there is a numeral n in the languageof S, and assume that

(i) there is a sentence G such that the sentence (G P r(G)) is true;(ii) there is an assignment of numbers to the formulas of the language of S and

a formula P r(v1) in the language of S such that for each formula X, S X ifand only if the sentence P r(X) is true, so in particular S G if and only if thesentence P r(G) is true.

(iii) every theorem of S is true.Then S G, G is true, and S G.

Proof. (1) Suppose that S G. (2) Then by (ii), P r(G) is true. (3) From(2) and (i), G is false. (4) From (3) and (iii), S G. (5) Since (4) contradicts (1),we have by reductio ad absurdum that S G (from assumptions (i), (ii), and (iii)).

(6) From (5) and (ii), P r(G) is false. (7) From (6) and (i), G is true.(8) From (7) and (iii), S G.

Remarks about this result:(1) This is a weak version of the first Godel incompleteness theorem since, while

assumptions (i) and (ii) are provable for weak systems of arithmetic, assumption (iii),soundness of the system with respect to truth in arithmetic, is a strong (highly non-finitistic) assumption, and unprovability of the Godel sentence holds from the muchweaker assumption that S is consistent. Godel sketches the proof of this weak formof the First Incompleteness Theorem in section 1 of his 1931 paper, and notes thatThe purpose of carrying out the above proof with full precision in what follows is,among other things, to replace the second of the assumptions just mentioned [everyprovable formula is true in the interpretation considered] by a purely formal andmuch weaker one. (van Heijenoort Sourcebook p. 599).

(2) In his introductory section Godel notes that this argument is closely re-lated to the argument for the Liar paradox. The argument does not lead to acontradiction since it starts from the assumption that G is provable, and so by RAAestablishes that G is not provable. Use of the Liar paradox also shows, as we shall

see, that unlike provability in a formal system, truth in a language of arithmeticcannot be expressed in the language.

(3) We shall establish how much basic arithmetic is required for arithmetizationof syntax, i.e. to prove assumptions (i) and (ii), to be made precise by the notionof 0-arithmetic, essentially just computations with addition and multiplication. Inparticular we dont need exponentiation. This shows that arithmetized syntax is aproper sub-part of what Hilbert meant by finitist mathematics. Hilbert never gavea precise characterization of finitist mathematics, but it is clear that it includes allprimitive recursive functions, so plus and times, but then also exponentiation and


20/125

LECTURE 1 19

so on. On the other had, both addition and multiplication are needed for incom-

pleteness: there is a complete theory of zero, successor, and addition (PresburgerArithmetic).

(4) The independence of the Godel sentence from formal arithmetic was unprece-dented. In the previous hundred years the independence of Euclids fifth postulatefrom the other postulates of geometry had been established. Godels result differsfrom this earlier one in two crucial ways. One was the technique used. The resultconcerning the fifth postulate was established by the construction of models. TheGodel result is purely syntactic (exploiting Hilberts insight). The other differenceis even more fundamental. The fifth postulate is neither true nor false, per se. Itis true in Euclidean geometry and false in non-Euclidean geometries. The Godel

sentence is demonstrably true, though not demonstrable in the system for which itis constructed.

Proposition 6 For any system S and sentence X, a proof that S X from theassumption of consistency is best possible.

Proof. If S is inconsistent, it proves everything, so in particular S X.

Remark. That the Godel sentence G for a system S is not refutable, i.e. Gis not provable, requires a stronger condition on S than consistency but a conditionmuch weaker than the soundness of S is sufficient.

1.2 The symbols and expressions of a languagefor arithmetic LE

A formal language is generated by combining symbols from a specified finite alpha-bet. For our formal language for arithmetic, as in [7], this consists of the following13 symbols:

0 ( ) f v = These formal symbols will be used with the following intended meanings:

The symbol 0 denotes the natural number zero

1

.The symbol denotes the successor functionThe symbols ( and ) are left and right bracketsThe symbols f and v are for functions and variables, to which numerical sub-

scripts in tally notation, i.e. iterations of the subscript are attached. The strings of

1Note that in this sentence I am being casual about the distinction between use and mention.

That distinction is easily but cumbersomely dealt with by using quotation marks, in which case

this given sentence would read: The symbol 0 denotes the natural number zero, which is fine,

though fussy, but the next sentence becomes just about unreadable if use vs mention is spelled out

in this way.


21/125

LECTURE 1 20

symbols f, f

, f

will denote the functions addition, multiplication, and exponenti-

ation, respectively, which we will write informally as +, , and exp or xy in the usualnotation. There are an infinity of variables v

, v

, v

, . . ., which we will usually write

as v1, v2, v3, . . .. If we want to signify a variable without specifying which variable itis, we will write vi, vj etc or use informal variable letters x,y,z,u,v,w.

The symbol for the propositional connectives negation and implication are and . The symbol for the universal quantifier is .

The symbols = and are for the two-place relations of equality and less thanor equals.

The symbol will be used to mark breaks between strings of symbols that areterms and formulas of the language (to be defined in the next lecture) in sequences

of terms and formulas.An expression in the language is (almost) any finite string of these symbols.

The set of expressions for the language LE is specified by the following recursivedefinition.

Definition 6 (expressions) basis: Each one of the symbols 0 ( ) f v = is by itself an expression.

recursion: If Ei and Ej are expressions, and Ei =, then the result of writing Ei

directly followed by Ej , which we call the concatenation of Ei and Ej and symbolizeas EiEj, is an expression.

Remark: Its for a technical reason (to do with our choice of Godel numbering)that we excluded from the class of expressions as here specified strings of more thanone symbol that begin with the symbol .

1.3 Godel numbering of the expressions of LE

.We assign Godel numbers to the expressions ofLE. This can be done in infinitely

many ways. The way we shall do it, following Smullyan following Quine, makes the

link between Godel numbering of expressions as strings of formal symbols particu-larly transparent. Godels original method involved coding by exponents of primefactors. On our method each number is the Godel number of an expression, whileon Godels method not every number is a Godel number. Having every number bea Godel number makes the formulation of some results a little simpler but is notessential.

Definition 7 (notation for expressions and Godel numbers) En =df the ex-pression with Godel number n.

E =df the Godel number of expression E.


22/125

LECTURE 1 21

Corollary 7 (of Definition 7) En = n.

We are used to the idea that numbers are denoted by numerals and that numeralsare not the same thing as numbers. The Roman numerals for the first five non-zeronatural numbers are I, II, III, IV, V, while the Arabic numerals are 1, 2, 3, 4 , 5. Thecrucial property of the Arabic numerals is that they are constructed on a place-valuesystem with a base of 10. That the system of numerals in common use is base 10 ispresumably down to the contingent fact (it could have been otherwise) that humanbeings have 10 fingers. Any other number greater or equal to 2 gives a perfectlygood numeral system with that base. Base 2 is used in machine code for computers,with 0 and 1 represented by current off and current on. The number we write as 15

in base 10 we write as 1111 in base 2 and as 13 in base 12. In the formal language forarithmetic we shall be using the numerals for numbers use a tally notation, rather

than place values. The formal numeral for the number n is the expression 0

n . . . ,

i.e. concatenation of the symbol 0 with n-many concatenations of the symbol .The following function plays a key role in our chosen system of Godel numbering.

Definition 8 (concatenation of base b numerals) For natural numbers m andn, we denote by mb n the number designated by the base b numeral that results fromconcatenating the base b numeral for m with the base b numeral for n.

Note that b is a function mapping pairs of natural numbers to natural numbers,and that natural numbers are not intrinsically in base b or any other base notation.The role ofb in this function is to specify the method of calculating this function.

Examples : For m = 673, n = 32 (written in base 10), m 10 n = 67332 andn 10 m = 32673. For m = 59, n = 0, m 10 n = 590 and n 10 m = 059 = 59.

Remark: As illustrated by these examples, b is not commutative. It is also notassociative, e.g. (17b 0)b 59 = 17059 = 1759 = 17b (0b 59). Non-associativity onlyarises when the middle value is 0, but since we will include 0 as a Godel number wecannot suppress parentheses in multiple computations with b except by adopting aconvention for reinstating them; we adopt the common convention of association to

the left, i.e. x b y b z = (x b y) b z.We assign Godel numbers to expressions by first stipulating the Godel numbers

of the symbols. We assign to these thirteen symbols the numbers denoted by thethirteen digits of base 13 notation, where the digits for 10, 11, and 12 (as we writethem in base 10) are taken to be , , and , respectively.

Definition 9 (assignment of Godel numbers to expressions) By recursion overthe recursive definition of expressions.

Base case: The assignment of numbers to symbols is specified by


23/125

LECTURE 1 22

0 ( ) f v =

1 0 2 3 4 5 6 7 8 9

Recursion: For expressions X and Y, XY = X 13 Y

By these stipulations, = 0, . . . = 0, and = 09 = 9 = . It is inorder for each expression to have a unique Godel number that we stipulated abovethat the class of expressions does not contain strings of more than one symbol thatbegin with the prime symbol .

There is a technical advantage in taking the base b to be a prime number butit is by no means essential. We can use base 10 and the operation 10 even with

thirteen symbols by, for example, assigning the thirteen symbols respectively thefollowing numbers (written in base 10):

0 ( ) f v = 1 0 2 3 4 5 6 7 89 899 8999 89999 899999

Of course on this assignment not every number is a Godel number. But we caneffectively tell the ones that are, i.e. we know that if an 8 or a 9 occurs its base 10notation it must occur within a string of the form 89, 899, 8999, 89999, 899999, andwe know which symbol is coded by counting the number of 9s in that string.


24/125

Lecture 2

Terms and formulas of the

language LE; expressibility ofdiagonal substitution in thelanguage LE

(Wednesday, 13 October 2010)

2.1 Terms and formulas of the language LE

2.1.1 Terms

Terms in the formal language LE are expressions that, on the intended interpretationof LE as a language for arithmetic, denote a number if the term does not contain afree variable, and if it contains one or more free variables, then the term that resultsfrom substituting a numerals for each variable denotes a number.

Definition 10 (Variables) v

is a variable, and if the expression E is a variablethen the expression E

, i.e. the concatenation of E and the subscript symbol

, is

a variable.

Remark: Formal variables are expressions of the form v, v

, v

, . . .. We will

abbreviate the string of symbols consisting of the formal variable symbol v followedby n subscripts as vn.

Further remark: Smullyans formal variables are of the form (v), (v), (v), . . .,i.e. formal variables as we have defined them enclosed in brackets (p. 15). Uniquereadability for terms within expressions does not require enclosing variables within

23


25/125

LECTURE 2 24

brackets in this way. The motivation for this unnecessary use of brackets might

be an artefact of whats simple to write in LaTeX. Evidently Smullyan producese.g. the substring v of his variable (v) by the LaTeX code v_{}. If this wasv2, then the seemingly natural way to write LaTeX code for the concatenation ofv2 with

would be $v_{}^{\prime}$, but this produces v, in which the sym-bol occurs interposed over the string v rather than concatenated at the end ofthat string. On the other hand, $(v_{})^{\prime}$ produces (v), in which

is correctly concatenated at the end. Nonetheless, more complicated LaTeX code(also with compounded subscript command to lower the apostrophe further so itlooks more convincing as a subscript) produces the required concatenation with vari-ables not enclosed in brackets, namely $v_{_{}}$\hspace{-.12ex}$^{\prime}$,

which compiles to the required concatenation of symbols v .

Definition 11 (Numerals) The symbol0 is a numeral (which denotes the numberzero). If the expression E is a numeral then the expression E is a numeral (whichdenotes the successor of the number denoted by E, so the expressions 0, 0, 0, 0, . . .are formal names of the natural numbers 0, 1, 2, 3, . . ., respectively).

Notation: For natural number n we write n for the numeral that denotes thenumber n, e.g. 7 = 0.

Corollary 8 (of Definition 11 and Notation) For any natural numbern, n + 1

is n, i.e. the numeral for the number n + 1 is the concatenation of the numeral forthe number n and the symbol .

Definition 12 (Terms) Among expressions of LE, the class of terms is specifiedby the following recursive definition:

Base clause: Each variable and each numeral is a term.Induction clauses: If t is a term, then t is a term. If t1 and t2 are terms,

then (t1f t2), (t1f t2), and (t1f t2) are terms. [Recall that the notations in LE foraddition, multiplication, and exponentiation are, respectively, f

, f

, and f

.]

Definition 13 (expression E1 occurs in expression E2) (i) Every expression oc-curs in itself.

(ii) Expression E1 occurs in expression E2 if there exist expressions E3, E4 suchthat E2 = E3E1E4 or E2 = E3E1 or E2 = E1E4Definition 14 (constant terms) A term in which no variable occurs is called aconstant term, or a closed term.


26/125

LECTURE 2 25

2.1.2 Formulas

Definition 15 (Atomic formulas) An atomic formula is any expression of theform t1 = t2 or of the form t1 t2, where t1 and t2 are terms.

Definition 16 (Formulas) The class of formulas is specified by the following re-cursive definition:

Base clause: Every atomic formula is a formula.Induction clauses: IfF andG are formulas, then F and(F G) are formulas,

and for every variable vi, the expression viF is a formula. [Note that the formula(F G) is enclosed in brackets, but that the other two formation rules do notintroduce new brackets.]

We will use logical equivalences between conjunction, disjunction, and existentialquantification and expressions in terms of , , and as abbreviations, i.e.

Definition 17 For formulas A and B,(A B) =df (A B);(A B) =df ( A B);viA =df vi A.

2.1.3 Free and bound variables; open formulas and sen-

tences

Definition 18 (a variable occurs free in a formula) (i) IfF is an atomic for-mula, all occurrences of variables in F are free.

(ii) For F and G formulas, if variable vi occurs free in F, then vi occurs free in F and in (F G) and (G F).

(iii) For any formula F, if vi occurs free in F, vi is free in vjF iff j = i.

Definition 19 (bound variables) A variable occurs bound in a formula F iff itoccurs in F and does not occur free in F.

Definition 20 (open formulas) A formula with one or more free variables is anopen formula

Notation: We write F(vi) to signify a formula in which the variable vi occursfree. Other unspecified variables may occur free as well unless we stipulate thatvi is the only variable free in F(vi). In the latter case we may say that F(vi) is aone-place formula. Similarly we write F(vi1, . . . vik) for a formula in which variablesvi1, . . . vik occur free, possibly with other free variables unless we stipulate that theseare the only ones.


27/125

LECTURE 2 26

[Note that this convention on possible occurrence of free variables other than

those explicitly shown is different from Smullyans: We write F(vi1, . . . vik) for anyformula in which vi1 , . . . vik are the only free variables. (p. 16). Since there aresituations, e.g. in stating the Induction axioms, in which we need to allow for thepossibility of other free variables than those explicitly shown, Smullyans conventionhas in those situations to be violated, e.g. F(v1) is to be any formula at all (itmay contain free variables other than v1) (Smullyan, p. 29). It seems to me morecoherent for the convention to allow other variables, which also then allows us underthe same convention to stipulate in a given situation that there are no other freevariables.]

Definition 21 (closed formulas a.k.a. sentences) A formula with no free vari-ables is a closed formula, also called a sentence.

Notation: Substitution of numerals for free variables in formulas: We writeF(n) to signify the result of substituting the numeral n for every free occurrenceof vi in the open formula F(vi). If vi is the only variable free in F(vi), then F(n)is a sentence. For numbers n1, . . . , nk, F(n1, . . . , nk) signifies the result of substi-tuting the numerals n1, . . . , nk for all free occurrences in F(vi1, . . . vik) of vi1, . . . vik ,respectively. Ifvi1, . . . vik are the only variables that occur free in F(vi1, . . . vik), thenF(n1, . . . , nk) is a sentence.

Definition 22 (regular open formulas) An open formula is said to be regular ifits k-many free variables are the first k variables.

For F(v1, . . . , vk) a regular open formula, the expression F(n1, . . . , nk) is unam-biguous; we dont need to stipulate which number is substituted for which variable.

Note that for F(v1) a formula with one free variable, F(n) is a sentence whileF(n) is not. The latter can be construed as shorthand for the statement that theopen formula F(v1) is satisfied by the number n, and that statement will be true ifand only the sentence F(n) is true.

2.2 Designation by terms in LE, truth of sentencesof LE, and expressibility of sets and relationsof natural numbers by formulas of LE

2.2.1 Designation

On the intended interpretation of the formal language LE each constant term des-ignates a particular natural number, by the following recursive specification:


28/125

LECTURE 2 27

(i) the numeral n designates the number n.

(ii) If constant term c designates n, then constant term c designates the nextnatural number after n; if constant terms c1 and c2 designate n1 and n2 respectively,then (c1fc2) designates the sum of n1 and n2, (c1fc2) designates the product of n1and n2, and (c1fc2) designates n1 raised to the power n2.

2.2.2 Truth

Truth for a sentence of LE in the structure of the natural numbers (the intendedinterpretation) can be defined by recursion over the recursive generation of thesentence in the usual way. For none of the results in this course do we require a

formal definition of truth, and I will take it as known informally what it means fora formula in the language of arithmetic to be true in the structure of the naturalnumbers.

2.2.3 Expressibility

Definition 23 (expressibility of relations) A formulaF(v1, . . . , vn) inLE is saidto express a relationR Nn iff for everyn-tuple < k1, . . . , kn > of natural numbersthe sentence F(k1, . . . , kn) is true iff < k1, . . . , kn > R. In such case the relationR is said to be expressible in LE.

Definition 24 (expressibility of functions) A function f(v1, . . . , vn) : Nn Nis expressible in LE iff the relation f(v1, . . . , vn) = vn+1 is expressible in LE.

Definition 25 (Arithmetical) A relation or a function is Arithmetical if it isexpressible by a formula in LE.

Definition 26 (arithmetical) A relation or a function is arithmetical [with alower-case a] iff it is expressible by a formula in LE in which the expression f(for exponentiation) does not occur.

2.3 Concatenation of numbers in a given base no-tation is Arithmetical.

Lemma 9 For a fixed number b 2, the condition that v1 is a power of b, whichexpression we abbreviate as P owb(v1), is Arithmetical.

Proof. P owb(v1) iff v2(v1 = bv2), or more formally

P owb(v ) iff v v = (bf v ).


29/125

LECTURE 2 28

Lemma 10 For b(n) the length of the base b notation for n, i.e. the number of

digits in the base b notation of n, the two-place relation bb(v1) = v2 is Arithmetical

Proof. This relation is expressed by the following condition on v1 and v2.

((v1 = 0v2 = b)(v1 = 0P owb(v2)v1 < v2v3((P owb(v3)v1 < v3) v2 v3))).

This equivalence is seen as follows: If v1 = 0, (v1) = 1 and b1 = b. The first

disjunct takes care of this case. Ifv1 = 0, then the length ofv1 (in base b notation) isthe least power of b > v1, e.g. 10(935) = 3, and 10

3 = 1000 is the least power of 10> 935. [Note that we need to treat these two cases separately, since (0) = 1, but

writing v3F(v3) for the least v3 s.t. F(v3)v3(10

v3

> 0) = 0 since 10

0

= 1 > 0.]That v2 is the least power of b greater than v1 is expressed by the two conditionsthat bv2 > v1, and for any v3 such that b

v3 > v1, v2 v3.The above condition is expressible in LE by Lemma 9 and the fact that v1 < v2

is equivalent to (v1 v2 v1 = v2).

Theorem 11 For any number b 2, the relation v1 b v2 = v3 is Arithmetical.

Proof. The relation v1 b v2 = v3 is expressed by the condition that

v1 bb(v2) + v2 = v3.

For example, 1570 10

365 = 1570365 = 1570000 + 365 = 1570 103 + 365 =1570 1010(365) + 365.

This condition is equivalent to

v4(bb(v2) = v4 ((v1 v4) + v2) = v3).

By Lemma 10, this relation is Arithmetical.

2.4 Substitution and diagonal functions, and theirarithmetization

The operation of substituting a numeral for a free variable lies at the heart ofthe construction of a self referential arithmetical sentence (This sentence is notprovable in the given formal system). To describe the formula F(n) obtained bysubstituting the numeral n for the free variable vi in the formula F(vi) is complicated(requiring recursion over the logical complexity of formulas). We follow Smullyanin utilizing a trick due to Tarski by which we construct a formula F[n] which is notthe same formula as F(n) but is logically equivalent to it for which there is a simplegeneral description that does not depend on the logical complexity of the formulaF(vi).


30/125

LECTURE 2 29

Definition 27 (quasi-substitution) For F(v1) any formula of LE with one free

variable and n any numeral, F[n] =df v1(v1 = n F(v1)).

Quasi-substitution is logically equivalent to substitution, i.e.

Lemma 12 (v1(v1 = n F(v1)) F(n)) is logically valid.

Proof. (i) Suppose v1(v1 = n F(v1)). Then by universal instantiation,(n = n F(n)). Since n = n is logically valid, by modus ponens, F(n).

(ii) Suppose F(n). Then by substitutivity of identity, (v1 = n F(v1)). Byuniversal generalization, v1(v1 = n F(v1)).

Note We could also have defined F[n] as v1(v1 = n F(v1)) since:

Lemma 13 (v1(v1 = n F(v1)) v1(v1 = n F(v1))) is logically valid.

Proof. Exercise.

Our first step in the arithmetization of syntax, i.e. showing that syntactic oper-ations on expressions ofLE can be reflected into arithmetically definable operationson their Godel numbers, is to show that the function

s(v1, v2) = v1(v1 = v2 Ev1)

is Arithmetical.

The value of the function s(v1, v2) is the Godel number of a formula logicallyequivalent to the substitution of the numeral of the number v2 into the expressionEv1 when Ev1 is a formula in which v1 occurs free. Note that for some values ofv1,Ev1 is a formula in which the variable v1 occurs free, and for other values it is not.Indeed for some values of v1, the expression whose Godel numbers if v1, i.e. Ev1 ,will not be a formula, in which case s(v1, v2) is the Godel number of an expressionwhich is not a formulaa dont care case. If Ev1 is a formula in which v1 does notoccur free then s(v1, v2) is the Godel number of a formula that has nothing to dowith substitution of the numeral for v2 into itanother dont care case.

Before we can establish that the two-place function (three-place relation) s(v1, v2) =v3

is Arithmetical, we must calculate the Godel numbers of the numerals.

Lemma 14 The Godel number of the numeralv2 is 13v2 (in our base-13 assignment

of Godel numbers ).

Proof. The numeral v2 is the expression 0

v2 . . . . The symbol 0 is assigned Godel

number 1, the symbol is assigned the number 0, so that whole expression is, by

concatenation, assigned the number written in base-13 notation as 1

v20 . . . 0, which

is 13v2 .


31/125

LECTURE 2 30

Theorem 15 The function s(v1, v2) = v1(v1 = v2 Ev1) is Arithmetical.

Proof. Let k = v1(v1 =, a particular number whose base 13 notationgiven our assignment of base 13 digits to the symbols of our languageis 965265(or if we use the base 10 G odel numbering also given in Lecture 1, whose base10 notation is 899652658999). Given that v2 = 13

v2 , = 8, Ev1 = v1,) = 3, s(v1, v2) = v3 iff v4(v4 = 13

v2 v3 = k v4 8 v1 3), which by left-hand association of 13 and repeated use of Theorem 11 is Arithmetical, i.e. thereis a formula S(v1, v2, v3) such that for all numbers m,n,k, S(m,n,k) is true iffs(m, n) = k. Expressing this argument from Theorem 11 more strictly, the formulaneeds to be the following:

v4v5v6v7(v4 = 13v2 k v4 = v5 v5 8 = v6 v6 v1 = v7 v7 3 = v3)

Definition 28 (diagonal substitution) The diagonal substitution functiond(v1)is s(v1, v1), i.e. d(v1) =df v1(v1 = v1 Ev1).

Remark. Since by Definition 27, v1(v1 = x Ev1) =df Ev1[v1], Definition 28means that d(v1) = Ev1[v1].

Corollary 16 (corollary to the proof of Theorem 15) The relation d(v1) = v2 isArithmetical.

Proof. Let D(v1, v2) be the formula that results by substituting v1 for v2 andv2 for v3 in S(v1, v2, v3). For all numbers m, n, S(m,m,n) is true iff s(m, m) = n.By the definition of d(v1), s(m, m) = d(m). So D(m, n) is true iff d(m) = n.


32/125

Lecture 3

The Diagonal Lemma;

expressibility of properties ofsequence numbers

(Tuesday, 19 October 2010)

3.1 The Diagonal Lemma

Theorem 17 (The Diagonal Lemma) For each one-place formula F(v1) in LE,there exists a sentence C in LE such that the equivalence (C F(C)) is a truesentence in LE.

Proof. (1) Let D(v1, v2) be the formula in LE from the proof of Corollary 16that expresses the relation d(v1) = v2.

(2) Let k =df v2(D(v1, v2) F(v2)).(3) Let C =df v1(v1 = k v2(D(v1, v2) F(v2))).(4) By Lemma 12, (C v2(D(k, v2) F(v2)).(5) By (1), (v2(D(k, v2) F(v2)) v2(d(k) = v2 F(v2))).

(6) By Lemma 12, (v2(d(k) = v2 F((v2))) F(d(k))).(7) By the chain of equivalences (4), (5), (6), (C F(d(k))).(8) By (2) and (3) and Definition 28, C = d(k) (fuller explanation below).(9) From (7) and (8) by substitutivity of identity, (C F(C)).

Explanation of step (8): k is the Godel number of a formula with one free variable,C is the formula that results from the diagonal substitution of quasi-substitutinginto that formula its own Godel number, and the one-place function d(v1) generatesfrom the Godel number of a formula with one free variable the Godel number of theformula that results from diagonal substitution into that formula.

31


33/125

LECTURE 3 32

The proof of the Diagonal Lemma is by a kind of double substitution into the one-

place formula for which a diagonal sentence is being established, first the substitutionof the Arithmetical expression of the diagonal function, and then the substitutionof the numeral for the Godel number of the formula that results from that firstsubstitution. Both of these substitutions are quasi-substitutions in this construction.It might make the idea of the proof more perspicuous if we consider how it goeswith actual substitutions if we have a term s(v1) in our language such that s(v1) =Ev1(v1). (We could have such a language, at the cost of taking more functions asprimitive.)

Theorem 18 (variant diagonal lemma) Given a formula F(v1) with one freevariable in a language for arithmetic L that has a term s(v1) such that for eachnumber n, s(n) = En(n), there is a sentence C in L such that (C F(C)) istrue.

Proof. Consider the formula F(s(v1)) formed by substituting the term s(v1) forthe free occurrences of v1 in F(v1). (1) Let k = F(s(v1)). (2) Let C =df F(s(k)).

(3) Then s(k) = C. (4) The numeral s(k) designates the same number as isdesignated by the term s(k), i.e. the equation s(k) = s(k) is true, so by substitutivityof identity, (F(s(k)) F(s(k))). (5) Hence by (2) and (3), (C F(C)).

3.2 Expressibility of properties of sequence num-bers in the language LE

3.2.1 Properties of sequences of digits

The first tool we need in order to code sequences of numbers in LE is to show thatwe can express in LE the relations that the base b notation of a number m begins,or ends, or is part of the base b notation of a number n.

Definition 29 (x begins y) x begins y in base b notation iff the base b notationof x is a (not necessarily proper) initial segment of the base b notation of y. We

write this as xBby.Examples. In base 10, 2 begins 20, but note that in base 13, 2 (as it is written

in base 10, and in base 13) does not begin 20, since 20 base 10 is written 17 base13. Other examples (base 10): The numbers which written in base 10 are 7, 76, 760,7600, 76007, 760074, and 7600748 all begin 7600748 in base 10. The last of theseexamples points up the fact that every number begins itself, i.e. an initial segmentneed not be a proper initial segment. Note that the number 0 does not begin anynumber except itself, i.e. we dont say that 0 begins 760748, even though 0760748= 760748.


34/125

LECTURE 3 33

Definition 30 (x ends y) x ends y in base b notation, which we write as xEby,

if the base b notation of x is an end segment (not necessarily proper) of the base bnotation of y.

Examples. In base 10, the following numbers all end 7600748: 7600748, 600748,748, 48, 8.

Given the notion of one number beginning another in a given base representationand the notion of one number ending another in a given base, we can define thenotion of a number being part of another in a given base in terms of these twonotions:

Definition 31 (x is part of y) x is part of y, in base b notation, which we write

as xPby, if x ends some number that begins y.Remark. Every number is a base b part of itself. Given a base b notation for

x, every proper sub-segment of the base b notation that does not begin with a 0 isthe base b notation of a number y that is a base b part of x. In base 10, the partsof 2600748 are all the numbers that begin or that end it, and 60074, and all thenumbers that begin or end it, and 7.

Theorem 19 For any b 2 the fol lowing relations are Arithmetical: (1) xBby, (2)xEby, (3) xPby and, for any natural number n 2, (4) x1 b . . . b xnPby

Proof. We will prove a stronger result, which we need later, that these relations

not only are expressible in LE, but also that this can be done using only boundedquantifiers, i.e. that these are finitary properties of numbers.1. If 0 does not occur in the base b numeral for y, then x begins y just in case

there exists z such that x b z = y. However, if a zero or a string of zeros occursin the base b numeral for y and the base b numeral for x is an initial segment ofthe base b numeral of y which ends just before the 0 or string of 0s in the base bnumeral for y or includes some but not all of those 0s, then the numeral of x hasto be extended by the remaining 0s before it can be concatenated with a numeralto result in the numeral for y. The extension of the base b numeral for x by therequired number of 0s is accomplished by multiplying x by b raised to the power ofhow many 0s need to be appended. This condition can be expressed in terms of thepreviously expressed notions P owb(w) and x b z = y, as follows:

xBby iff (x = y (x = 0 (z y)(w y)(P owb(w) (x w) b z = y)))The bounds on the quantifiers hold from the fact that if z is part of y, then

z y, and any number of the form 10...0 in base b with a string of 0s of a string of0s in y is y.

2. xEby iff (x = y (z y)(z b x = y)For this case we dont have any complications from the occurrence of zeros.3. xPby iff (z y)(xEbz zBby)4. x1 b . . . b xnPby iff (z y)(x1 b . . . b xn = z zPby).


35/125

LECTURE 3 34

3.2.2 Sequence numbers

Treating sequences of expressions as expressions with Godel numbers is very conve-nient. It requires making sequences of expressions into single expressions, which isdone by expanding the langauge of arithmetic by introducing the symbol to markout the beginning and end of a sequence of expressions and the boundary betweentwo successive expressions. Thus among the primitive symbols of LE (Lecture 1) is, which did not enter into the rules for the formation of terms and formulas of LE.The use of this symbol is to allow us to concatenate a finite sequence of formulas ofLE into a single expression of the language, by serving as a marker between differentformulas in a sequence of formulas. When that is done, a sequence of formulas, asan expression, will have a Godel number.

Definition 32 (sequence number) x is a sequence number if it is the Godelnumber of an expression of the form Ei1Ei2 . . . E ik in LE where each expressionEij does not contain the symbol ,

3.3 Coding of finite sequences of Godel numbers

Recall the assignment of the first 12 digits of base 13 representation of naturalnumbers to the 12 symbols that enter into formulas of the language LE:

0 ( ) f v = 1 0 2 3 4 5 6 7 8 9

Thus if a number is the Godel number of a formula in LA on the particularGodel numbering we have adopted, then the 13th digit, , will not occur in its base13 representation. Call the class of such numbers N.

A formal proof is a finite sequence of formulas, so to code a proof by a numberit suffices to find a way of coding finite sequences of numbers in N. We code such asequence (a1, . . . , an) by the number 13 a1 13 13 a2 13 13 . . . 13 13 an 13 . Infuture I shall mostly suppress the explicit notation of base 13 (or more generally base

b for any b 2) concatenation and write v1 = v2v3 for v1 = v2 b v3, i.e. symbolizethe concatenation relation by concatenation itself.

There are several points about the concatenation relation that need to be bornein mind. (1) It is a three place relation and not a two-place function. (2) It isa relation between numbers, and numbers are expressible in base b notation for allb 2 but are not in base b notation. The situation is similar to what it is in numbertheory generally. When we compute with natural numbers we do so using their base10 notation (or in the case of computers, base 2 notation). But when we provesomething about numbers what we prove is proved using properties of numbers that


36/125

LECTURE 3 35

are not specific to decimal notation, even if what is being proved specifically refers

to decimal notation, as in if the digits of a number in its base 10 notation add upto 9 then the number is divisible by 9, which is proved from general properties ofthe congruence relation and the fact that 10 1 (mod 9).

Proposition 20 (sequence numbers) A natural numbern is a sequence numberiff n = a1a2 . . . an for ai N.

Proof. Immediate from the definition of sequence number above.

Proposition 21 The property of being a sequence number, Seq(v1), is Arithmetical,i.e. it is expressible in LE.

Proof. The property of being a sequence number is expressible by the followingformula:

(Bv1 E v1 = v1 Pv1 (v2 v1)(0v2P v1 Bv2))

The first four conjuncts characterize the required occurrences of the digit in thebase 13 representation of v1. The last conjunct rules out the occurrence of a stringof zeros of length greater than one. The reason for this requirement is that sequencenumbers code sequences of numbers in N. Since 00 = 0, if, e.g., 00 were allowedas a sequence number it would code the sequence (0) but which is also coded by0. Since 00 = 0, if we allowed, e.g. 00 as a sequence number, any sequencewhich includes the number zero would have more than one (indeed infinitely many)sequence numbers.

Definition 33 For v1 a sequence number, we say that v2 is in v1, symbolized asv2 v1, iff v2 is one of the numbers coded by v1

Proposition 22 v2 v1 is Arithmetical.

Proof. v2 v1 iff (Seq(v1) v2P v1 P v2). It is a necessary condition for

v2 v1 that v2P v1 but not sufficient since numbers of the form a1a2 satisfy it;the condition P v2 rules out those cases.

In expressing the condition that a number is the Godel number of a proof ina formal system we need to be able to express the condition that one part of asequence number occurs earlier in the sequence than another. We do this as follows.

Definition 34 v2 v1v3 iff v1 is the sequence number of a sequence in which v2

and v3 occur and the first occurrence of v2 in the sequence is earlier that the firstoccurrence of v3 in the sequence.


37/125

LECTURE 3 36

Proposition 23 The three-place relation v2

v1v3 is Arithmetical.

Proof. v2 v1v3 iff (v2 v1 v3 v1 (v4 v1)(v4Bv1 v2 v4 v3 v4))

Note that the formulas v2 v1 and v3 v1 each contains the condition Seq(v1),so we dont need a separate conjunct Seq(v1).


38/125

Lecture 4

A formal system PAE for

arithmetic; an Arithmetical proofpredicate for PAE; a weak versionof Godels first incompletenesstheorem for PAE

(Wednesday, 20 October 2010)

4.1 A formal system PAE for arithmetic

We now begin the investigation of formal first-order axiomatic systems of arithmetic.A theory is first-order if its formal language is first-order. A formal language is first-order if its quantifiers range only over the objects in its domain of interpretationand not over collections (pluralities) of those objects. A second-order language hasquantifiers that range over collections (pluralities) of objects (possibly also over

relations between objects). Formal systems of first-order logic are complete (whichGodel proved in 1930, in his doctoral thesisthe incompleteness theorem was hisHabilitation thesis). There is no complete axiomatization of full second-order logic.Accordingly, where we are interested in properties of what can, or cannot, be provedin formal systems, we shall be concerned only with first-order systems. In this coursewe will investigate properties of a number of different formal systems for arithmetic.

All the formal systems we investigate in this coursewill explicitly or by assumption contain a com-

37


39/125

LECTURE 4 38

plete axiomatization of classical first-order logic

with identity.

We now set out a formal axiomatic system for arithmetic, PAE in the languageLE. PA stands for Peano Arithmetic, a standard misnomer

1 The subscript Esignifies that in this system exponentiation is taken as primitive, i.e. it is governedby its own axioms. In Lecture 5 we shall see that exponentiation need not be takenas a primitive and that via coding of ordered pairs of natural numbers the relationxy = z can be expressed in terms of zero, successor, addition, and multiplication.

We shall follow Smullyan in the specification of PAE. The axioms are in fourgroups, the first two of which, with two rules of inference, is a formal system of

first-order predicate logic, and the third and fourth groups are axioms specific toarithmetic.

Group I are axiom schemata for propositional logic: These are the standardaxioms for propositional logic with and as primitive. L1 and L2 are exactly theaxioms for required to establish the Deduction Theorem (taken as proved in aprevious course). L3 establishes the classical logic of . These axioms are completefor truth-functional validity.

Group II are axiom schemata for first-order predicate logic. The axiomatizationof first-order predicate logic by the Group II schemata is highly unnatural in termsof establishing formulas as logically valid. Its virtue for us is that it is very easy to

arithmetize since it involves no substitution of terms for free variables, which morenatural axiomatizations of predicate logic with identity do. Note that L6 strictlyshould be written vi vi = t. To prove the following valid formulas from theseschemata is non-trivial: vi = vi, (vi = vj vj = vi), (viF(vi) F(t)) for t anyterm of LE not containing a variable that is quantified in F(vi) within the scopeof which vi occurs. For proofs of these formulas from these schemata see DonaldKalish and Richard Montague, On Tarskis formalization of predicate logic withidentity, Archiv fur mathematische Logik und Grundlagenforschung 7 (1965), pp.81-101, Lemmas 2, 3, and 8 on pp. 85-87.

Group III are axioms specific to each of the primitive non-logical notions of the

languageGroup IV is all instances of (a version of) the axiom schema of induction. Ausual formulation of the induction schema is

(F(0) (v1(F(v1) F(v1)) v1F(v1)))

.

1It was Dedekind who established the first axiomatization of arithmetic, in 1888, which Peano

took over in his publication a year later. Peano cites Dedekind 1888 as the source of his axioms.

It seems to have been Russell who introduced the misnomer Peano Arithmetic.


40/125

LECTURE 4 39

However, two formulas within this schema are generated by substitution, namely

F(0) and F(v1), and for ease of arithmetization we want to use quasi-substitutioninstead of substitution. We cant use quasi-substitution directly on F(v1) to expressF(v1), since F[v

1] would be v1(v1 = v

1 F(v1)), in which no variable occurs

free, so not equivalent to F(v1). We could change the variable in the auxiliaryquantification, say to v2 if v2 does not occur free in F, i.e. v2(v2 = v

1 F(v2)),

which is logically equivalent to F(v1). But this involves substitution of the variablev2 for all free occurrences of v1 in F, which would defeat the purpose of avoidingsubstitution. However, we can use a quasi-substitution to obtain from F(v1) aformula logically equivalent to F(vi), namely v1(v1 = vi F(v1)) where vi is anyvariable that does not occur in F(v1). Then we can use quasi-substitution to obtain

a formula without any substitutions that is logically equivalent to F(v1), namelyvi(vi = v

1 v1(v1 = vi F(v1)). We abbreviate this formula as F[[v

1]]. Its

easily seen that F[[v1]] is logically equivalent to F(v1). This logical equivalence only

requires that vi does not occur free in F(v1), but the sufficient condition that it doesnot occur at all in F(v1) is easier to express in arithmetized syntax, and thats thecondition we take.

Definition 35 (the system PAE) The axioms and rules of inference of PAE arethe following:

A. Logical axioms and rules of inference

Group Ipropositional logic. All instances of the following schemata:L1 (F (G F))

L2 (F (G H)) ((F G) (F H))L3 (( F G) (G F))Group II predicate logic. All instances of the following schemata:L4 (vi(F G) (viF viG))L5 (F viF), provided vi does not occur in F.L6 vi(vi = t), provided vi does not occur in t.L7 (vi = t (X1viX2 X1tX2)), where X1 and X2 are any expressions such

that X1viX2) is an atomic formula and t is any term of LE.Rules of inference

R1 From F and (F G), infer G. [Modus Ponens]R2 From F, infer viF. [Generalization]B. Non-logical axioms

Group III axioms specific to each of the primitive non-logical notions of thelanguage

N1 (v1 = v

2 v1 = v2)

N2 0 = v1

N3 (v1 + 0) = v1N4 (v1 + v

2) = (v1 + v2)


41/125

LECTURE 4 40

N5 (v1 0) = 0

N6 (v1 v2) = ((v1 v2) + v1)N7 (v1 0 v1 = 0)N8 (v1 v

2 (v1 v2 v1 = v

2))

N9 (v1 v2 v2 v1)N10 v

01 = 0

N11 vv2

1 = (vv21 v1)

Group IV axiom schema of mathematical inductionN12 (F[0] (v1(F(v1) F[[v

1]]) v1F(v1))),

where, for vi any chosen variable that does not occur in F(v1), F[[v1]] is

vi(vi = v1 v1(v1 = vi F(v1))). Recall that when we write a schematic formula

F(vi), unless we stipulate otherwise, variables other than vi may occur free in it.These other free variables are referred to as parameters.

Definition 36 (proof) A proof in PAE is a sequence of formulas each one of whichis either an axiom of PAE or follows from an earlier formula in the sequence by therule of Generalization or follows from two earlier formulas in the sequence by ModusPonens.

Definition 37 (provable) A formula F of LE is provable in PAE, symbolized asP AE F, if there exists a proof in PAE of which F is a member.

4.2 An Arithmetical proof predicate for PAE

Each numbered paragraph in this section is both a definition of a property or relationof numbers, and a proposition that that property or relation is Arithmetical, theproof of which is established by the formula that follows. Because it will be neededfor later results we prove, in all but cases (4) and (6), a stronger result than is neededfor the present theorem, namely that the expressing formulas from LE require onlybounded quantifiers. The quantifications in (4) and (6) can also be bounded, butthese cases are considerably more complicated. The property of being the Godel

number of a provable formula requires an unbounded existential quantifier.Both for ease of reading and of typesetting in the following I will, after the first

case, abbreviate the abbreviation 13 for base 13 concatenation by using concatena-tion itself, i.e. for vi 13 vj I will write vivj, and for 4 13 5, which is an abbreviationfor 0 13 0

and could also be written f 13 , I will write 45.(1) V ar(v1): Ev1 is a variable, an expression of the form v , i.e. the variable

symbol followed by a finite string of subscript symbols. Recall that the Godelnumber of the subscript symbol, on our chosen Godel numbering, is 5, and that ofthe variable symbol is 6.


42/125

LECTURE 4 41

(v2 v1)((v3 v1)(v3P13v2 0P13v3) v1 = 0

13 v2)

In this formula I write out the formal numerals 0 and 0 rather than ab-breviating them as 5 and 6, respectively, to bring out the fact that the relation0P13v3) is between numbers, in this case between the number 5 (as we write itin base 10, and also, as it happens, in base 13 notation) and some other numberand not a relationship between numerals, though the relation between numbers isdetermined to hold or not by going from the number to its, in this case, base 13representation. The number required to exist by the quantification over the variablev2 is the Godel number of a string of subscripts, by the condition that the subscriptsymbol is a part of every part of that expression. All of which is to way that whenthe formula V ar(v1) is written in the primitive notation of LE, the numbers in it

will be expressed by numerals, i.e. 0 for 5 and 0 for 6, and similarly in the restof the formulas expressing arithmetized syntax.

(2) N um(v1): Ev1 is a numeral, i.e. an expression of the form 0...

P ow13(v1)(3) Seqt(v1): Ev1 is a formation sequence for terms, i.e. a sequence of expressions

each one of which is either a variable or a numeral or the result of applying one ofthe four functions of successor, addition, multiplication, or exponentiation to anexpression or expressions occurring earlier in the sequence, i.e. of the form t or(t1f t2) or (t1f t2) or (t1f t2).

(Seq(v1) (v2 v1)(v2 v1 (V ar(v2) N um(v2) (v3 v1)(v3 v1v2 v2 =

v30)(v3 v1)(v4 v1)(v3 v1 v2v4 v1 v

2(v2 = 2v345v43v2 = 2v3455v43v2 =

2v34555v43)))))(4) T m(v1): Ev1 is a term.v2(Seqt(v2) v1 v2)

Note: The formula Seqt(v2) in (4) is obtained from the formula Seqt(v1) in (3)by changing the free variable from v1 to v2. In changing the free variable in thisway corresponding changes of bound variables in Seqt(v1) must be made so that v1is free for v2 in a logically equivalent transform of Seqt(v1), e.g.

(Seq(v1) (v5 v1)(v5 v1 (V ar(v5) N um(v5) (v3 v1)(v3 v1v5 v5 =

v30)(v3 v1)(v4 v1)(v3 v1 v5v4 v1 v5(v5 = 2v345v43v5 = 2v3455v43v5 =2v34555v43)))))

If we had given the formula in (4) as the logically equivalent formula v5(Seqt(v5)v1 v2), the only change needed to obtain Seqt(v5) from Seqt(v1) is to replace alloccurrences of v1 by v5.

Note: The formula in (4) above contains an initial unbounded existential quan-tifier. This quantifier can be bounded by the correlate in arithmetized syntax ofProblem 2 on Problem sheet 1, i.e. decidability of whether or not an expression is


43/125

LECTURE 4 42

a term, but its a delicate question in which languages, i.e. with what primitives,

that bound can or cannot be expressed.(5) AF(v1): Ev1 is an atomic formula, i.e. of the form t1 = t2 or t1 t2 for t1, t2

terms.(v2 v1)(v3 v1)(T m(v2) T m(v3) (v1 = v2v3 v1 = v2v3))(6) Seqf(v1): Ev1 is a formation sequence for formulas, i.e. a finite sequence of

expressions each one of which is either an atomic formula or of the form E for Eoccurring earlier in the sequence or of the form (Ei Ej) for Ei and Ej occurringearlier in the sequence or of the form viE for vi any variable and E occurring earlierin the sequence.

(Seq(v1)(v2 v1)(v2 v1 (AF(v2)(v3 v1)(v3

v1

v2v2 = 7v3)(v3

v1)(v4 v1)(v3 v1v2 v4 v1

v2 v2 = 2v38v43) (v3 v1)(v4 v1)(v3 v1v2 V ar(v4) v2 = 9v4v3))))

(7) F m(v1): Ev1 is a formula.v2(Seqf(v2 v1 v2))

Note: The remark as at (4) above applies here also. We know by Problem 2on Problem sheet 1, that we can determine by a finite search whether an expressionis a formula, but its a delicate

Documents

Incompleteness Theorems MT 2010