CALIJ:i'ORNIA STATE UNIVERSITY, NORTHRIDGE

GODEL'S " "ON FORMALLY UNDECIDABLE ••• " •

A thesis submitted in partial satisfaction of the requirements for the Degree of Master of Sciences in

Mathematics, Option II

by

Mary Margaret Pepe

June, 1976 Pepe is approved:

Date '

Dr. Donald H. Potts Date

yalifornia State University, Northridge

ii PREFACE

Any of arithmetic encompassing the addition and multi­

plication of positive integers and zero contains arithmetical proposi­

:tions which can neither be proved or disproved within the system.

Kurt Godel, a young Austrian mathematician who had imigrated to the

United States and became a member of the Institute for Advanced Study

at Princeton, announced his discovery to the Vienna Academy of Sciences

·in 1930. He published his detailed proof in his paper, "Uber formal

,unentscheidbare Satze der Principia Mathematica und verwandter Systeme

,I," in the Monatshefte Fiir Mathematik und Physik, Volume 38, pages 173-

:198 (Leipzig, 1931). Godel intended to write a second part to his

paper, but illness prevented him from publishing it.

Godel's is sometimes regarded as the most decisive result

1 in mathematical . In the paper which follows we shall familiarize

;ourselves with the () required to understand

•this theorem.

iii TABLE OF CONTENTS Page(s) PREFACE iii TABLE OF CONTENTS iv-v ABSTRACT vi-vii I. INTRODUCTION l

A. THE STATE OF MATHEMATICS @ 1930 l B. THE CREATION OF METAMATHEMATICS - 3 PROOF

II. SUPPORTING DETAILS FOR THE PROOF 8 A. GODEL NUMBERING - ARITHMETIZATION OF 8 METAMATH11MATICS B. SELF-REFERENTIAL STATEMENTS 15

1. Formulation of the Richard Paradox 15

2. The Richard Paradox 16 3. Resolving the Paradox 17 C. THE FORMAL SYSTEM 19

1. Basic Symbols 20

2. Well-Formed Formulas (WE'F' s) 21 3· (Initial Formulae) 23 4. "Immediate Consequence of" 24 D. PRIMITIVE RECURSIVE FUNCTIONS 24

1. Significance 24

2. Definitions 25 3. Definition by Induction - Recursive 27 Definition 4. Primitive Recursive 28

iv TABLE OF CONTENTS

~age(s)

5. Examples of Primitive Recursive Functions 31 6. Historical Utilization 33

III. GODEL' s PROOF 34

A. OF THE THEOruMS 34

B. TARSKI' S EXAMPLE :;4

C. GOOEL 1 S ASSUMPTIONS 35

D. LEMMA 1 38

E. DEFINITION: ¢ (x,y) 38

F. EXPRESSIBILITY 39

G. "Bew(x)" 39

H. LEMMA 2 4o

I. UMMA 3 4o

J. GODEL Is SECOND THEOREM 41

K. CONCLUSIONS 41

IV. EPILOGUE - IN MATHEMATICS 43

ANNOTATED BIBLIOGRAPHY 45

INDEX OF DEFINITIONS 48 Photograph of Kurt GOdel by Orren J. Turner viii •rables:

Table 1 - Godel Numbers of Constant Signs 9 Table 2 - Godel Numbers of Type 1 Variables 9 Table 3 - Godel Numbers of Type 2 Variables 10

Table 4 - Godel Numbers of Type 3 Variables 10

v ABSTRACT

GODEL'S

11 0N FORMALLY UNDECIDABLE PROPOSI'l'IOOS ••• 11

by Mary Margaret Pepe Master of Sciences in Mathematics, Option II

Godel's theorem is composed of two parts: Theorem 1: For suitable formal systems L, there are undecidable pro+ I positions in L; that is propositions F such that neither F nor not F is provable.

Theorem 2: For suitable L, the simple of L cannot be proved in L.

Part One of the thesis examines the state of mathematics @ 1930 and !the creation of - metamathematics.

I I Part Two presents the supporting details for the proot including:

l) GOdel numbering - the establishing of a one-to-one correspon~ dence between the finite expressions of the system and the natural numbers - is presented. I 2) Belt-referential statements - statements which talk about them The Richard Paradox is formulated, I selves - are described. l

vi examined, and resolved. -----· ··-· --~

3) The Formal System L is described - its basic signs, well-formet· formulas, axioms, and the relation "Immediate consequence of."

4) Primitive Recursive Functions are defined, examples shown,

and historical utilization described.

Part Three presents a Rosser version of the proof of 1 and

2 stated above. Reference is made to primitive recursive functions and numeralwise expressibility.

Part Four, an epilogue, considers ramifications regarding truth in mathematics.

An index of definitions and an annotated bibliography follow the thesis.

____ _jI

vii

I. INTRODUCTION 1 I A. THE STATE OF MATHEMATICS @ 19.30 At the second mathematical congress held in Paris in 1900, David I ! Hilbert posed twenty-three problems which he thought would be among I those occupying the attention of the twentieth-century mathematicians. I ! This paper is in partan answer to Hilbert's second query as to whether: the axioms of arithmetic are consistent. I I Before we continue we need to have a common understanding of what i is meant by consistent, complete, incomplete, decidable, undecidable, I provable, and disprovable. We will give naive definitiorJs and when I I necessary make these definitions more explicit when we an9.lyze the I metamathematics of the formal system. I

Con:::t:::tem of axioms is said to be consistent if a finite number of I logical steps based on the axioms would never lead to a contradiction. ~.~! I Inconsistent

Not consistent.

I Complete I in :~:·:::t:: :~: ::g::.:o::::::u::c:v::yt::-:.:::::ment expressible I

Incomplete j A system is said to be incomplete if ?£t eve!Y tru~ statement of the! system is deducible from the axioms. Decidable

A formula P written in the vocabulary of a given of axioms is

(1) (2)

said to be decidable if there is an (see page 5) for determining whether P or not P is deducible from the axioms.

Undecidable

A formula P is said to be undecidable if neither P nor not P is a theorem.

Provable

A formula P is provable, i.e., is a theorelii, iff there is a of the formula ending in P. A formal proof of a formula P is a finite column of formulas each of whose lines is an or may be inferred from the preceding lines by specified rules of inference

(such as modus ponE:ms : A, A--") B, ;. B).

Unprovable

Cannot be proved.

Bertrand Russell and Alfred North Whitehead (1861-1947) had made an elaborate e.ttempt to develop the fundame.ntal notions of arithmetic

/ from a precise set of axioms. Their text was in the tradition of

Leibniz, Boole, and Frege and was based on Peano's axioms. Russell and Whitehead intended to prove that all of pure mathematics could be derived from a small number of fundamen-tal logical principles. If they had succeeded their proof would have supported Russell's view that mathematics is indistinguishable from logic.

In 1926 Finsler published his paper "Formal proofs and undecid- ability." Finsler presented a that although false is formally undecidable. In 1930 Presburger published a decision pro- cedure applicable to every proposition of a system of arithmetic which used only addition, but not multiplication. Presburger proved

that every one of its propositions is decidable - either provable

or unprovable.

Hilbert had hoped to prove the consistency of elementary number

theory with the possibility of reducing other portions of classical

mathematics to that of the natural numbers via models.

Work along these lines carne to somewhat of a halt in 1931 when

GOdel demonstrated the impossibility of proving the consistency of I any formal theory by constructive methods formalizable within the

theory itself. This includes the formulas of the Natural numbers.

Godel proved that if the fot'mal .system is consistent then it is

incomplete. Nineteenth-century mathematicians had hoped that the

axiomc; for ari thmetio· were complete or could be made complete by the ..

addition of a finite number of further axioms. ~1e discovery that

this.·. was not so was more of a shock than Godel 1 s demonstrating the

existence of undecidable propositions.

. B. THE CREATION OF METAMATHEMATICS - PROOF THEORY

Metamathematics, which is the study of rigorous proof in mathe­

matics, was developed by Hilbert during the 1920's-1930's. ­

ematics looks at mathematics from the outside. It is concern~;d with

the interpretations of signs and rules and not with the operatlons of

arithmetic.

Godel's theorem is a result which belongs to metamathematics.

Godel put the notion of a formal system at the very center of his

investigations~ We thus need to define the following formal notions: (4)

Formal

When we use the word formal we are referring only to the forms of

certain expressions and not to their .

•Formal Theory

A formal theory is a completely symbolic built according

'to certain rules from an alphabet of specified primitive symbols.

String

A finite sequence of formal symbols.

Object Language

The symbolic ( f\)nnal) language in which the statements of the formal! i :theory are written. A statement in the is a statement !.2! the theory. I I I iHetalanguage - Language i I The language used to present the formal theory. A statement in i the is a statement about the theory. It is used to discus$ i the fonnal theory, which includes defining its syntax, specifying its

.axioms and rules of inference, and analyzing its properties.

Meta theorem

A theorem about a formal theory (written in English). A theorem of the theory is written jt1 the symbolism of the theory. A

'is a statement that is proved in the metalanguage (English) about the· theory.

'Metatheorz

Metatheory is the theory of formal and systems and their interpretations. It takes formal languages and systems and their in-

:terpretations as objects of study, and consists in the body of (5)

and conjectures about these objects. Among its main problems are

problems about the consistency, , and i!!,­

_?ependence of sets of formulas.

Admissible Methods

The methods used in the are restricted to finitary

methods. They employ only intuitively conceivable objects and per,

formable processes. No infi.alte clas::> may be regarded as a completed

infinite.

Denumerable: A set is said to be denumerable iff there is a one~to­

one correspondence between it and the set of natural numbers. The

natural numbers will be defined as. 0.,1,2,3, ••• (Logiciane 1 . definition).

Countable: A set is countable iff it is either denume:t'able or finite.

Uncountable: A set is uncountable iff it is neither denumerable l'lo:t'

finite.

Effective Procedure

By rm effective procedure we mean one of the following:

1) There must be exact instructions, finitely long, explaining

how to execute the procedure. The instructions should demand

no cleverness on the part of the person following them. The

idea is that a person who knows no mathematics or a computing

m~>chine (which does not think at all) should be able to execute

the procedure by mechanically following the instructions:

2) The procedure must avoid random devices (such as flipping a

coin), or any such device which can, in practice, only be

approximated.

3) If we are asking for a decision procedure, the procedure b------~------~ (6)

must produce a "yes" or a "no" <.::u31;vr;r after a finite number of

step~..;.

Effective Enumeration

An enumeration (listing) \olhich is finite or for which there is an

effective procedure for telling what the n!£ term is for each positive

integer n.

Effectively Decidable

If an effective procedure is found for a of propositions, this class is said to be effect:i,vely decidable ..

Formal Proof To prove a theorem

To prove a theorem means to produce a proof whose last step is the theorem.

Every axiom is a theorem. The statement of the axiom itself is the only one thus, the last, step of the proof.

A formal proof leaves nothing to the imagination. There is an ef- fective procedure for deciding \vhether or not a formal proof is correct.

Total

A function of n arguments from natural numbers to natural numbers is total iff its domain is the set of all ordered n-tuples of natural numbers. (If n=l, its domain is the set of all natural numbers.)

Computable

A function of n arguments from natural numbers to natural numbers is computable iff it is total and there is an effective method for computing the value of the function for each member of its domain. (?)

,------~.,.t-~v~-.t--.

there is an effective method for computing the sum of any t, ;:. n;ttural

numbers, so the sum functivit for natural numbers is a computttble

function. 'l't1t~ product function for natural numbers is anotr;, r com~

putable function.

Example: A non- h.

Now let be an effective enumerati:Al of tb.> formulas 0 1 1 2 of a set s and hl (f ,f ,f , •• be the corresponding enumera­ 1 0 1 2 1

tion of the functions determined by the formulas A0 ,A1 ,A2 , .... Let h be the function defined by the rule:

h(n) 1 if f (n) Q = n = [ h(n) = 0 otherwise

I.emma: The h is not compuCable.

Proof: Assum;:, h is computable. 'l"hen it must be one of the f. 1 s, J.

Then h(k) = fk (k) for some k.

if fk(k) ::: 0 then h(k) = 1

if fk(k) /z 0 then h(k) :.:: 0 so fk(k) /z h(k) which contradicts our assumption.

Therefore, h is not in our enumerati~,,~ and is thus not computable.

If it were we would have fk(k) = 1 iff fk(k) == o.

Now that we have a commv> nnderstanding of some of the major de-

finitions we are ready to proceed witb the supporting details for

the proof. II. SUPPORTING DETAILS FOR THE PROOF

A. GODEL NUMBERING - ARITHMET!ZATION OF METAMATHEMATICS

The following ingenious device or "trick" was discovered by GOdel..

This device is called tlh; arithmetL;;ation of metamathematics. We shall set -1p a one~·to-one correspondence between finite expressions and the natur<

GOdel-·numbers. GOdel used this trick to bypass the Richard Paradox

(Sec.:tion IIB).

The one-to-one -:; ;·r·esponrh::nce can be established as follows:

J) Assign numbers t. 11 the symbols, pdr:iitive or defined

\-Jh:i ctl occur in the language.

2) Numt;ers will <•l.sc be assigned to numbers. ('l'hese numbers

will not, in g<:meral, be the same as those numbers, or else

all of the numberr' would be uc;ed up in numbering numbers.)

:;) Strings of symbols can be arithmetized as follows:

Write down the prime numbers in ascendjng order for each succes

sive of the sentence. Attach the numbers of those sym­

bols v.n exponents to the prime numberr:•. The number of the

whole sentence will be the product of all those primes raised

to various powers.

The following is an example of one way to set up the one-to-one correspondence:

--····------·------'

(8) Table 1

Godel Numbers of Constant Signs

Constant signs Godel number Meaning

1 Zero

llfll 2 The succ~:s ;or of

l!i'J11 3 Not

II V !I 5

11 i 11 7 There exists an x

II v II 9 For all

11(11 11 Left Parenthesis ")" 13 Right Parenthesis

11-tll 17 If .... then 19 Equals Co nun a " ' " 23 11 "j x 11 is logically equivalent to """ V' ><- ~"'-' ". We assigned "3 a

Gddel number so that our examples would be easier to write•

Among his primitive symbols Gadel u.scd three types of Variables.

Godel variables of the first type are for individuals; i.e., natural numbers including o. These are nmnericnl vari '

Godel numbers of Type 1 Variables

Variables, T,ype 1 Numerical Gorlel number Possible

0 "x1 " 29 fx (The successor of x ) "y1 " 31 1 1 ffx (The successor of x "z1 " 37 1 2 (10)

Note that the variables of Type 1 are given GOdel numbers of the form pn where n=l and p is a prime number 7 23. Variables of type n will be assigned Godel numbers of the form p n , with PI 23. Type 2

') variables will be of the form pL, with p / 23, etc.

Variables of the second type (for classes of individuals) are called the sentential variables. They are the variables for which sentences may be substituted.

Table .2.

Godel Numbers of Type 2 Variables

Variables, Type 2 Sentential GOdel number Possible Interpretation 2 "x II 29 ,..._,. (fx =0) (No number has 2 1 zero for a successor) 2 lly II 2 31 ( fxl =fy1) ~ (xl = y1) ') 3?L ( ''z 2 " (g ~) :x:l::: fyl)

Variables, Type 3, are for classes of classes of individuals. We will call them predicate variables for which predicates such as Prime,

Composite, Greater Than, etc. could be used.

Type 3 variables will be assigned Godel numbers of the form p3, where p / 23.

Table Lt

GOdel Numbers of Type 3 Variables

Variables, Type 3 Predicate Godel number Possible Interpretation

ttx " :Prime 3

"Y II Composite 3

"z II Less Than 3 (11)

We may continue assigning Godel numbers for every natural number j I I type. It is presupposed that for every variable type denumerably many

signs are available. Variables of the type n are given Godel numbers n of the form p where p is a prime number/ 23. Thus, we have estab-

lished a one:..to-'one correspondence between every finite sequence of I basic signs (including formulas) and the natural numbers. We can now

map these finite sequences of natural numbers on to natural numbers by ~ n2 ~ 1 letting the number 2 • 3 •••• I\: correspond to the sequence

·~ • n ..... I)c, where pk denotes the k-th prime in the order of in- 2

sequence of such signs. The one-to-one correspondence between GOdel

:numbers and

Ilof the fundamental theorem of arithmetic, namely that every natural

number greater than 1 may be uniquely expressed up to order as the

product of prime factors. Example:

Let us examine the axiom '' .rV (fx =0)." 'l'his states that zero is not 1 the successor of any natural number. We determine its Godel number.

The numbers associated with the symbols are 3,11,2,29,19,1,13, respec-

tively. The one-to-one correspondence is shown below:

/'V ( f xl = 0 ) l X t J t t $ 3 11 2 29 19 1 13 We will assign a single number to the formula (axiom). We will write I down the prime numbers in ascending order, and attach the Godel numbersi I of those symbols as exponents. The number of the whole sentence will b,

.,,., '' _) (12)

the product of all the primes raised to their corresponding power. The

Godel number of the above formula becomes:

For simplicity's sake we will call this number a. Similarly, every

finite sequence of elementary signs may be assigned a Godel number.

Let us consider a sequence of formulas for which we would like to

determine the Godel number.

Consider the formulas:

(1) r-v( fx :::0 ) 1 (2) ( J xl )(xl = fyl)

(3) (3 x )(x fO) 1 1 = These formulas might occur as steps in a proof. The sequence must then have a Godel number. The Godel number of formula {1) is ~·

'Formula (2) has the following correspondence:

( 3 x ) 1 ( xl = f yl ) t J X t 1 t t r x J 11 7 29 13 11 29 19 2 31 13 Its Godel number is:

'Let us call this number b. And the number assigned by replacing the

1y with zero let us call .£• We would like to have a single number to 1 represent our sequence. Since it is three formulas in length, we agree to associate the first three primes, in order of magnitude, with the sequence. If we let a represent this number. Then the Gode1 number of the ,sequence is: ,a=Fxrxra b c In essence, every permissible expression of the system can be (13)

assigsted a unique Godel number.

Note: The Godel number of formula (3) differs only where "1", the

.Godel number of replaces "31" the Godel number of y • The Godel o, 1 i nwnbt·,· of formula (3) is thus: 11 7 22 13 11 29 19 2 1 13 > X 3 X 5 X 7 X 11 X 13 X 17 X 19 X 23 X 29

\ i (14)

GODEL NUMBERS - POST SCRIPT

We assigned natural numbers to sequences of signs of L, and to

; sequences of sequences of signs of L. Given a sequence, the number

assigned to it can be effectively calculated. Given a number, we can

!effectively decide whether the number is assigned to a sequence. If

'it is, we can write down the sequence. Not all numbers are GOdel

numbers. The one-to-one correspondence established by the rule is

between those natural numbers which are Godel numbers and the basic

signs, series of signs, and series of series of signs which are

arithmetized. (15)

B. SELF-REFERENTIAL STATEMENTS

The Richard Paradox is a self-referential statement. Self-

referential statements are statements thut talk about themselves.

GOdel's argument is closely akin to the Richard Paradox. We pre-

sent the formulation of the Richard Paradox. We then show how

Gpdel used this ·formulation, but resolved the paradox and discov-

ered his "unprovability" theorem.

1. ·Formulation of the Richard Paradox

Consider all. expressions of the English larigUage which are defini-

tions of properties of integers. Specifically, let the alphabet

include the usual 26 letters, ?- blank spi>ce to separate words,

and the comma. When we are talking about an expression in the

language we are talking about any finite sequence of our 28 sym-

bo1s that does not begin with a blank space. The number of these

expressions is denumerable (see Hunter p. 219 for proof) and we

can therefore arrange them in an infinite sequence:

(a) ~,R2 , ••• Rn,••• We can agree that R. precedes R. if R. contains less letters than ~ J ~ R.. If both expressions have the same number of letters, R. pre• J ~ cedes R. if R. precedes R. in the lexicographical ordering of J ~ J vJords.

For arbitrary integers m and p one of the following cases must

occur:

(b) m possesses the expressed by R , p (c) m does not possess this property.

We write in the case (b) ~ (m) and in the Rp- ease (c)-1- R (m). (16)

Consider now the property of an integer m expressed by the for- mula

/V 1- R (m) • m- This property has been defined in the English language and must therefore coincide with one of the properties (a).

Hence there is a d such that for each .!.!!. the conditions

Taking m = d we obtain a contradiction:

Richardian:

A number x is Richar~ if x does not have the property expressed by the sentence R in (a) above. X The following examples will make the use of the word "Richardian" clear.

Example a:

Suppose the definition "not divisible by any integer other than 1 and itself" is the 31£1 expression. Then the number 31 has the property defined by the expression• 3l.is said to be .!!;2i

Richardian.

Example b:

Suppose the defining expression "being the product of some integer by itself" is correlated with 20. 20 does not possess this pro- perty. 20 is said to be Richardian.

2. The Richard Paradox The defining expression for the property of being Richardian de- scribes a numerical property of integers. The expression must be one of the R.'s stated above. Let us assume that it is R, the Cl-7)

n!h expression. The number correlated with this expression is n.

Is n Richardian? ]'or n is Richardian iff n does not have the pro­

perty designated by the defining expression with which n is corre­

lated. In short, n is Richardian iff n is not Richardian. The

statement, "n is Richardian", is both true and false.

3. Resolving the Richard Paradox

Certain passages in the reasoning of the Richard Paradox leading

to the paradox are not translatable into the language of (t).

GOdel modified and translated these passages and thus obtained

his famous theorems.

Consider all expressions of (L) which are definitions of proper­

ties of integers. Consider all expressions (sentences) with one

free variable. The number of such expressions is denumerable

(see Hunter, p. 220) and can be arranged in a sequence: (a) ... If we try to continue the analogy as in the Richard Paradox we

have to consider the following property of an integer n:

{b) n does not possess the property expressed by Rn.

We have a problem. What does it moan for an integer to possess a

property expressed by a sentence containing one variable? Let us

consider the positive integers 1,2,3, ••• with n~th numeral~· It

is intuitively obvious that an integer satisfies a sentence in one variable if and only if the sentence R(~) obtained by substituting n for the nth varir::tble is true.

We do not know formally what a true sentence ls. But we know that the notion of formal provability is closely related to the intuitive notion of truth. 0.8)

We then substitute for (b) the following property of n:

(c) The sentence Rn..., (n) is unprovable in (L). Statement (c) does not possess the same intuitive meaning as (b).

But we can express (c) with any desired degree of precision.

Formula (b) is intuitively obvious but cannot be made precise.

We need to identify (c) as one of the R.'s. This is difficult ~ because (c) contains words like "sentence'' and "unprovable."

These are names of notions which belong to the grammar o;f (L).

We have no right to ma.intain that the property (c) is identical

with one of the properties expressed by the Ri's.

Godel now uses his trick of th.e arithmetization of metamathematics.

The property (c) is definedwith the use of grammatical terms. If

we obtain a new definition which is expressed in purely arith- metical terms and is identical with the former.

Let (p,m) be the Godel number of the sentence R (m). This is a ¢ p- non-arithmetical definition of a function ¢ with two arguments. We will see that this is equivalent to a purely arithmetical de-

finition expressible in (L); provided that (L) contains a suffi-

ciently large portion of arithmetic. Let T be the class of the

GOdel numbers of theorems of (L). This definition does not be- long to arithmetic but can be shown to be equivalent to a purely

arithmetical definition in each sufficiently strong system (L).

The property (c) is equivalent to the following:

(d) ¢ (n,n) non E. T

and hence to an arithmetical definition expressible in (L). It follows that there exists a sentence Rd of (L) which expresses (19)

in (L) the property (d). We can explicitly find this Rd when we

write the arithmetical sentence (d) in the language of the system

(L) •

For the final step, we substitute the d-th numeral for the unique

free variable of Rd. This gives us the ::;entence Rd (d) v1hich

corresponds to the sentence constructed in the Richard Paradox.

Rd (g) intuitively says that the number d hus the property ex­

pressed by Rd. Now Rd was a formalization of (d), and 1dnce

Rd(£) says the same as ¢ (d,d) non~ T and means the same as:

Rd (_s!) is unprovable. The sentence Rd (g) thus says of itself that

it is unprovable in (L). This seems very paradoxical. But by

the arithmetization of metamathematics for every sm'ltence L of

(L) there exists an arithmetical sentence L' expressible in terms

of the system (L) which says that L is unprovable. There is

nothing paradoxical in the fact that for a suitably chosen L

the sentence L' turns out to be identical with L.

We have resolved the paradox. (1) We shifted from "true" to

"provable." (2) We arithmetized our sentence. Then ipso facto

the paradox is resolved. Throughout this section we have been

informally talking about the formal system (L). We next briefly

examine the components of the formal system.

C. THE FORMAL SYSTEM

Formal Syste,m~

Definitions: A formal system contains a finite collection of sym­ bols and perfectly precise rules for manipulating these symbols to form certain combinations called theorems. The rules are given in in­ formal mathematical language. The completely explicit rules require that no infinite process be used, since it can't be coded into a com-

puting machine. Questions concerning infinite sets are replaced by

questions concerning the combinatorial possibilities of a certain

formal game. If we restrict ourselves to finite processes we will then be able to say that certain statements are not decidable with given formal systems.

GOdel gives an exact description of his formal system by, specify- ing its:

(l) Basic Signs (symbols)

(2) Well-formed formula - rules for forming "grammatically

correct" finite sequences. of symbols

(3) Axioms (initial formulae)

(4) The relation "Immediate Consequence of"

He states that his system is essentially obtained by superimposing

Peano's Axioms upon the logic of Principia Mathematica.

The following is an example of a formal system:

1. Basic Symbols

a. Logical Symbols

The logical symbols defined in the Godel numbering section:

11fll If If II II 11 II II v II 11(11 11)11 II II 11-11 II II , fV , v , 3 , w , ' ' ~ ' - , , The logical symbols always play the same role in translating

to and from English. b. Nonlogical Symbols

Symbols Intended Meaning

p,q,r, ••• Propositional symbols

x,y,z, ••• Variables (21)

Symbols Intended Meaning

0,1,2, ••• Natural numbers

f(O),f(l), ••• Successor function

x+y Sum function

x.y Product function

xy Exponentiation

X=:; y Identity function

X"- y Less than function

c. Notations as abbreviations:

Disjunction (or): V Logically equivalent to (A- p.....,q)

Conjunction (and): 1\ Logically equivalent to ,v (p__., N q)

Universal : V A means for every member of the

universe A

Existential q1;1antifier: ] x mean.'~ there exists a number x

logically equivalent to ..-v If x...... ,

Biconditional: ~ means iff (if and only if) logically

equivalent to (p~q)A (q-+p)

2. Well-Formed Formulas (WFF's)

a. Definition of WFF

i. Any propositional symbol is a WFF.

ii. If A and B are WFF' s, then ("-' A), (AV B) and

(A...,B) also are.

iii. No expres~ion is a WFF unless it is so by one of

the above.

b. Examples of WFF' s

i. p (22)

ii. ;v p

iii. (q ~ p) c. Examples of Non-WFF's

i. q - p (No parentheses)

ii. x (x was not defined as a symbol) 2 2 iii. (p~ q (No right parentheses) d. Definitions for. WFF's

Free and Bound Occurrences of Variables

An occurrence of a variable v in a formula A is bound if and only if it follows a quantifier or is in the scope of a qua1'ltifier.

Atomic Formula

A WFF without quantifiers. The variables are free.

Notation: If A is a WFF, t a term, v a variable, then At/v is the WFF obtained from A by substituting t for all FREE occurrences of v in A.

Closed WFF

A closed WFF is a WFF in which there are no free occurrences of any variable. A WFF that is not closed is open.

Closure

0 The closure of a WFF A is designated .by A • If A is a WFF in which the variables v v , ••• vn have free occurrences, then 1 , 2 A preceded by 'v' v , V v ~, ••• V v is a closu:re of A. A sen- 1 e n tence A = A0 is true because a sentence is a closed WFF. Any closure of a closure of A is a closure of A. (23)

Scope of a Quantifier: In V vA if A is a WFF then A is the scope of the quantifier.

Statement

A statement is a formula with no free variable.

Predicate

A predicate is a formula with one or more free variables.

3. Axioms (Initial Formulae)

The axioms will be the following formulas listed by Godel

(Davis, p. 50).:

a?" Axioms concerning the notions of :

1. ((p...,. q)_.., (q-r))- (p- r)

2. (("'p)~p)-~op

3. p.- ((- p)-""q)

4. (p 1\ q) = ( "' [< "'-' p) v ( ,.._. q) 1 )

5. (p v q) - (( "'-p)~ q)

6. (p S qj S ((p--'!> q) A (q- p))

7. (paq).-, (p-q)

8. (p;sq)--.(q-p)

b. Axioms concerning the notion of identity:

1. X :: X

2. (x = y)-(g(x) = g(y)) (where g w a function)

3. (x=y)A(y=z)-(z=x)

c. Axioms corresponding to Peano's axioms for the natural numbers:

1. AJ (O = f(x)) f(x) means the successor of x

2. f(x) = f(y)~x = y (24)

4. "Immediate Consequence of"

A formula B shall be called an immediate consequence of A and

A~ B if B follows directly from A and A~B by our rule of

inference.

Modus ponens:

l.et S be the set of WFF' s of our syL>tem (L). For any A and B £ S

,.rom the formulas A and A__... B we may infer B•

.;onstructive

By modus ponens there is an effective procedure for determining

whether a given formula B is an immediate consequence of formulas

~, ••• An. We need only apply modus ponens a finite number of

times.

D. PRIMITIVE RECUHSIVE FUNCTIONS

1. Significance

An analysis of GOdel's incompleteness theorem would not be com­

plete without some Uki rstanding of primitive recursive functions.

We will not attempt to prove GOdel's Theorem using primitive re­

cursive functions. Wt~ '"ill simply give an introduction to the

use of primitive recursive functions. To fully understand each

of GOdel's eleven Propositions and forty-six definitions would

require a step-by-step development of the relations and functions

which is beyond the scope of this paper. The following pages are

offered as a prelude to an understanding of the original text.

GOdel begins this section by stating:

We now turn to some considerations which for the present have

nothing to do with the formal system. We are concerned exclu­

sively with functions whose arguments and values are natural (25)

numbers. We now call these functions primitive recursive. They are by their very nature effectively calculable. They are decidable since it can be determined by a finite procedure whether a relation does or does not hold, since the representing function is computable.

The first of GOdel's 46 definitions defines "x is divisible by y 11 as x/y ;:. ( i z) [ z '.!: x" x y • z] which might be translated as there exists a z less than or equal to x such that x = y • z. By restricting the quantifier z GOdel's function is recursively defined starting with z = 0, z = l, ... z = x, we obtain

X = y • o, X = y • l, ••• X = y • X • We have a finite number

(x + l) of equalities. GOdel states that the functions x + y, x • y, xY, and the relations x< y, and x = y are readily found to be recursive. He starts from these and defines a series of functions (and relations) l-45. Each of these is de­ fined from the previous ones by operations named in his Propo­ sitions I-IV (p. 180). Each of his functions is therefore recursive.

2. Definitions

We will present a few of the definitions basic to an understand­ ing of primitive recursive functions.

Natural Numbers - non~negative integers will be denoted by x,y,z,.

Finite sequences of natural numbers- x , ••• xn 1

Y1' • • •Ym Functions of one or more natural numbers whose values are natural numbers will be abbreviated by Greek letters ~~ r ~~ ,... (26)

Classes of or relations among natural numbers will be abbreviated by capital Roman letters, R, S, T, •••

R(x) stands for the proposition that x is in the class R.

S(x , ••• xn) for the propositions that x , ••• xn stand in the 1 1 relation S.

Classes may be considered as relations with only one ~·

Relations as classes of ordered n-tuples.

Representing Function ¢ "There shall correspond to each class or relation R a representing function¢ such that ¢(x , ••• x ) 0 . 1 n = if R(x , ••• xn) and ¢(x , ••• xn) = 1 if rV R(x , ••• xn). 1 1 1 Notations - as Abbreviations: p, q are to be replaced by any propositions p s q p ib equivalent to q, i.e., (p~ q) /1 (q ...... ,.p)

Recursiveness

A recursive definition is the specification of each number in a sequence of numbers by means of specification of the first number and a rule to specify the k + 1 - st number. The importance of recursiveness in metamathematics is that recursive definitions enable us to specify every number in a recursively defined infinit sequence to be constructed according to a rule, so that a remark about the infinite sequence can be construed as a remark about the rule of construction and not as a remark about a "completed infinite.''

For the proof of GOdel 1 s ''Unprovahility" Theorem, the importance of recursiveness lies in the facL that every statement of a re- cursive relationship holding between given numbers x , x , ••• xn 1 2 (27)

is expressible by a formula F of the forrrkd. system P which is

"provable" within P if the state11kmt is true and "disprovable"

within P (i.e., the "negation" of F, rvF' is "provable" within P)

if the statement is false.

3. Definition by Induction - Recursive Definition

Consider a number-theoretic function ¢(y) or pre,;Lcate P(y). We

are given first ¢(0) or P(O), the value of the function or predi-

cate for 0 as argument n. Then for any natural number y, ¢(y 1 )

1 or P(y ) , the next value after tfi, ' for y, is expressed in terms

of y and ¢ (y) or P(y) ( th

conclude that undt:'t the circumstances the value ¢(y) or P(y) of

the function or predicate is defined for every 11atural number y.

For the two parts of the definition enable us to generate any

natural number y, at the same time to determine the value ¢(y)

or P(y).

Example: Consider the pair of equations

(l)f 'fl (0) = p ~ r (xI ) = X (x' If (x))

to express the definition of the function tjl ( 0) by induction on

x, where Pis a given natural number, and /((x,y) is a given

number - theoretic function of two variables.

To generate t.f (3), we successively generate 0,1,2,3.

If (0) = p

If (0') =: t (1) = k: (0, f (O)) =X (O,P) q; = r<2> = x: <1, r <1>> = x c1, x co,P)) r = r o> = "k<2,r<2>> = r.. <2,X<1, x co,P>>> "'_,______(28)

We have a process by which for each natural number x, on the basis of the generation of x in the natural number sequence, a corres- pending number f (x) is determined. Since a number L}' (x) is thus associated with x, .for each x, a particular number-theoretic function lf' is defined with the number lf(x) as its respective value.

The function satisfies equation (l) and is determined for succes- sive values by the equations. Any function r satisfying the equations must have the values selected.

In other definitions by induction the function defined depends on additional variables x2, ••• xn.called rarameters, which have fixed values throughout the induction y.

What number-theoretic functions are defined by induction?

To make this question precise, we must specify what functions are to be taken as known initially, and what operations, including v;hat forms of definition by induction, are to be allowed in de- fining further functions.

We shall now select the specifications with a view to obtaining functions definable by induction in an elementary manner. These functions will be called "primitive recursive."

4. Primitive Recursive

A function is primitive recursive if it is definable by a series of applications of the 5 operations of definition below:

Designation Examrle: s (I) ¢(x) = x' Successor function ¢(0) = 0 1 = 1

(II) ¢(x , •• xn)~q Constant function 1 (29)

Designation Example:

U~ (III) ¢(x , •• xn) =xi Identity function ¢(x , •• x ) =x 1 1 5 3 s~ < r , 1: 1 , • • X m> Dfn. by substitution (IV) ¢ (x , •• xn) 1.(1 ( -:( (x , •• xn) ••• "}{ m(x , •• xn)) 1 = 1 1 1 The. expression for the a1ub i guous value of Y-' is obtained by sub­ stituting expressions for the ambiguous values of X ~ •• m 1 ?,:' for the varL':.bles of lj1 • ·The function ¢ defined by an application of this schema we sometimes write as Sn ( m (Va) ¢(o) = q ¢(y') ='KCv, ¢(y))

(Vb) ¢(0, x2 , •• xn) = ~ (~ 2 , •• xn) ¢(y'' x2, •• xn) =\ (y, ¢(y, x2, •• xn)'

(Va) constitute~'' the case of V for n = 1, and (Vb) for n 7 1.

We refer tb the above equations and equation pairs (I) '- (V) as schemata. They are anologous to postulates, vJith (!) - (III) in the role of axiom schemata and (IV) and (V) in the role of rules o inference.

Compound

The functions ¢(x1 , ... xn) shall be compound with respect to

(x , • .xm) and x (x , •• x ) (i = 1, • we m) if, for all natural 1 1 1 0 numbers x , •• x • . 1 n

¢(x. 1 , •• x n ) shall be said to be recursive with respect to lJ' (x , •• xn .. ) and K (x , •• xn+l) if for all natural numbers 1 1 1 (_30)

(b) ¢(o,x2, •• xn) :.:: r (x2, •• xn) ¢(k+l,x , •• xn) .::: X(k, ¢(k,x , •• xn) x , •• xn) 2 2 2 Class of Recursive :functions

We define the class of recursive functions to be the totality of

functions which can be generated by substitution according to

schema (a) and , according to the schema (b) from the

successor function x + 1, constant function f(x , •• xn) c and 1 = identity function UX: (x , •• x ) = x. (1 :::0 j 6 n) .. J 1 n J In other words, a function ¢ shall be recursive if there exists a

finite sequence of functions ¢ , ••¢n which terminates with¢ 1 such that each function of the sequence is either the successor

function x + 1 or a constant function f(x , •• xn) c, or an 1 = identity function ~ (x , •• x ) = x or is a compound with re- J. 1 n j spect to the preceding functions, or is recursive with respect

to p:cc~ceding functions.

Recursive Relation

A relation R shall be recursive if the representing function is

recursive.

Initial Function

A function ¢ is called an initial function, if ¢ satisfies equa­ tion (I) or equat;i.on (II) for a particular n and q, or equation

(III) for a particular n and i.

Immediate Dependent

A functlt'>n ¢ is called an im.tnediate d•t12er1dent of other functions 1 if ¢ satisfies equation (IV) for a particular n and m, with

, •• m as other functions or equations ( Va) for a 1(/ , }: 1 X

------··-···-···------' (31).

particular q with:( as the other function, or equations (Vb) 1 for a particular n with (f' , k. as the other functions. Explicit Definition,

'I'he explicit definition of a function consists· in giving an ex- pression fo:r· its ambiguous value constructed syntactically from lts independent variables (with no other variables occurring free) and symbols for functions, constants, operators, etc. which are either pd ; t.1.ve or previously defined.

5. Examp:Les of Pril_!!j.tive Recursive Functions

Example ~: Predecessor function The predecessor of x, pd(x) is defined by pd{O) = 0, pd(x') = x. It L primitive recursive by virtue of the primitive recursive derivation:

Apply the constant functiondefinition (II) c;{x) = 0 Constant func don ~(x1 , •• xn) = q where q=O and n=l ui(x,y) = x Ident·i f.J function ~(x1 , •• xn) = x1 where i=l and m':"':

Example .!2..: Proper Subtraction Proper subtraction is def:;i.ned by:

X..:.. Y' = pd ex~ y)

That is, x..:... y =X - y if X 7'/ y

x....:... y = 0 if X.<_ y

This is a primitive recursive function. To verify this initially v,,,,ite o (y ,x) for x ~~··· y ,md obtain the following primitive derivation i'or: (32)

U~ (y,z,x), pd(z), f(y,z,x) = pd(U~(y,z,x)), U~(y,z,x), U~ (y,z,x) = z Identity function

b (O,x) = U~ (y,z,x)

6 (y' ,x) :.:; f(y, f> (y,x),x)"

By listing the predecessor function as the initial function in- stead of a derivative for it, '-"e have taken a shortcut. To obtain X..!... y D.S the Value of .:.... at (x, y) instead Of <;'y ,X) !iS in b we UGe the follovJing three steps: 2 2 ? 2 2 u (x,y), u <::c,y), s2 ( cr(x,y), u (x,y), u (x,y)) 2 1 2 1 In particular, we say that the function¢ is defined e'SE~;L.Qitly from functions , •• and con;:;tants q , •• qs if an exprussion f 1 ~L 1 for its ambiguous value ¢(x , •• xn) can be given in terms of the 1 variables x , •• x , the constants q , •• qs and the functions 1 . n 1 The use of the identity function lf: in the analysis ~1, •• (/JL. 1. of explicit definitions is du~ to G8del 1934.

Schema ( v) is the. schema of primitive recursion, \>Ii thout para- meters (Va) or the parameters (Vb).

The functions defined in (Va) and (Vb) are sometimes written as

R1 (x) of (Va) or Rn( ~ ,}i) for (Vb). However, in speaking of a q primitive recursion we shall now understand that the application of (V) may be lumped with it in some steps of explicit definitions

Example £: Addition

To analyze the primitive recursion for a+b, write ¢(b,a) for a+b. 1 ¢(O,a) =a The ident~ty function (a) =a u1 ¢(b' ,a) = (¢(b,a))' X (b,c,a) = c' = S(U~(b,c,a))

where'X (b, ¢(b,a),a) = (¢(b,a)) 1 (33)

,.------~-·-·------... This fits schema (Vb) when the right member::; are expressed as

shown in brackets. So we accomplish the definition thus:

(a) S(a) = a' Schema I

(b) ui

(e) { ¢(O,a) = ui

¢(b' ,a) = K (b,¢(b,a),a) Schema (Vb), n=2, steps 2 and 4

This shows that a+b considered as ¢(b,a) (i.e .. , ¢ = ba a+b) is

primitive recursive , with S, ui ~ U~, {' , ¢ as a primitive recur-'- sive description. We obtain a+b as ¢ (a,b) (i.e., ¢ ab a+b) 1 1 = by three more steps. Symbolically,

br, a+b 2~ 3 (s, u 3 ) A = R .< 1' s1 2 Jab 2 2 tt- 2 u2) a+b = S2(R(l' s{

and Ackermann (1927) used recursive functions in their research

into the foundations of mathematics. In 1934 Godel extended the

class of primitive recursive functions to that of general recur-

sive. Church and Kleene (1932-35) defined a class of functions ·

(the~-definable functions) which they proposed would include all

functions \~hich could be cl·assified as effectively calculable. Church's thesis can be stated as: "Every decidable set of

natural numbers is recursive." Church proposed (1935) that the

intuitive, informal notion of effectively calculable functions (34)

of positive integers be identified with the precisely defined mathematical notion of general recursive funct;ions. Turing de­ fined a class of computable functions in terms of what now are known as Turing machines. Post (1943) gave two analyses of com­ putability (Post combinatorial systems and Post normal systems).

Markov's theory of functions computable by followed in

1951. All of these difr,:rent definitions sj_nce GOdel's 1934 de­ finition have turned out to be equivalent in the sense that they all define the same class of number-theoretic functions.

··------"------~----· III. dODEL' S PROOF

A. STATJ!MENT Oli' THE THEOREMS

.Theorem 1: For suitable L there are undecidable propositions in

L; that is, propositions F ,_,uch that neither l•' or.-F is prova.ble.

(Either F, or ""'F is true, theref(y;·c) there are true sentences that are not provable.)

Theorem 2: b'or suitable 1,, the simple consistency of L cannot be proved in L.

DEFINITIONS

Simply Consistent

L is simply consiste·nt if the:r:-e ic; no proposition F L-. · ;:hat

both F and tv F are provable.

If Lis not simply consistent, then some provaole proposition of

L must express a false sentence.

B. TAH.SKI 1 S EXAMPLE

.fiPa:f¥ample of a Provable Proposition Ex:pressing a False Statement:

Alfred Tarski (1933) shmved that some provable propositions of L may express fal:.':ie sentences even if L is simpl.y consistent. Tarski showed this by constructing a Logic L which was simply consistent but in which one could prove the propositions expressing each sentence of the following infinite set (with R properly chosen):

Not all positive integers have property R

1 has property R

2 has property R

3 has property R (35)

This system is -inconsistent. See the following definition.

&T-Consistent

A logic L which is "simply consistent" and in \vhich provable pro- positions of L may not express a false sentence.

For Suitable L

An exact formulation of these assumptions was to constitute the

second part of GOdel's (1931) paper. Godel has never written this

second half. However, S. C. Kleene ("General recursive functions of natural numbers", Mathematiche Annalen, Vol. 112 (1936), PP• 727-742) gives an exact statement of the set of assumptions sufficient for his proof of Godel's first theorem. They are phrased in terms of general recursive functions, and are illuminating only to someone who is thor­ oughly familiar \vith the theory of general recursive functions.

C. GODEL' S ASSUMPTIONS

1. There are two :

a. The Logic of Ordinary Discourse in which the proof is

carried out, and

b. The Formal Logic L about which the theorem is proved:

i. Propositions of L are formulas built according to

certain rules of structure (theory of propositional

calculus) ..

ii. Each formula consists of a finite number of symbols

chosen out of finite or denumerably infinite set (WFF 1 s).

iii. Any symbol may be used more than once in any

formula.

iv. The symbols have meanings attached to them through (36)

which we make the allowe.ble translations of propositions

of L.

v. The rules of structure of the propositions of L are

such that the interpretations of the propositions of L

will be declarative sentehceD. ('rhe sentences formed are

not necessarily true in "English.")

vi. If A is a proposition of L, and a certain sentence

is the .Lnterprccation of A, then A is said to be the

"expres:>ion in L" of that sentence or any sentence equi­

valent to it.

vii. Not all c:;entence? cw, be expressed in L. (There is

no apr:,trent way of saying such sentences as "Work is fun 11

and "Have a nice day", etc.)

2. The symbol ""' Where A..- A, the contradiction of A is assumed to exist.

3. Interpretations:

a. Integers - For each positive integer there must be a par-

ticular formula in L which denotes that integer.

b. Variables - If a formula A of L expresses a sentence S

and if A contains symbols called variables v ,v , •• v then S 1 2 8 contairtl3 variables. S is an open sentence.

c. If a formula W of L with the symbol v, cculed a variable,

expresses in L the sentence "x has the property Q", with the

variable x corresponding to v, and if F is got from W by re­

placing all the v 1 s of W by the formula denoting the number n,

then F expresses in L the sentence "n has the property Q." (37)

d. In General - If B is the formula got from A by replacing

various of the v. 's of A by other symbols, then the sentence ~ which B expresses is got from S by making corresponding re-

placements for the variable of S.

· 4. Provable - There must be a proces~> whereby certain of the pro-

positions of L are specified as "provable." The definition of

"provable" is always supposed to be made without referring to the

meanings of the formulas. However, it was a~ways hoped that the

set of provable propositions of L, would coincide with the set of

propositions of L which express true sentences.

5. If F and ~ F are both provable in L, then all propositions of

L are provable. So if L is not simply consistent, it is not

vi-consistent. So t.r -consistency implies simple consistency.

In fact, the non-provability of a single formula of L implies

the simple consistency of L.

6. There is a symbol~ , of L such that if the formula A ex-

presses the sentence S and the formula B expresses the sentence

T, then A - B expresses the sentence "If S, then T." Also the

definition of "provable" shall be such that if A and A---+ B are

provable then so is B.

When Gddel numbers have been assigned to formulas, statements

about formulas can be replaced by statements about numbers. That

is; if P is a property of formulas, we can find a property of

numbers, Q, such that the formula A has the property P iff the

number of A has the property Q. Throughout the rest of the proof,

P will signify a property of formulas, and Q will signify the (38)

corresponding property of numbers. That is, Q will be the pro­

perty of numbers such that we can use the statements "A has the

property P11 and "the number of A has the property Q" inter­

changeably.

Many statements about mtmbers can be expressed in L, even though

all cannot. In particular, if P is properly chosen, we can often

express "x has the property Q" in L. If 25....)~0 taken to be the

number of a formula of L, we are expressing in L a statement

about a formula of L. Lemma 1 exhibits this circularity and

appears confusing.

D. UMMA 1

Let "x have the property Q" be expressible in L. Then for suit­ able L, there can be found a formula F of L, with a number n, such that F expresses "n hp.s the property Q." That is, F expresses "F has the property P."

We assume £c,[· suitable L that "z = ¢(x ,x)" is expressible in L, where ¢(x,y) is the function described below.

E. DEFINITION: ¢(x,y)

¢(x,y) is the Godel number of t~e formula got by taking the for­ mula with the Godel J;J.umber x and replacing all occurrences of v in it by the formula of L which denotes the GOdel number of y.

Proof: Assume "x has the property Q" and "z = ¢ (x ,x)" are ex­ pressible in L. Then "¢(x,x) has the property Q" is expressible in L becauue it is equivalent to the statement "there is a z such that z = ¢(x,x) and z has the property Q." G is the formula, and n is the Gode number assigned to that formula. We now get F from G by replacing all (39)

v's of G by the formula of L which denotes n. Then F denotes "¢(n,n) has the property Q. 11 (See Assumption 3 above. ) However, from the definition of ¢(x,y), ¢(n,n) is the Godel number ofF because F was obtained by taking the formula whose Godel number was n and replacing all occurrences of v in it by the formula of L which denotes n. F thus expresses "the Godel number of F has the property Q" that is

11 F has the property P."

In order to use Lemma 1, we must know that 11 z = ¢(x,x)" is ~­ pressible in L. Godel proves this for a large class of functions of

L. He proves:

(1) ¢(x,y) is "rekursiv" (Se~ G8del pp. 179-188 for proof).

11 (2) If lfCx1 ,x2 , •• xs) is "rekursiv", then "z =If (~,x 2 , •• xs) is expressible in L. (.ill£., Proposition V, p. 55).

We previously defined "rekursiv" in our section on primitive recursive functions. We now define expressibility.

F. EXPRESSIBILITY

We say that an open sentence R expresses a given property of in- tegers n1 , •• ~ if it satisfies two conditions:

(C1 ) If the n1 , •• ~ have the property, then R(~,··~) is provable.

(C ) If the n , •• ~ do not have the property, then the negation 2 1 of R(ni, ••~) is provable.

R(ni, ••~) denotes the sentence obtained from the open sentence

R by substituting the numerals n1 , ••~ for the free variables in R. G. 11 Bew(x) 11

For Lemma 1 Godel chooses for P the property of not being provable (4o)

in L. If "Bew(x) 11 denotes "the formula with the number x is provablu in L", then "x has property Q" is equivalent to "not-Bet-J(x)."

By the extensive <:=.trgument involving "rekursiv" functions, GOdel shows that for a large class of L's:

(1) "Bew(x)" (and hence "not-Bew(x)") is expressible in L.

(2) If L is lu- -consistent and if the formula expressing "Bew(x)"

is provable, then "Be"' (x)" is true.

(3) If "Bew(x)" is true, then the formula expressing "Bew(x)"

is provable. For Lemma 1, let us find a formula F with the

number n, such that F expresses "not-Bew(x)."

H. L:EMMA 2

If L is simply consistent, then F is not provable in L.

Proof: Suppose F to be provable. 'l'hat is, the formula with the number n is provable •. That is, Bew(n)o So by (5), the formula which expresses 1 Bew(n)' is provable. However, F expresses 'not-Bew(n)', and so F expresses "Bew(n)" (Assumption 2). So IV F is provable.

However, we assumed F provable, so that L is not simply consistent since F and ""-' 1<, are provable. So if L had been simply consistent, F

'-'.IOUld not have been provable.

I. LEMMA 3

If L is lv- -consistent, then ,.,_, F is not provable in L.

Proof: Suppose L to be L4r -consistent and pretend that ""-" F is provable. "'-" F expresses "Bew(n)." So by (4) Bew(n). That is, F is provable. So L is not simply consistent. However, w- -consistency implies simple consistency (Assumption 5). So our pretense that~ F could be provable has to be false. (41)

As l.r-consistency im1Jlies simple consistency, Lemma (2) and (3) together give Godel's theorem. For suitable L, there are undecidable propositions in L; that is, propositions F such that neither F nor

tv F is provable.

J. GODEL I s SECOND THEOREM

For suitable L, the simple consistency of L cannot be proved in L.

Proof: Let A be a provable proposition of L, and let m be the

Godel number of "' A. If Be\v(m), then both A and :.v A are provable, and L is not simply consistent. On the other hand, if L is not simply consistent, all propositions of L are provable, including /V A, so that

Bew(m). Hence "not-Bew(m)" and "L.is simply consistent" are equiva­ lent. So Lemma 2 is equivalent to

"If not-Bew(m), then not-Bew(n)", since n is the Godel number of F. Let Mip be the formula of L which expresses "not-Be\.,r(m) ." F is the formula of L which expresses 11not­

Bew(n)."

So Mip _..., F expresses Lemma 2 in L (Assumption 6). Now the proof of Lemma 2 can be carried out in a great many logics, so that in those logics

Mip...:., F is provable. Then if Mip were provable, F would be provable (Assump­ tion 6). So Mip is not provable if L is simply consistent (by Lemma

2) which is what Godel's Second Theorem states.

K. CONCLUSIONS

1) ~ metamathematical proof is possible 12£ the formal consis­ tency of a system comprehensive enough to contain the whole of arith- (42)

metic; unless, that is, the metamathematical proof employs rules of inference whose consi.stency is as doubtful as is the consistency of the Trm1.sformation Rules used in deriving theorems within the system ..

2) Principia, or any other system within which arithmetic can be developed, is essentially incomplete. In other words, given any consistent set of aritwnetic axioms, there are true arithmetical statements which are not derivable from the set. Even if the set of

Hathematician.s of the 19th Century had hoped that c:J.l mathematical truths ~tJould be provable. Godel established the existence of an unde­ cidable proposition. Neither ,:;_ proposition F nor """'F \vas provable within the formal system. Since either F or""' F is true, we have a true proposition of the formal system th<~t is not provable. Ottr for­ mal system is incomplete because there exist a true proposition that is not provable. (We are assuming that the system is consistent.)

GOdel demonstrated in his proof that the consistency of the system could not be proved, i.e., \ve cannot prove th.::.tt a finite number of logical steps would never lead to a contradiction. Godel further sho~tJed that even if we augmented the system 'N'ith this true but un­ provable proposition·the augmented system would still contain an un­ decidable proposition. The system is therefore essentially incom­ plete. Godel established:

"If '1ri thmetic is consistent, then it in incomplete. 11

What impact did this Dtatement have on the mathematical world?:

We must accept the fact that mathematics will probably never be established as a system of absolute truths. We must consider this kind of incompleteness as an inherent characteristic of formal mathe­ matics as a whole. The notion of truth cannot be identified with the notion of provability in a formal system.

Even though we cannot define truth in mathematics to coincide with our intuitive definition of truth, we can still create mathematics.

Mathematicians are still piling theorem upon theorem upon theorem. No mechanical procedure can be found for solving all mathematical problems. No computer can be programmed to spew out all mathematical truths. We should be grateful. We still need the mathematician for his creative processes. Perhaps, someday, in some higher mathematical system, truth can be defined. (45)

ANNOTATED BIBLIOGRAPHY

(1) Boyer, Carl B., A History of Mathematics, John Wiley and Sons, Inc. , New York, 1968. Chapter XXVII, "At;pects of the Twen­ tieth Century", Hilbert and Godel pp. 65Lt-662.

(2) Bulloff, Jack J., Hahn, S. W., Holyoke, 'fhornas C., editors, Foundations of Mathematics: §ymposium Pape£_s~ ,Cn!nmemmorating the Sixtieth Birthdccy of. Kurt Godel, Springer~ Verlag New.· York, Inc., 1969. (The photograph of Kurt Godel is by Orren J. Turner. Biographical information used from the Preface and Introduction pp. IX and X.)

(3) Cohen, Paul J. , and the .Continuum Hypothesis, . W. A. Benjamin, Inc., New York, 1966. Information was used from Chapter I, "Genernl Background in Logic", Introduction, Formal Languages, pp. 1-7, ExarnpJ.es of Formal Systems, pp. 20-26, Primitive Recursive Functions, Godel's Incompleteness Theorem", PP• 27-32, PP• 39-46. (4) Davis, Martin, The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems, and Computable Functions, Raven Press, Hewlett, New York, 1965. Parts used from Kurt GOdel's papers, "On Formally Undecidable Propositions of the Princiuia Mathematica and Related Systems, I", pp. 4-38; and "On Undecidable Propositions of Formal M8thematical Systems", pp. 39-74, J. B. Rosser's, "An Informal Exposition of Proofs of Godel's Theorem arid Church's Theorem", pp. 223-230. An excellent anthology on fundamental papers dealing with unde­ cidability and unsolvability containing papers by Godel, Church, Turing, Rosser, Kleene, and Post.

(5) Enderton, Herbert B., : An Introduction to the ­ theory of Standa.rd First Order Logic, University of California Press, Berkeley and Los Angeles, California, 1971. Parts of Chapter One, ''Sentential Logic", were used for the paper. (Well-formed formulas, and Induction and Recursion section, specifically pp. 14-30).

(6) Findlay, J., "Goedelian Sentences: A Non-Numerical Approach", Mind, Vol. 51 (1942), pp. 259-265. Parts of the article were u;ed for informal comments on the construction of G0del 1 s sentences.

(7) Godel, Kurt, trans by B. Meltzer, intra by R. B. Braithwaite, On Formally Undecidable Propositions of the Principia Mathe­ matica and Related Systems, Basic Books, Inc., New York, 1962. The introduction by Braithwaite was quite helpful. Parts of the thesis relied heavily on pp. vii, 1-13, 23-32, and the translation of G6del's paper by Meltzer for useful analysis and cogent comments. (46)

(8) Hanson, N. R., "Godel's Theorem: An Informal Exposition", Critique by A. Church, Journal of Symbolic Logic, Vol. 27, pp. 471-472, December 1962. (Bl J685)

(9) Helmer, Olaf, "Significance of Undecidable Sentences", Journal _£f Philosophy, Vol. 31+, pp. 490-lt<)4, 1937. Several ideas were used from this article for commentary throughout the thesis.

(10) Hunter, Geoffrey, Hetalogic: · f.}_n, .l_ntroduction to the Metatheorz of Standard First Order Logic, University of California Press, Berkeley and Los Angeles, California, 1971. This is an ex­ tremely readable book. Part J!our, "First Order Predicate Logic: Undecidability", was borro1ved a great deal from, es­ pecially pp. 219-228.

(11) Kleene, Stephen Cole, Introduction to Het

(12) Mostowski, Andrzej., Sentences Undecidable in~alized Arith­ t!!,etic: An Exposition of the Theory of Kurt G

(13) Nagel, Ernest, and Nevnnan, James R., Godel's Proof, New York University Press, Ne'.v York, 1958. Sections on Godel numbering, Arithmetization of Metamathematics, and the Heart of Godel's Argument, pp. 69-97 vvere adopted for use throughout the thesis.

(14) Nagel, Ernest, and Netvman, James R., i'G6del's Proof", Scientific American, June 1956, pp. 72-86. Ideas for organizing the sec­ tion on GOdel numbering were borrowed from the article.

(15) Newman, James R., ed., The World of Hathematics, Vol. 3 & 4, Hahn Hans, "Infinity", pp. 1591-1611, and the famous Newman and Nagel rerun, "GOdel's Proof", pp. 1668-1695, Simon and Schuster New York, 1956. Pages 1685-1695 were used for background dati,_ and commentary.

(16) Rosser, Barkley, "An Informal Exposition of Godel 1 s Theorems and 1 Church's Theorems ', Journal of Symbolic Logic, Vol. 4, No. 2, 1939, pp. 53-60. ThL> article was included in the Martin Davis anthology.

(17) Stoll, Robert R., Set Theory and Logic, W. H. Freeman and Com­ pany, San Francisco, 1961. Parts from "Chapter 9, First Order " were used. Specifically, pp. l1·0l-L~o4, "Hetamathe­ matics", pp. 426-428 on the process of the arithmetization of the metalanguage, and pp. 446-452, 1'Gddel' s Theorem." ( 4'?)

(18) Tarski, A., Mo.stot-Iski, A., and Hobinson, R. M., Undecidable Theor:Les, Amsterd;;un, North Holland Publishing Company, 1953. Not directly related to the thesis. Used for background information.

11 (19) Ushenko, A., "Undecidable Statements ~md Netalanguage , ~' Vol. 53, July 1944, pp. 258-262.

(20) Van Heijenoort, Jean, Frege and Godel, Harvard University Press, Ccunbridge, Hassachusetts, 19i6. Godel's articles, pp. 83-108.

(21) Van Heijenoort, J., From Frege to Godel- A Source Book in Mathe­ matica.l Logic 2 1879-1931. Ccunbridge, Mam:>., 1967. The una­ bridgement of the above book.

(22) Wilder, Raymond L., Introduction to the Foundations of Mathem2- tics, John Wiley & Sons, Inc., New York, 1952. "Chapter XI, Formal Systems; Mathematical Logic", pp. 264--279. Used for background data. (48)

INDEX OF DEFINITIONS

Admissible Methods , 5 Proof Theory Arithmetization of Metamathematics, 8 (see metamathematics) Axioms, 23 Provable , 2 Axiom Schemata, 29 Relations , 26 Bew(x), 39 Representing function , 26 Church's Thesis, 33 Recursive definitions , 27 Classe~ 26 Recursive functions , 30 Complete , 1 Immediate dependenc0.,30 Compound , 29 Initial function , JO Computable , 6 Primitive recursive Consistent , 1 functions , 28 inconsistent , 1 definitions , 28 tv--consistency , 35 examples , 31 simply consistent , 34 Recursive relation , JO Constructive, 24 Richard Paradox , 16 Countable , 5 Self-referential Decidable , 1 statements , 15 decidable set , 1 String, 4 undecidable, 2 Symbol , 20 undecidable set , 2 Syntax language Decision procedure (see metalanguage) (see effective procedure) Total , 6 Denumerable , 5 Truth ,4) Domain , 6 Unprovable , 2 Effective , 5 Variable Effective Enumeration , 6 Free , 22 Effectively decidable , 6 Bound , 22 Effective Procedure (method) , 5 Well-formed formulas Expressibility , 39 (WFF' s), 21 Formal , 4 Closed WFF , 22 Formal Proof, 6 Formal System , 19 Formal Theory , 4 "For Suitable L" , 35 Godel Numbering , 8 GOdel 1 s Proof, 34 "Immediate Consequence of" , 24 Incomplete , 1 Metalanguage , 4 Metamathematics , 3 Metatheorem , 4 Metatheory , 4 Natural Numbers , 25 Numeralwise Expressible (see expressibility) Object Language, 4 Parameters, 28