Godel's Theorem Simplified 081913869X, 9780819138699

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GODEL'S

THEOREM

SIMPLIFIED Harry J. Gensler

UNIVERSITY PRESS OF AMERICA

f[S

Copyright © 1984 by University Press of America�11 Inc. 4720 Boston Way Lanham, MD 20706 3 Henrietta Street . London WC2E 8LU England All rights reserved Printed in the United States of America

ISBN (Perfect): 0-8191-3869-X ISBN (Ooth): .�-8191-3868-1

TABLE OF CONTENTS

I II III

INTRODUCTION

1

SYSTEM A:

5

................. ADDITION . . . . . . . . . . .

SYSTEM B: LOWER ARITHMETIC

IV

SYSTEM C: ARITHMETIC

V

ID NUMBERS AND THE GODEL FORMULA

VI VII

...

.........

17

29 41

.......... ............

57

CONSTRUCTING THE GODEL FORMULA'S FA'I'HER .....

71

FURTHER QUESTIONS

iii

CHAPTER I INTRODUCTION In 1931 Kurt Godel showed that it is impossi­ ble to reduce basic arithmetic to a system. This surprising result ranks as one of the great discov­ eries of the century; and Godel's argumentation is fascinating and ingenious. Unfortunately though, the proof as usually presented is immensely diffi­ cult and demands a good deal of mathematical sophistication. My goal in this little book is to explain and prove this important but difficult theorem in as simple and direct a way as possible, so that more people may come to under,3tand and appreciate the theorem. Most of the book presumes only an under­ standing of high school algebra. Occasionally I use some symbolic logic; but I explain this as I go along and most of the book should be understandable to a person with little or no background in logic. The only really technical part is Chapter VII; to understand this, you need a good knowledge of 1uan­ tificational logic and a lot of patience. But you may omit this chapter if you wish -- you will still grasp the main argument (although you won't then know how certain things are worked out from a t3ch­ nical standpoint). In addition, I should mention that I restrict myself to showing that a particular set of axioms does not formalize arithmetic com­ pletely; but the same argument could be used regardless of the specific axioms chosen. Godel's theorem establishes the following: ARITHMETIC CANNOT BE REDUCED TO AHY AXIOMATIC SYSTEM. To explain what this means, we need to answer three questions: (1) "What is arithmetic?," (2) "What is an axiomatic system?," and (3) "What would it mean to say that arithmetic is reduced to a given axio­ matic system?" 1

(1) What is arithmetic? Very roughly, "arithmetic" here is high school algebra (but lim­ ited to positive whole numbers) and includes such truths as "2+2 = 4," "If x+y = z then y+x = z," and "If More techni­ xy = l8 and x =2y then x = 6 and y =3. " cally, "arithmetic" is defined as the set of truths and falsehoods expressible using a certain speci­ fied vocabulary: symbols for positive whole numbers (1, 2, 3, . •. ) , "plus," "times," "to the power of," parentheses, "equals," variables (x, y, z, . ••) , and some simple logical concepts ["not " "and " "or," "if -then," "for every (number) ," 'and "f�r some (number)" J. (2) What is an axiomatic system? Very roughly, an "axiomatic system" is a system in which some principles (axioms) are just taken to be true while other things (theorems) are proven to be true using these principles; high school geometry is an example of such an axiomatic system. More techni­ cally, an "axiomatic system" is a specified set of formulas (axioms) together with a specified set of logical operations (inference rules) for deriving further formulas (theorems). [I regard as implicit here another technical qualification: that in an axiomatic system there must be a finite and mechan­ ical notational procedure for checking whether a string of symbols is an instance of an axiom or an inference rule or a proof. ] (3) What would it mean to say that arithmetic is reduced � a given axiomatic system? It would mean that the axiomatic system contains as a theo­ rem each arithmetic truth but no arithmetic false­ hood -- that any arithmetic truth is derivable from the axioms but that no arithmetic falsehood is so derivable - - that for any arithmetic truth or fal­ sehood A: If A is true, then

AXIOMATIC SYS'l'EM

If A is false, then

AXIOMATIC SYSTEM

'

Can the goal of reducing arithmetic to an axiomatic system be reached? Godel's theorem says NO - - the goal is impossible. For each axiomatic system you construct, either of two evils happen to it: either 2

some arithmetic truths are not be provable or else some arithmetic falsehoods are provable - - there are either unprovable truthsor else provable fal­ sehoods - - either your system fails to prove "2+2=4" (or some more complex arithmetic truth) or it proves "2+2=5" (or some more complex arithmetic falsity) : AXIOMATIC

...;.:t

SYSTEM

�

an arithmetic truth 0r an arithmetic falsity

Our proof tries to show this result in detail for a specific axiomatic system to be called "SYSTEM C. " We will start slowly to work toward the proof of GodeJ.'s theorem. Chapter II will present an extremely simple axiomatic system - - SYSTEM A -­ which gives a way of symbolizing and proving formu­ las involving the addition of specified "positive whole numbers ("addition formulas") . SYSTEM A uses symbols for positive whole numbers, "plus," paren­ theses, and "equals. " SYSTEM A's proof mechanism contains one axiom and two inference rules and suf­ fices to prove each truth (but no falsehood) expressible in the system. Hence the arithmetic of addition is reducible to SYSTEM A. [Note that Godel's theorem does not say "No part of arithmetic can be reduced to any axiomatic system"; rather it says "Arithmetic (as a whole) cannot be reduced to any axiomatic system. " J Part of the goal of Chap­ ter II is to give an intuitive feel for what axio­ matic systems are and how they work. Chapter III will present a richer axiomatic system - - SYSTEM B - - which incorporates the symbo­ lism and proof mechanism of SYSTEM A and more besides. SYSTEM B adds symbols for "times" and "to the power of" and gives a way to symbolize "lower arithmetic" (formulas involving the addition, mul­ tiplication, and exponentiation of specified posi­ tive whole numbers). SYSTEM B's proof mechanism adds four inference rules and suffices to prove each truth (but no falsehood) expressible in the system. Hence lower arithmetic is reducible to SYSTEM B. system

Chapter IV will give an even richer axiomatic SYSTEM C which incorporates the

3

symbolism and proof mechanism of SYSTEM B and more besides. SYSTEM C adds symbols for variables ( x, y, z, •••) and some simple logical concepts [ " not," "and," " or," " if-then," " for every ( number)," and "for some ( number)"]; hence SYSTEM C is able to express any formula of " arithmetic," as we defined "arithmetic" above. SYSTEM C's proof mechanism adds ten axioms and two inference rules but does not suffice to prove each truth ( but no falsehood) expressible in the system. In Chapter V we will try to show this: that arithmetic cannot be reduced to SYSTEM C or (by extension of the argument) to any axiomatic system at all. This is what Godel's theorem establishes.

Chapter VI will discuss some broader ques­ tions relating to Godel's theorem. Finally, Chap­ ter VI I will fill in some technical details that were conveniently skipped in the argumentation of Chapter Vin order to make this argumentation more understandable.

Godel's theorem -- that .!!2 possible axiomatic system suffices to prove each truth ( but no false­ hood) of arithmetic -- at first may seem difficult to believe. Arithmetic seems to be an area where everything is cut i:i,nd dried, where ideas are pre­ cise and everything can be proven decisively one way or the other. But, according to Godel's theo­ rem, what " seems" to be the case here is simply not so. While some limited parts of arithmetic (e.g. SYSTEMS A and B) are cut and dried in this way, arithmetic itself ( in e.g. SYSTEM C) is not and never will be so cut and dried. Godel's theorem shatters our preconceptions.

4

CHAPTER II

SYSTEM A: ADDITION

Our first system -- the very simple SYSTEM A covers addition formulas ( formulas expressible using only symbols for positive whole numbers, The set of "plus," parentheses, and "equals"), truths and falsehoods expressible in such addition formulas is called " the arithmetic of addition." SYSTEM A formulas uses these five symbols: SYMBOL +

I ( )

MEANING

one plus ( left parenthesis) (right parenthesis) equals

A formula of SYSTEM A is defined as any string ( whether meaningful or not) of one or more of these five symbols, So "= /=" ("equals one equals") is a formula of SYS'rEM A, even though it doesn't make any sense. The legitimate grammatical strings of SYSTEM A ( the formulas which do malce sense) are called "numerals" and "terms" and "wffs" and are specified by grammar rules ( 1) to (3), The first grammar rule gives a way to con­ struct numerals ( which are strings of /'s): A string consisting in one or more instances of "/" is a NUMERAL and is also a TERM.

( 1)

Examples include the following:

I

II Ill Ill/ //Ill

(" 1" ("2 " ("3" (" 4 " ("5"

or "one") or "one one" ) or "one one one") or " one one one one") or " one one one one one") and so forth

5

Thus we have a simple way of expressing any posi­ tive whole number -- we just write out a string of /'s of the corresponding length. People often use this primitive way to express numbers when they tabulate the results of a vote or a questionnaire; sometimes they write 11 1'f.J./ 11 (instead of "/////") for "5."

Every numeral is a term, but the next grammar rule gives a way to construct comolex terms that • are not numerals: (2)

A string consisting in "( " and then a term and then 11 +" and then a term and then ")" is itself a TERM.

Applying rule (2) to numerals gives complex terms like the following: (/+///) (//+//)

(" 1+3 11 or "one plus one one one") ( "2+2 11 or "one one plus one one")

Note that a pair of parentheses is always to be used with each "+." Applying rule ( 2) to the terms constructed above gives this more complex term: ((/+/ // )+(// +//))

["( 1+3)+(2+2)" J

Repeated uses of rule (2) would yield terms of greater and greater complexity.

Every complex term is "equivalent" to some numeral, in the sense that both refer to the same number -- for example, "(/ /+//)" ("2+2") is equiva­ lent to "// //" ("4"). One simple way to find the numeral which is equivalent to a complex term is to rewrite the complex term, replacing every instance of "(" or "+" or ")" with a blank, and then rewrite the resulting string of l's to eliminate the blanks. Let me give two examples: ("2+2 ")

(I /+I/)

("4 ")

II II !Ill

'

[ So 11( //+//)" is equivalent to "////. 11] (( /+///)+(//+// ))

I Ill II II I/II/Ill

[ "( 1+3)+(2+2) 1 1 J ( II

6

8")

[ So "((/ +/// )+(// +/ /)) 11 is equivalent to "//////// . 11]

of So far in SYSTEM A we have seen examples ements stat not but rs) numbe g ntin rese (rep terms t numbers). �esenting true or false claims abou state­ form to way only our third rule gives the A: M SYSTE in wffs) ments (or "=" (3) A string consisting in a term and then and then a term is a WFF ( "well formed formula").

So every SYSTEM A wff is an equation. Examples of wffs would include the following (note that we do not use a pair of parentheses with each 11="): (" l=l") ("1 =2'') ("2+2=4 ") ("2+2=5")

/ =/ / =// (// +// )=//// (/ /+// )=/ ////

Here the first and third are true, while the second and fourth are false.

A simple way to determine whether a wff is true or false is to rewrite the wff, replacing every instance of "(" or "+" or ") 11 with a blank, and then rewrite the resulting string of l's to eliminate the blanks; if the resulting wff consists in "=" flanked by an equal number of / 's on each side, then the original wff is true -- otherwise it is false. Let me give two examples: ("2+2=4")

(I/+//)=/// /

("4 =4")

II II =/Ill /!//=////

[So "(//+// )=// //" is true.] (//+/ /)=/// // //// =/////

II II =//Ill

("2+2=5") ("4=5")

[So "( //+//)=/////" is false.]

Notice how simple and straightforward it is to test whether a SYSTEM A wff is true or false! Now we begin to set up an axiomatic proof mechanism for SYSTEM A. This will give us another

7

way to test SYSTEM A formulas -- a way which is more difficult to apply· but which will help us understand more concretely what axiomatic systems are and how they work. The axiomatic system here is very simple and will prepare the way for the more difficult systems to be u sed later on. R ecall that an "axiomatic system" is a speci­ fied set of formulas (axioms) together with a spec­ ified set of logical operations (inference rules) for deriving further formulas (theorems). Our axi­ omatic system here has one axiom (Al) and two inference rules (ARl and AR2); any true wff of SYSTEM A is able to be derived from this axiom using these two inference rules -- but no false wff of SYSTEM A is so derivable. We consider every instance of Al to be an axiom (where an instance of Al is the result of substituting any string of /'s for the "x" in "x = x"): x =x

Al

So the following instances of Al are axioms: /=/ ("1 =1") //=// ("2 = 2") ///=/// ("3 = 3") (" 4=4") ////=//// and so forth Note that Al specifies a whole family of wffs to be axioms -- each member of the family consists in one instance of a numeral, followed by " = " followed by [To be pre­ another instance of this same numeral: cise, when I said that o�r system has "one axiom " I should rather have said that it has "many axiom� all of which are instances of one axiom-form" _,: but I will generally slur overthis," written between two terms, to indicate that (any or all of the occurrences of) these two terms may be substituted one for the other. Every instance of ARl indicates an allowable substitution (where an instance of ARl is the result of substi­ tuting any string of /'s for "x"): ARl

(x+/)

x/

So the following instances of ARl indicate allow­ able substitutions: ("l+l < = > II ("2+1 Ill ("3+1 < = > /Ill ("4+1 < = > /Ill/ and so forth

(/+/) (//+/) (///+/) (////+/)



2") 3") 4") 5")

ne By ARl, ,for a term representing a-number-plus-o repal numer the itute subst [e,g, "(///+/)"] we may [e.g. one-more-than-the-first-number resenting as such ARl of nce insta an Note that "////"], < => /Ill" is not itself a formula of "(///+/) M A's SYSTEM A (since "< = >" is not part of SYSTE saying vocabulary) but rather is a shorthand way of "////," that term"(///+/)" may substitute for term and vice versa. Our second inference rule is more complex, Any instance of AR2 indicates an allowable substi­ tution (where an instance of AR2 is the result of substituting any string of /'s for "x" and then substituting any string of /'s for "y"): AR2

(x+/y)

Instances which substitute (/+//) (/+///) (/+////) (/+/////)

< =>

< =>

"I"

(x/+y) for "x" include:

(I/+/) (//+//) (//+///) (//+////) and so forth

("1+2 ("1+3 ("1+4 ("1+5

< =>

2+1") 2+2") 2+3") 2+4")

Instances which substitute "//" for "x" include: (//+//) (//+///) (//+////)

(!/+Ill/I)

< =>



(///+/) (///+//) (///+///) (///+////) and so forth

9

("2+2 ("2+3 ("2+4 ("2+5



< =>

3+1") 3+2") 3+3") 3+4")

By AR2, for a term adding two number s (where the second is greater than "/" ) we may substitute the term adding one-more-than-the-first-number to one­ less-than-the-second-number.

We can u s e the s e two inference rules to sim­ plify a term down to its equivalent numeral. Starting w ith an easy example let us simplifY " (// +// )" ("2+2" ) down to "Ill!"' ("4" ): 1 2

3

from 1 by AR2 from 2 by ARI

(//+//)

(///+/)

Ill/

Formula 1 is the formula to be simplified. Formula 2 result s from formula 1 by making a substi tution allowed by rule AR2 : (/ /+// )

(/// +/ )

("2+2

3 +l" )

Formula 3 result s from formula 2 by makin g a sub­ stitution allow ed by rule ARI: (/// +/ )

/Ill

It sh ou 1 d be pretty clear

(" 3+1 4")

w hat

is happening here. We now try a more complex example the simplification of "(///// +/////// )" ("5+ 7") down to "Ill/!///////" (" l2" ): 1 2 3 4 5 6 7 8

(///// +/////// ) (////// +////// ) -- from 1 by AR2 (/////// +///// ) -- from 2 by AR2 (/////// /+//// ) -- from 3 by AR2 (/ //////// +/// ) -- from 4 by AR2 (////////// +// ) -- from 5 by AR2 (/////////// +/ ) -- from 6 by AR2 II/!////!//! -- from 7 by ARI Note that each use of substitution rule AR2 in steps 2 to 7 move s the "+" one "/" to ���!l the�e is only one "/" to the righthe right t of the«;; P 7), then w e use s ubst ion rule ARI t 0 eliminate the "+" and parentheitut s altogether thus arriving at the equivalent numese ral (step 8). ' t p f er 0w s q t t; ����c:d \ o �w� r:1:� �!� f :;� S �i1 b �al�p : i �i ep b step manner (here " x" and " y 11 repre s ent s trings oYf one or more l's):

r\

10

(ar2)

(arl)

If the last formula has a term of the form " (x+/y), " then write out a new for­ mula substituting a term of the form " (x/+y)" [e.g. for " (// +/ I)" substitute "(/// +/ )"]; then go back to (ar2). If there is no such term, then go to (arl). If the la s t formula ha s a term of the form " (x+/), " then write out a new for­ mula substituting a term of the form "x/" [e.g. for "(///+I)" substitute "/ I//"]; then go back to (ar2). If there i s no such term ) then you are done.

Thi s " mechanical s implif ication method" would even­ tually eliminat e all in s tance s of "(," "+, " and ")" and produce an equivalent formula without the s e. The method would yield the two simplification s given above.

This mechanical s implification m e thod can al s o be used to test whether a wff i s true or false; in this ca s e we apply the m e thod, not to a t.fil'..!!l [like " (// +// )"], but to a s tatement or wff A SYSTEM A wff is true-rf [like "(/l+/l)=///1"]. and only if the method would eventually simplify it down to a s imple identity (an instance of axiom Al). The method would yield this s implification of wff "(ll+//)=IIII" ("2+2= 4") down to the s imple identity "llll=/111" (" 4= 4"): 1 2 3

Ull+l)=IIII -- from 1 by AR2 /Ill =/Ill -- from 2 by ARI

(/ I+/l)=IIII

Here again, formula 1 i s the formula to be s impli­ fied, formula 2 re s ult s from formula 1 by making a s ubstitution allowed by rule AR2 , and formula 3 re s ults from formula 2 by making a s ub s titution allowed by rule ARI. Thi s again should be r e adily under s tandable.

and more dif ficult example is this of "(lllll+ll///l/)=II//II///III" identity s imple the to down (" 5+7= 12" ) "ll/l////llll=I/I/II//IIII" ("12= 12"): A s econd s implification 1 2 3 4

(l//ll+ll//lll)=//II///I/I// (/l/l/l+l///ll)=IIIIII/III// (/l/lll/+/l/ll)=I/I///I//I/I (llllll/l+////)=III/I/III/I/ 11

from 1 by AR2 from 2 by AR2 from 3 by AR2

5 6 7 8

(/////////+///)=//////////// (//////////+//)=//////////// (///////////+/)=//////////// //////////// =////////////

from 4 from 5 from 6 from 7

by by by by

AR2 AR2 AR2 ARl

Simplification of terms and wffs get carried out in a very similar waybyour mechanical simplification method.

Now we move from simplifications to proofs. We define a " proof" as a vertical sequence of wffs, each of which is either an instance of axiom Al (a simple identity) or else follows from previous wffs of the sequence by inference rules ARl or AR2 . A " theorem" is defined as ani wff of a proof. We give, as an example of a proof, the fol­ lowing proof of theorem " (//+//) =////" ("2 =2+ 4" ) :

1 2 3

(///+/)=//// (//+//) =////

/Ill =/Ill

by Al -- from 1 by ARl -- from 2 by AR2

This is a proof because each line instance of axiom Al or else follows lines by means of inference rules Hence the last formula (the formula prove) is a theorem.

is either an from previous ARl or AR2 . we wanted to

SYSTEM A is now defined as the system con­ taining grammar rules ( 1) to (3) in addition to the proof mechanism given above (axiom Al, inference rules ARl and AR2 , and the definitions of " proof" and "theorem" ) . SYSTEM A is an example of an " axi­ omatic system" as defined in Chapter I: a specified set of formulas (axioms. -- Al) together with a specified set of logical operations (inference rules -- ARl and AR2) for deriving further formulas (theorems) . Note that the structure of SYSTEM A is specified in a notational way -- such and such typographical strings are defined to be terms or wffs or axioms or proofs oi theorems or whatever -­ we need not use or even know the intended meaning of the symbols to apply these rules. In the defi­ nition of "axiomatic system" given in Chapter I, when we speak of a " specified" set of formulas (axioms) and inference rules this is to be under­ stood as a "notationally specified" set (a set " specified in terms of the typographical qualities of the various strings and not in terms of their intended meanings"). 12

Let us compare simplifications with proofs. Our simplification of " (//+//) =////" down to a sim­ ple identity is on the left, while our proof of this formula is on the right (in both cases I omit line numbers and justifications -- these are not strictly part of the simplification or proof but rather are comments to help us understand what is happening) PROOF

SIMPLIFICATION (//+//) = //// (///+/)=////

(///+/)=//// (//+//)=////

/Ill =/Ill

/Ill =/Ill

One sequence of wffs is the reversed form of the other. The first starts with "(//+//) = ////" and ends with " //// = ////, " while the second goes in the opposite direction. If we have a simplification (to an equivalent simple identity) and turn it upside down, we get a proof. To see anotner exam­ ple of how this works, let us compare the simplifi­ cation of "(/////+///////) =////////////" ("5+7 = 12 ") with the corresponding proof: SIMPLIFICATION

(/////+///////) = //////////// (//////+//////) = //////////// (///////+/////) = //////////// (////////+////) = //////////// (/////////+///) =//////////// (//////////+//) = //////////// (///////////+/)=////////////

///Ill/II/I/ =/////Ill/I/I PROOF

(///////////+/) = //////////// (//////////+//) =//////////// (/////////+///) =//////////// (////////+////) = //////////// (///////+/////) = //////////// (//////+//////) = //////////// (/////+///////)= ////////////

//I/Ill///// =//I/Ill/////

[Here the steps which go between " (///////////+/)" and " ////////////" use rule ARl; all the other transitions are by means of AR2 . J The point is

13

that a simplification can be reversed mechanically to yield a proof.

So proofs can be constructed mechanically from simplifications to simple identities just as simplifications to simple identities can' be con­ structed mechanically for true wffs. From these two facts, it follows that proofs can be con­ structed mechanically for true wffs -- and hence that � wffs are provabl�Theargument can be put as follows: 1

2

3

If a wff of SYSTEM A is true, then it can be simpli­ fied down to an equivalent simple identity.

If a wff of SYSTEM A can be simplified down to an equi­ valent simple identity, then it can be proven.

Hence if a wff of SYSTEM A is true, then it can be proven.

[Using rules ( ar2) & ( arl)]

[Reversing simplification to get proof]

This is not, as before, a proof in SYSTEM A ( for then the lines would consist in wffs and not in English sentences), but rather is a proof about SYSTEM A. The proof shows that the proof mechanism built into SYSTEM A is strong enough to prove all that we want it to prove.

In addition, it should be clear intuitively that if a wff can be proven in SYSTEM A then it is true ( obviously instances of the axiom Al are all true and if you check out inference rules ARl and AR2 you will see that they yield only true conclu­ sions when applied to true premises). But then we can establish that a SYSTEM A wff is provable if and only if it is true: 1

2 3

If a wff of SYSTEM A is true, then it is provable.

If a wff of SYSTEM A is provable, then it is true.

Hence a wff of SYS'rEM A is provable if and only if it is true.

14

( Last paragraph)

( This paragraph)

So, using the terms given in Chapter I and equating "the �rithmetic of addition" with the set 9f truths and falsehoods expressible in SY STEM A, we can say: THE AR ITHMET IC OF ADD IT ION CAN BE REDUCED TO AN AX IOMAT IC SYSTEM.

Thus we can say "Mission accomplished!" with regard to the arithmetic of addition. Our simple 11 ttle axiomatic system -- using a single axiom ( Al) and two inference rules ( ARl and AR2) -- is capable of proving any truth involving the addition of specified whole numbers but will not in turn prove any falsehood. To use a little jargon, SYSTEM A is "sound" ( all provable wffs are true) and "complete" ( all true wffs are provable).

Before going to SY STEM B, let us note that we could have got equivalent results using certain other well-chosen combinations of axioms and infer­ ence rules. For example, instead of using one axiom ( Al) and two inference rules ( ARl and AR2), we might have used three axioms ( A'l, A'2, and A'3) and one inference rule ( AR'l). Then any instance of these four would be a legitimate axiom ( for A'l, A'2 , and A'3) or substitution rule ( for AR'l) -­ where an instance of one of these is the result of substituting any strings of /'s uniformly for "x" and for "y" throughout: A'1

A'2

A'3

AR'l

(x+/) =x/ (x+/y) = (x/+y)

/ =/

If x = y, then: x y

Rule AR'l in effect says that equals may be substi­ tuted for equals -- if a previous step (e.g from A'l) says that " ( ///+/) = ////" and we have a previ­ ous step containing " ( ///+/), " we may derive a fur­ ther step containing "////" instead. Here [A'l + AR'l] serves in place of ARl -- just as [A'2 + AR' 1 J serves in place of AR2 . We can get every instance of Al ( except "/ =/ , " which is provided for by A'3) by using [A'l + AR'l] [for example, A'l would give us " ( /+/) =//" and by AR'l this allows us to substitute "//" for "(/+/)" and hence get "// = //"]. The proof of "(//+//)= ////" in the revised sys�em would look like this:

15

l 2 3

(/// +/ )= //// (// +// )= (/// +/ ) (// +// )= ////

by A'l by A'2 from 1 & 2 by AR'l

The important point here is that quite different axiomatic systems often give similar results, in the sense that anything provable in one is also provable in the other. In this case the choice between systems was made on a pragmatic basis. I chose [Al + ARI + AR2] over [A'l + A 1 2 + A'3 + AR'l] because this results in more intuitive and often shorter proofs, Other than such pragmatic considerations ( which are of no great theoretical importance), it would not have mattered which of the two equivalent systems was chosen,

16

CHAPTER III

SYSTEM B: LOWER ARITHMETIC

Now we consider a second system -- SYSTEM B containing SYSTEM A and more besides. SYSTEM A covered addition, but SYSTEM B also includes multi­ plication and exponentiation.

We define "lower arithmetic" to be the set of truths and falsehoods expressible using symbols for positive whole numbers, "plus, " "times, " "to the power of, " parentheses, and "equals," Basically, "lower arithmetic formulas" are single-equation formulas of arithmetic without variaples single-equation formulas involving the addition, multiplication, and exponentiation of specific whole numbers, Lower arithmetic is able to be reduced to an axiomatic system -- our SYSTEM B, SYSTEM B uses the five symbols of SYSTEM A, plus one new symbol: SYMBOL *

MEANING times

We express "to the power of" as "** " -- this is done to keep down the number of distinct basic sym­ bols· so "* " translates as "times" while "** " tran�lates as "to the power of." 'rhe legitimate terms and wffs of SYSTEM B are whatever is speci­ fied to be such by its grammar rules: ( 1) to (3) of SYSTEM A, plus rule ( 4): ( 4)

A string consisting in "(" and then a term and then "* "-or-"** " and then a term and then ")" is itself a TERM.

Terms generated through rule ( 4 ) include: (///* //) (///* (//+//)) (///** //) ((///* //)** //)

17

["3* 2"] ["3*2 (2+2)"] ["3 n J ["(3* 2) 2 "]

N-ew wffs include:

(///*//) =////// (//+ //) = (///*////)

(/ll**ll)=//III/I//

"[ 3* 2= "6 ] "[ 2± 2= 3*4" ] "[ �3 =9 "]

Wffs constructable using 1) ( to (4) will also be called" lower-arithmetic statements,"

SYSTEM B adds two new notions to those of SYSTEM A, namely multiplication and exponentiation. We will deal with multiplication first,

Chapter II gave a simple way to find the numeral equivalent to a te rm adding one numeral to another [such as " (//+ //) "]; we simply rewrite the term, replacing every instance of " "( or 11+ 11 or 11 with a blank [here getting " // // " ], and then rewrite the resulting string of /'s to eliminate the blanks [here getting "////"], We can give an analogous, but more complex, way to find the numeral equivalent to a term multiplying one numeral by another [such as" (///*////) 11 Suppose ) " -- where the we have a term of the form " ( * first and second blanks are filled with strings of one-or-more /'s, Then to find the equivalent numeral we can follow these instructions: )"

],

Write out the first string of /'s horizontally a number of times ( one above the other ) to cor­ respond vertically to the number of /'s in the second string, Draw a line below this, Rewrite the ' l s in the above rows to form a single string, The last string is the equiva­ lent numeral. FOR EXAMPLE:

u 11•/ 111>

Equivalent numeral

Ill Ill Ill Ill

.//I/I/I/I/II

[ So (///*////) = //////////// -- i, e, 3 *4=1 2 ,]

So we have procedures to find the numeral which is equivalent to a term which adds or multi­ plies two numerals, It is easy to extend these procedures to enable us to determine whether a given wff of SYSTEM B which doesn 't involve expo­ nentiation is true or false:

18

for each te r m in it of the Take the wff and, )" ( where the * ) 11 or "( + form 11 of one-or-more gs strin with d fille e r a s blank al arrived numer alent equiv substitute the ) /'s h, at u�ing the directions in the last pa r agrap mula fo a with up end you until � this Keep doing on consisting in "= " flanked by a string of / s r ight; then the on 's / of g strin a and left the true if the original lower-arithmetic wff is have the and only if these two strings of /'s 's, / of r same numbe (

FOR EXAMPLE:

(///*(//+ // ) =//////////// ( ///*////) =////////////

///////!////=///////I/Ill

The fi r st line here is the formula to be tested [" 3*( 2+ 2) =1 2"], g /� r we get the 11second line by substituti� , 4) " (/ /+ //)" ( 2+ 2") its equivalent , above shown is e -- the equivalenc ng for we get the third line by substituti 1 alent equiv its ) 4" 3* " ( " (///*////)" 1 2") -- the equivalence is shown above, Since the right and left strings are equal, the o riginal formula is true, 11////

(

1//////­

//// / /"

( 11

A NOTHER EXAMPLE:

(/ /+ //) = (/ //*////)

////=!II/I/I/Ill/

The fi r st line here is the formula to be tested [ ti ( 2+ 2): ( 3 I4) II ] ' we get the . second line by subst�tuti� g r� r " (//+ / /)" "( 2+ 2") its equivalent , //// , ( ,. 4 ) and by substituting for "( / //*////) " 11 3* 4" ) its equivalent "//// //////// " ("1 2 11 Since the r ight and left strings are not equal, the original formula is false, ),

(

This should be simple and intuitive,

At this point we begin to expand the axio­ matic system of the last chapter to cover multipli­ cation formulas, To do this, we need multiplica­ tion ana�ogues of inference rules ARl and AR2 , Recall rule ARl: ARl

(x+/)

19

x/

By this rule we may, for example, move from " (/+/)" to " //" or from " (///////+/)" to " ////////. " The purpose of ARl is to eliminate "+" in .formulas where we are adding a number to one [ e. g. "(/+/)" or "(///////+/ )" ] . The corresponding multiplica­ tion rule would eliminate " *" in formulas where we are multiplying a number by one [ e. g. "(/ */)" or " (/// */)"]. Since a number x multiplied by one still equals x, the corresponding rule is very sim­ ple (here any string of /'s may substitute for " x" ) : BRl

(x* /)

X

The following instances of BRl indicate allowable substitutions: (/ */) (// */) (/// */) (//// */)

I (" l*l II (" 2 * 1 Ill (" 3 *1 Ill/ ("4 *1 and so forth

Recall rule AR2 : AR2

( x+/y)



1" ) 2" ) 3 ")

4" )

(x/+y)

By this rule we may, for example, move .from "(/+//)" to " (//+/)" or from "(////+////)" to " (/////+///)." The purpose of AR2 is to move to an equivalent addition formula with one fewer " /" to the right of " +" -- so that we can keep applying AR2 until we have only a single " /" to the right of " +" and then finally eliminate " +" using ARl. The corresponding multiplication rule would allow us to move to an equivalent multiplication formula with one fewer " /" to the right of " *. " Since (x *(y+l))= (x *y)+x, the multiplication rule is as follows (here any string of l ' s may substitute for " x" and any striqg of / 's may substitute for " y" ) : BR2

(x* /y)

( (x*y) +x)

Instances which substitute " /" for " x" include: (/*//) (/ *///) (/ *////) (/*/////)



((/*/)+/) ( " 1 *2 B)" in our notation] is equivalent to " -(A * -B)" [" Not both (A and not-B)" ] ,

This equivalence holds because to say that i f you have A then you have B is to say that you don't have A without B -- o r, equivalently, that what you don't have is A-but-not-B. We look upon sequences using "v" (for "o r") and " >" (fo r "if-then" ) , not as SYSTEM C formulas and w ffs, but rather as abbre­ viations for SYSTEM C fo rmulas and wffs.

There are va rious combinations of axioms and inference rules which suffice to capture the logic of "not" and "and" (and indirectly capture the logic of "o r" and " if-then" ) . Any of the standard systems of propositional logic would do the job. Here we use three axioms and one inference rule.

The first axiom holds regardless of what wff is uniformly substituted for " A" : CPl

-(A * -(A * A) )

Using our def ined term " >, " this axiom would be: (A > (A * A) )

[ " If A, then A and A."]

[" If it is raining, then it is raining and it is raining."]

Our second axiom holds rega rdless of what wf fs a re uniformly substituted for "A" and "B": O r using " >":

CP 2

-((A * B) * -A)

((A * B) > A)

[" If A and B, then A."]

[" If it is raining and it is windy, then it is raining."] 31

Our third axiom ( the least intuitive of the three) holds regardless of what wffs are uniformly substi­ tuted for " A" and "B" and "C": CP3

Or using

- ( - ( A * -B) * -- ( - ( B * C) * -- ( C * A)))

" >" :

( ( A > B) > ( - ( B

*

C) > - ( C

*

A)))

[ " If ( if A then B) then, if not- ( B and C) then not- ( C and A)."]

[ " I f ( if it is snowing then it is cold) then, if it is not the case that ( it is cold and it is windy) then it is not the case that ( it is windy and it is snowing)."]

The inference rule is the following, which holds regardless of what wffs are uniformly substituted for " A" and "B": CPR

Or using '' >"

:

From "- ( A * -B) " and "A" one - may infer "B."

From " ( A > B)" and "A" one may infer "B."

[ "From ' If A then B' and 'A' one may infer 'B.'"]

This is all the proof mechanism we need for propo­ sitional logic. For more information, consult just about any textbook on symbolic logic.

Now we come to the machinery needed for quan­ tificational logic. Here the only additional sym­ bol we use is "n." We use " n , " "nn," "nnn," and so forth as variables; these are used in place of " x , " "y," " z," and so forth in order to keep down the number of distinct basic symbols. We use parenthe­ ses in a new way to symbolize "every"; so " ( n)" is used for "for every ( number) n" [ similarly " (nn)" is used for "for every number nn," and so forth]. We call " ( n ) , " (nn)," " (nnn)," and so forth uni ve r­ sal quantifiers. The two new grammar rules for quantificational logic are ( 7) and ( 8 ) : 32

(7)

( 8)

A string consisting in one or more instances of "n" is a VARIABLE and is also a TERM.

A string consisting in " (" and then a variable and then ")" and then a wff is itself a WFF.

Rule ( 8) in effect says that the result of prefix­ ing a wff with a universal quantifier is itself a wff.

Let us consider some new wffs able expressed in SYSTEM C : n = (// +// ) n =n -n =n

(n)n = (/ / +// ) ( n)n = n ( n)-n = n

[" n = 2+2"] ["n = n" J ["nln"]

to be

[ " For every n, n = 2+ 2"] [" For every n, n =n"] [" For every n, nln"]

In the first group of three wffs, the variable "n" is free -- it does not refer back to any quantif ier to ""'e'x'press "every." V arious instances of these wffs are true or false ( where an instance of these wffs is the result of replac ing the free variable "n" with some specif ic numeral) , The f irst formula [ " n = (// +// ) "] has both true and false instances and so is not considered either " true" or " false"; the second formula [ " n =n"] has only true instances and so is considered " true," while the third formula [" -n =n"] has only false instances and so is consid­ ered " false." The last three wffs start with th � universal quantifier [" ( n) " -- "for every ( number , n" ] and are respectively false, true, and false; in these the variable "n" is not free -- since it refer� back to the quantifier"to express "every•" Wffs with free variables ( e.g. the first three) are considered "statement-forms" but not "statements"; wffs with no free variables ( e.g. the last three) are considered " statements." Let us consider four mo�e SYSTEM C wffs:

- ( n =/ * n =// )

(n)- ( n =/ * n =// )

[" Not both (n = l and n = 2)" L e. " Either nil or nl2 or both"] ["For every n, not both (n = l and n = 2)" ] 33

( n +n n)= ( nn+n) ( n) ( nn) ( n+nn) = ( nn+n)

[" x+y [" For x+y

y+x"]

e v e ry x = y+x"] =

and y,

In the last two, w e use th e mor e conventional vari­ ables ["x" and " y"] and quantifiers [" For every x and y," or " ( x) ( y)"] to t ranslate th e p eculiar va riabl e s [" n" and "nn" J and quantifie r s [" ( n) ( nn)"] of SYSTEM C ,

Although SYSTEM C contains no special symbol for " fo r some ( numbe r)," still w e can express t his id e a indi r e ctly in the system as follows ( wher e " x" r epre s ents some variable): " For som e ( numbe r ) x" [" ( Ex)" in ou r notation] is equivalent to " - (x ) -" [" Not ev e ry ( numbe r) x is not"] ,

This equivalenc e holds b ecause to say "Something is " is to say " Not everything isn't • • • " So to say "Som e number is equal to 2" [" ( En)n =//"] is to say " Not e v e ry number is distinct from 2" [" - ( n)-n = //"] . So whe.never w e want to expr e ss this " some is" notion we can expr e ss the sam e thing using " not everything isn't , " Again this symbol " E" ( us e d to translat e " some" ) is not an official part of SYSTEM C but rather is used to abb r eviate the corr e sponding SYSTEM C formulas . We call " ( En)," " ( Enn)," " ( Ennn)," and so forth existential quantifiers .

In Chapte r I we d e fined " arithmetic" as the s e t of truths and falsehoods expressible using a c e rtain sp ecified vocabulary: symbols for positive whole numbers, " plus," " times," " to the power of," parenthes e s, "equals," variabl e s, and som e simple logical concepts [" not," " and," " or," " if-then," " for every ( number)," and " for some ( number)"] , Each of thes e vocabulary items is expressible in SYSTEM C; so each truth or falsehood of " arith­ m e tic" ( as w e defined it) is expressible in SYSTEM

c.

Let us see how to expr e ss the thr e e e xamples of arithmetic truths that we gave in Chapter I : "2+2= 4" is expr e ssible as " ( //+//) = //// . " 34

" If x+y= z then y+x = z" is e xpr e ssib le as " ( ( n+nn) = nnn > ( nn+n) = nnn)" --i . e . " - ( ( n+nn) = nnn * - ( nn+n) = nnn) ,"

" If xy = l8 and x =2 y then x = 6 and y =3" is expr e ssi ble as " ( ( ( n* nn) =////////////////// * n = ( //* nn)) > ( n =////// * nn = ///))" --i . e . " - ( ( ( n* nn) = ////////////////// * n = ( //* nn)) * - ( n = ////// * n n = ///)) ,"

Simpl e though its vocabulary may b e , SYSTEM C has a gr e at pow e r of e xpr e ssion . L e t us se e how to expr e ss a few other arith­ metic formulas in SYSTEM C notation: n is ev en ( i . e . n is 2 , or 4, or 6, or ••• ) = For s ome number x, n=2x ( Ex)n =2 x = ( Enn)n = ( //* nn) = - ( nn)-n =(//* nn)

n is som e pow e r of 1 0 ( i . e . n is 10, or 1 0 0, or 1 2 0 0, or •• . ) = For som e X num b e r x, n = lO = ( Ex)n = lO ( Enn)n = ( //////////** nn) = - ( nn)-n = ( //////////** nn)

n consists in all9 's ( i . e . n is9 , or99 , or999 i or , , , ) = lO For som e number x, n+l ( Ex)n+l=lO x ( Enn) ( n+/) = ( //////////** nn) - ( nn)- ( n+/)= ( //////////** nn)

n consists in all l's ( i . e . n is 1, or 1 1, or 1 1 1,X or . . . ) b e r x,9 n+l= l0 w r . �� : r � � ) ; �: ( Enn) ( ( /////////* n )+/) = ( //////////** nn) = - ( nn)- ( ( /////////* n)+/) = ( ////////// ** nn)

SYSTEM C formulas become long and unwieldly very quickly , So we make th e following conc e ssion to human nature . We will at tim e s write out SYSTEM C formulas in abb r e viat ed ways; we call this " using loos e notation ," So w e might us e "2+2= 4" as loos e notation for its equivalent " ( //+//) = ////" ; simi­ la rly we might use " n c onsi st s in all l's" o r

35

" (E x ) 9n+l= l0 x " as loose notation for the equivalent When " -(nn) -((/////////* n) +/) =(//////////** nn) ." we speak of a loose notation string as being a " theorem, " this is to be understood as meaning that the corresponding s trict notation formula (which this loose notation string abbreviates) is a theo­ rem. When we use loose notation we will always somehow make clear how to move to an equivalent strict notation formula of SYSTEM C which uses only the eight basic symbols: " /, " "+, " "* , " " (, " " ) , " " = , " " -, " and " n." There are various combinations of axioms and inference rules which suffice to capture the logic of quantificational logic (the logic of variables, Any of the standard systems " all, " and " some " ) . would do the job. Here we use three axioms and one inference rule.

Before getting to the quantificational axi­ oms we need to give a technical definition of the ter� " free." An instance of a variabl e is " free" in a given wff if it is not part of any wff therein beginning with a quantifier using that variable; so each "n" in " n =n" is free but no " n" in "(n) n = n" is free. Intuitivel y, a "free" variable is one that does not refer back to a quantifier to e xpress " every " or " some."

holds axiom quantificational The first regardless of what variable is substituted for " u , " what term for " x , " and what wffs for " • • • x..." and " • • • u • • • " (provided that the wff substituted for " ...x..." is like that substituted for " • • • u • • • " except for having the term substituted for " x" instead of each free occurrence of the variable substituted for " u" and every instance· of a vari­ abl e in the term substituted for " x " in the corre­ sponding substituted instances is free in the sub­ stitution for " • • • x..."): Or using

CQ 1

- ( (u ) • • •u• • •

* -• • • x• • • )

" >" : ( (u) • • ,u• • •

> • • • x. . . )

[ " If every u is such that u is • • • , then x is • • • "]

The second axiom holds regardl ess of what variable 3 6

substitutes for " u" and what wffs for " B" and " C" : CQ 2

Or using

-((u) -(A

" >" :

((u) (A

-B)

*

*

--((u) A

*

-(u) B))

> B) > ((u) A > (u) B) )

[" If everything which is A is B, then if everything is A then everything is B , "]

The third a x iom holds regardless of what variable is substituted for " u" and what wff for " A" (pro­ vided that the variable substituted for " u" does not occur freely in the wff substituted for " A") : Or using " >" :

CQ3

-(A * -(u) A)

(A

> (u) A)

[ " If it is raining , then for all x it is raining " -- the " for al l x " here occurs vacuously,]

The inference rule governing " every " is as follows (which holds regardless of what variable substi­ tutes fo r " u" and what wff for " A" ) : CQR

From " A" one may infer "(u) A."

This rule in effect says that if such and such is a theorem [ e.g. "n = n " ], then we may derive that this theorem holds for every value [e.g. "(n) n= n" ] ; if we derive a theorem then we can stick a universal quantifier in front of it.

We define "lower logic" (often cal led " first order quantificational logic") to be the set of truths and falsehoods e x pressible using "not, " " and, " " every, " small letters (to represent indi­ vidual objects) , and capital letters (to represent statements , properties of these objects, and rela­ tionships between these objects) -- in addition it being specified that we can · use quantifiers with the smal l letters only (so we in effect can speak of " every object x " but we cannot speak of " every property F") ; it can be shown that a statement of lower logic is " logica l ly true" (i.e. " true rega rd­ less of what objects, statements, properties, and r e lations the letters represent" ) if and only if it 37

is provable f rom the axioms, def initions, and inference rules given so far in this chapter. (Actually Kurt Godel himself was the f irst to prove that lower logic could be reduced to an axiomatic system.) So, just as lower arithmetic is reducible to an axiomatic system, so also . lower logic is reducible to an axiomatic system. However joining docile lower arithmetic to docile lower logic pro­ duces a rebellious couple that cannot be reduced to any axiomatic system at all.

We include also the old axioms and inference rules of SYSTEMS A and B, but broadening the idea of an "instance" of these so that now any substitu­ tion of any terms at all (not just numerals) f or "x" and "y" is allowed, so long as the result is a legitimate term or wff. Also we include this ver­ sion of the "Equals may substitute for equals" principle (which holds regardless of what terms substitute for "x" and "y" and what wffs for " • • • x • • • " and " · · · Y · · · " (provided that these two wffs are alike except for the interchange of one or more instances of the substitutions for "x" and "y" and every instance of a variable in the substitu­ tions for "x" and for "y" in the corresponding interchanged instances is free in these wffs): Cl

-(x = y

Or using ">":

(x = y

* -- ( . .

. x...

* - . . .y•

• • ))

> ( • • • x. • . > • • • y • • • ))

Axiom Cl is a standard principle of identity logic (the logic of " =") and is analogous to inference rule AR'l given at the end of Chapter II.

Now we add three "mixed axioms, " combining notions from lower arithmetic with notions from propositional and quantificational logic. The f irst axiom is very simple (here any term may sub­ stitute f or "x"): CMl

- (x+ /) = /

In loose notation this would be: x+ V/

Our second mixed axiom is also fairly simple (here any terms may substitute unif ormly f or "x" and for 38

"y" ) :

-((x+/) =(y+/) * -x = y)

CM2

In loose notation this would be:

((x+/) =(y+/) > x =y)

[" If x+l = y+l then x = y."]

O ur third mixed axiom is a version of the principle of mathematical induction [here any variable r:iay substitute for "u" and any wffs for " • • • u • • • , " " • • • (u+/) • • • , " and " • • • / • • • " -- provided that the substitutions f or these three wffs are alike except that, wherever the substitution for "u" occurs free in the f irst, "(" followed by the substitution for "u" followed by "+/)" occurs in the second, and "/" occurs in the third] : CM3

-(( • • • / • • •

(u) -( • • • u • • •

*

* -(u) • • • u• • • )

In loose notation this would be: ( ( • • •/• • •

*

* - . . . (u+/) • • • ))

(u)( • • • u • • • > • • • (u+/) • • • ))

>

(u) • • • u • • • )

["If a given formula holds for 1 and if, for every number u, if this formula holds for u then it also holds for u+l, then this formula holds f or every number."]

We could add more "mixed axioms" if we wished. By Godel's theorem, we could never add enough such axioms; for any set of true mixed axioms that we added, there would be arithmetic truths which would be provable only if we added still more axioms.

A "proof" of SYSTEM C is def ined as a verti­ cal sequence of wffs, each of which either is an instance of one of the eleven axioms [Al + CPl + CP2 + CP3 + CQl + CQ2 + CQ 3 + Cl + CMl + CM2 + CM3 ] or else follows from previous wffs of the sequence by means of one of the eight inference rules [ ARl + AR2 + BRl + BR2 + BR3 + BR 4 + CPR + CQR] . A "theo­ rem" of SYSTEM C is defined as any wff of a proof. Note that proofs and theorems of SYSTEMS A and B are also proofs and theorems of SYSTEM C; but SYSTEM C of course contains many additional proofs

39

and theorems,

We will soon begin to show t hat SYSTEM sC has either � arithmetic truth are a fatal flaw : falsehooc1s not provable in it or some arithmetic . how this resu l t provable in it . And we could s er axioms in a similar way regardless of what furth strengthen and inference rules would be added to accidental SYSTEM c . The flaw resul ts not from an ence rules infer and s defect in our choice of axiom of ms syste such that fact the from r bu t rathe talk to sense in certa a in arithmet ic are able about themselves,

are

40

CHAPTER V

I D NUMBE RS AND 'r HE G ODEL FO RMULA

This chapter is crucial and difficult; it might take several slow readings to get a feel for what is happening, The basic idea is that we are going to give a way of interpreting SYSTEM C equa­ tions as encoded messages about SYSTEM C formulas -- and one such SYSTEM C equation will in effect encode the message a bout itself that says "� very formula is not provable" ; this equat ion (the " Godel formula" ) asserts its own unprovability and is the key to the proof of G odel's theorem, Let us take a simple example to illustra te the use of numbers and equat ions to encode mes­ sages, Suppose we go into a small fast food res­ taurant and f ind that they sell only eight items -­ each itern being given a corresponding ident ifica­ tion number: 1

3 4 2

Hamburger French Fries Cheesebu rger Coke

5 6 7 8

Hot Dog Chocolate Malted Sausage Sauerkraut

Then we could talk about the items by talking about the corresponding ID numbers. " I want an i tern 3" would become an encoded mesaage that really means " I want a cheeseburger," And " I want items 1, 2, and 6" encodes " I want a hamb urger with french fries and a chocolate malt ed."

The I D numbers here follow certain pat terns. Som ething is a meat item if and only if it has an odd ID number. So " G ive me something with aJ odd I D number" would encode the message " Give me some­ th ing with meat," And something is a meat -less item if and only if its I D number is even. So " G ive me an item with an even I D number" (or equiv­ alently " Give me an item with some I D number n such that n sat isfies the equation 'Some number times tw o equals n ' ' [ ( Ex ) 2x=n ' ]" ) would encode "Give :me a meat-less l t em," The point is that we could talk

41

about food items ( e.g. " cheeseburger" ) by talking about ID numbers (" 3" ) and we could talk about groups of food items ( e.g. " meat-less items" ) by talking about items whose ID numbers satisfy certain equations [" n is an even number" or "( E x ) 2 x = n"] .

Now w e will d o with SYSTEM C formulas what w e did with fast food items -- assigning I D numbers to formulas and then noting how we can talk about groups of formulas ( e.g. numerals or variables or wffs) by talking about formulas whose ID numbers satisfy certain equations.

ID numbers are assigned to SYSTEM C formulas as follows. First each symbol of SYSTEM C is assigned a one-digit ID number : SYMBOL

ID NUMBE R

I + *

2

1

3

4

5 6 7

n

8

Thus " /" has ID # 1 and " +" has ID # 2. Then each formula is assigned the ID # which results frorn replacing each of its symbols with its one-digit ID # ( replacing " /" with " l," " +" with "2 , " and so forth). Let me give two e xamples : The ID # for " / = /" is 1 6 1.

The ID # for "( //+//)" is 4 112 1 15.

If we want to find the formula with a given ID # , we simply reverse the process - - we write out the # and then replace each digit with the corresponding symbol ( replacing " l" with " /," " 2" with " +," and so forth). Let me give two e x amples : The formula with ID # 4 8331 1 1 5 is "( n * *///)." 42

· The formula with ID # 11111 is " /////."

The ID #s follow certain patterns. For exam­ ple, the SYSTEM C numerals have ID #s consisting in all l's : Numerals : " /, " " //," " ///," " ////, " • • • Corresponding ID #s : 1, 11 , 111 � 11 1 1 , •••

So we can say :

n is the ID # for a numeral

if and only if

n consists

in all l's

Recall that " n consists in all l's" can be put as the equation "( nine-times-n P.1 us one) equals some power of ten" or "( E x)9n+l = l0 x.'' Hence the numerals are those formulas whose I D #s satisfy the equation " ( E x)9 n+l = l0 x ... So we can interpret this equation as in a sense encoding the message "n is th� ID # for a numeral . " "( E x 9n+l= ) l0 x" encodes "n is the ID # for a numeral" in the sense that, given our way of assigning ID #s, both are true in the e xact same cases ; each is true if n equals 1 or 1 1 or 1 11 or • • • -- otherwise each is false. So there is a cer­ tain equation which encodes " n is the ID # for a numeral" or, equivalently, " Formula # n is a numeral . " Likewise SYSTEM C variables have ID #s con­ sisting in all 8 1 s: V ariables: " n, " " nn," " nnn," " nnnn, " • • • Corresponding I D #s : 8, 8 8 , 8 8 8, 8 8 8 8 , • • •

So we can say :

n is the ID # for a variable

if and only if

n consis ts

in all 8 1 s

In a similar way, but less intuitively, we can e x press " n cons ists in all 8 's" as the equation "( E x ) 9n+ 8= 8 *1 0 x... Hence the variables are those for­ mulas who�e ID #s satisfy the equation 43

11 (E x9 ) n+ 8= 8*1 0 \" So we can interpret this equation as encoding the message "n is the ID # for a vari­ able" or "Formula # n is a variable." Again, "(E x 9 ) n+ 8= 8*1 0 x " encodes "Formula # n is a vari­ able" in the sense that, for every value of n, one is true if and only if the other also is true; each is true if n equals 8 o r 8 8 or 8 8 8 or ••• -- other­ w ise each is false.

So we a re interpreti ng these two equations as encoded messages about SYSTEM C formulas. But these two equations themselves may be put into strict SYSTEM C notation. So, for e xample, "(E x) 9 n+ l=l0 x " (for "n consists in all l's" ) may be written as the SYSTEM C equation 1 1 -(nn) -((/ ////// // * n ) + /)=(//// //// //** nn ) ." This SYSTEM C equation would likewise be true, fo r any value of n, if and only if n is a number consisting in all l's ( when put in decimal notation ) -- a nd hence this equation would likewise be true, for any value of n, if and only if n is the ID # of a nume ral; hence this SYSTEM C equation would l ikewise encode "Formula # n is a numeral." Similarly a slightly mo re comple x SYSTEM C equation would encode the message "Formula # n is a variable." . Thus the assignment of ID #s to formulas enables us to inte rpret various SYSTEM C equations as encoded messages about SYSTEM C for­ mulas; so SYSTEM C ca� talk about itself. The I D #s for numerals (1 , 1 1 , 11 1, lll l , •••) and va riables (8 , 8 8, 8 8 8 , 8 8 8 8, • • • ) follow simple patterns. The I D # s for terms fo llow more complex patterns. We could define a "term" as whateve r is specified to be such by these three rules: (1 ) (2 ) ( 3)

" ! , " "II , " "n, " "nn, " The result "(x+y ) " terms is

" Ill , " " /Ill , " etc. are " nnn, " " nnnn, " etc. are of replacing " x" and o r "( x * y ) " or "( x** y ) itself a term.

terms. te rms. " y" in " with

S imila r ly we could define an "ID # of a term" as whatever is spec ified to be such by these three rules: ( 1) (2 )

( 3)

" 1, " "11 , " " 1 1 1 , " "1 1 1 1 , " etc. are I D # s of terms. " 8 , " " 8 8 , " " 8 8 8 , " 1 1 8 8 8 8 , " etc. are I D # s of terms. The result of replacing " x " and n y n in 44

" 4 x 2 y 5" or " 4 x 3y5" or " 4 x 3 3y5" with I D # s of terms i s itself the ID # of a term.

So there is a complex pattern of which #s a re or are not ID #s of terms. We can reduce this pattern t o a complex equation (actually a combination of equations i n a single fo rmula) which can be expressed i n SYSTEM C. I will for the moment spare you t he technical de ta ils -- we wi l l get to these in Chapter VII. But for now le t us justx say that there is an equation -- like "(Ex9 ) n +l= l0 " ( for "n consists in all l's") but much more complex -- such that n is the ID # of a term if and only if n sat ­ isfies this equat ion. This equat ion encodes the message "Formula # n is a te rm." If we le t "n is · · · " rep resen t this equat ion (which we will see in Chapter VI I), then we can say: n is the I D # for a term

if and on ly if

n is the ID # fo r a wff

if and only if

EJ

Analogous comments apply to wffs and their ID #s• Here too we can (and w ill)�e up wi th a equat ion (express ible in SYSTEM C ) wh ich i s satis­ fied by n if and only if n is the I D # for a wff. This equat ion encodes the message "Formula # n is a " represent this wff•" If we aga in let "n is equat ion, then we can say :

EJ

Simila rly the I D #s for ax i oms f ollow def i­ nite (but comple x) pat terns. Cons ider ou r first axiom : , Al

x=x

The i nstances of A l which subs t i tute nume fo r " x " are ca 1 1 ed "simple iden t i t ies" ; these, rals and the correspondi ng I D #s, would go as follows: Simple Identi t ies: "/= /, " " //=//, " " !//=///, " Correspond � ng ID #s: 1 61 , 1 1 6 1 1 , 1 1 1 6 11 1 , ••• 45

We s ee that the I D #s fo l low a s imp le pattern : n is the ID # for a s imple i dent ity

n cons ists in a number o f l' s and then a 6 a n d then th is same number of l's

if and o nly i f

The pattern for the other i nst ances of ax iom A l (wh i ch subst itu te a term other than a numera l for "x" i n "x =x") . wou l d be more comp lex -- and even more complex wou l d be the patterns invo l ved in the other ax ioms. Th is pro je ct o f r e du c ing thes e pat ­ terns to an equ at ion ( ex press ible in SYSTE M C ) is very d i f f i cult. I ref er you to Chapter VI I for det a i ls. But aga i n there is an equat ion wh i c h is sat is f ie d by n if a nd o n ly if n is the ID # fo p an ax iom , Th is equat ion encodes th e message " For�u la # n is an ax i om . " If we aga i n l �t "n is • • • " rep­ resent th is equa t ion, then we c an say: n is the ID # for an ax iom

if and only if

B

Sim i l ar ly the I D #s for SY 3'r EM C theorems fol low def init e (but even more comp lex) patterns. The ma jor aim of Chapter V I I is to redu c e th is pat ­ tern to an equat ion (express i b le in S Y STEM C) -- to find an equat ion wh i ch is sat is f ied by n if a nd only if n is the ID # for a th eorem. If we aga in let "n is . . • " r epresent th is equat ion, then we can say: n is the ID # for a theorem

if and only i f

B

To ma k e th is point more i ntu it ive, let us suppose for a mome nt someth ing that is not true -­ name ly that the pattern of w h i c h nur.1bers are and are not ID # s for theorems is very s imple -- the theorems hav i ng even ID #s and the non-theorems hav i ng odd I D #s : 46

I D # s fo r Theorems: 2, 4, 6, 8, 1 0, 1 2, • • • ID # s for Non-Theorems: 1, 3, 5, 7, 9, 1 1,

B B

( If only life were that simple ! ) say: n i s the ID # for a theorem n i s the ID # for a non-theorem

if and o nly if

Then we could

if and only if

For "n is eve n" a nd "n is odd" we cou l d here sub­ stitute the corre sponding equat ions " ( Ex) 2x = n" and " - ( Ex)2x = n" [ i n stri ct notation these woul d be Then a nd " (nn) - ( //*nn) = n" ]. 11 - ( nn) - ( //*nn) = n" " - ( Ex) 2x = n" ( for " n is o dd") woul d enco de the mes­ s age "Formula # n is not a theorem" ; i. e. " - ( Ex) 2x = n" wou l d be true of a ny n if and only if formula # n is not a theorem. So then " - ( Ex)2x = l 6 1 1" (for " 1 6 1 1 is o d d" ) wou l d enco de the message "Formu la # 1 6 1 1 (" /= //") is not a theorem" ; i. e. " - ( Ex) 2x = l 6 1 1" wou l d be true if and only if " /= //" is not a theorem , A ctua l ly, of course, we are j ust pretend i ng certa i n th ings to make the po i nt more understanda ­ ble. The equat ion wh i ch i s true of n i f and only if n is the ID # of a non-theorem is mu ch more com­ p lex than the equat ion for "n is. o dd. " But the bas i c i dea is correct. In part i c ular, ther e is a definite equat ion (here repre sented by "n is • • • ") wh i ch i s sat is f ied by n i f and only if n is the I D # o f a non-theorem: n is the I D # for a non-theorem

i f and only if

B

And i f we rep la ce "n" with a spe c i fi c numera l (e. g. " 7 4858 6 1 1") we get an arithmet i c statement (here repre sente d by " • • • ") which i s true i f and only if t he fo rmula w ith tha t I D # is a non-theorem : 47

7 4 85 8 6 1 1 is the ID # for a non-theorem

if and only if

D

So, j ust as a simple equation [ " (Ex )(( 9 * 1 1)+ 1)= 1 0 x" -- for " 1 1 consists in all ls" ] encodes the message " Formula # 1 1 is a numeral, " so now a certain com­ ple x equation would encode " Formula # 7 4 858 6 1 1 is not a theorem." To make another point more intuitive, let us again suppose for a moment something that is not true -- namely that the equation represented by the box above t o the right (the one t hat encodes " For­ mula # 7 4 858 6 1 1 is not a theorem" ), when e xpressed in strict SYSTEM C notation, turns out to be the very simple equation " - ( n)n = //," Let us subs titute this equat ion for the " .,," in this right bo x (as we also rephrase slightly t he words in the left box ) and see what happens: Formula # 7 4 858 6 1 1 is not a theorem

if and only if

Now t he right box is true if and only if the left bo x is also true -- " -(n)n =n" encodes " Formula # 74 85 8 6 1 1 is not a theorem" ; so: ( 1)

" - ( n)n = / /" is true if and only if formula # 7 4 85 8 6 1 1 is not a theorem.

B u t notice that formula " -(n)n = / /" happens t o have I D # 7 4 85 8 6 1 1: (2)

" -( n)n = / /" is formula # 74 858 6 1 1.

But, since " - ( n)n =n" ( formula # 7 4 858 6 1 1) encodes t he message " Formula # 7 4 85 8 6 1 1 is not a theorem, " in effect " - ( n)n =n" encodes the message that it i t s elf is not a theorem ! So from ( 1) and ( 2 ) conclude ( 3 ) -- that this formula is true if and only if it is not a theorem:

we

48

( 1) (2) (3)

" - ( ri)n = / /" is true if and only if formula # 748 5 8 6 1 1 is not a theorem,

" - ( n)n = / /" is formula # 7485 8 6 1 1.

Hence, " - ( n)n = / /" is t rue if and only i f " - ( n)n = / /" is not a theorem.

But then " - ( n)n = / /" is either a true �-t heorem or a fal s e theorem: (3)

(4)

(5) (6)

is true if and " - ( n)n = / /" " - ( n)n = //" is not a theorem.

only

if

So if " - ( n)n = //" is true, then "- ( n)n = / /" is not a theorem ( t hen it is a true non­ the�m) ,

So if " - ( n)n = / /" is not true " - ( n)n = / /" is a t heorem'"°-rthen it false theorem) . Hence " - ( n)n = //" is either a theorem or a false theorem.

then is a

true non­

Hence our system contains either a true non-t heorem ( a non-provable arithmetic truth�o�a false t heorem ( a provable arithmetic falsehood); we can put this in a diagram : an arithmetic truth or an a rithme t i c fals i ty

In neither case is arithme tic reducible to SYSTEM C.

This argument may be hard it get s to the conclusion in I t may ta k e a couple of readings is going on here and t hroughout chap ter.

to grasp at first a surprising way, t o underst and what the rest of ,this

The argument was based on an admit tedly false sup posit ion -- namely that the SYSTEM C equat ion encoding " I q) (Eu)

[" Eithe r p or q" ]

= df

[" I f p then q" ] = df

[" F'or some (numbe r ) u" ]

-(-p * -q)

-(p * -q) = df

-(u)-

We need to stipulate what things may substitute for the lette rs used i n these and late r definitions. We stipulate that any wffs or abb reviations for wffs may substitute fo r "p" or " q" ; any va riable may subs titute for "u" ; any te rms or abbreviations fo r te rms may substitute for " w" th rough " z" and for these with subsc r ipts ; the smallest va riable not free in the defined sequence is to substitute for " a " the second smallest such variable fo r " a " ;nd so on through the a's and then the b's an! ' so on to " f" ; symbols define d late r in this chapte r are always to be eliminated before symbols defined earlie r ; and instances of the same defined symbol are to be eliminated starting from the beginning of the sequence and wo rking to the end. Next we define " greater than" :

[" x is g reate r than y" ]

GR(x, y) = df

(Ea)x = (y+a)

[" x equals y-plus-some-number" ]

The "nume rals" of SYSTEM C are strings of one o r mo re l 's ( " I , " " II , " " Ill , " and so f orth) and have I D numbe rs which consist in all 1 s (1 , 1 1 , 111 and so fo rth). S o , as we saw in Chapte r V, for�ula # n is a nume ral if and on ly if n consists in all 1 's ; and so we encode " Formula # x is a numeral" as follows: [" Formula # x is a nume ral" ] or [" x is a nume ral #" ]

N(x) = df

9 * x)+l)= (lO * *a)) (Ea)((( 72

["x consists in all l's" ] or [" For some number a, 9 x+l= l0 a" ]

The next definition is somewhat ana logous: [" 1''ormu la # x is the nume ral whic h represents y" J NUM(x, y ) = df (( 9 *x)+ 1 )= ( 1 0 * *y ) ["9 x+l= 1 0 Y n J

An example might clarify this : formula # 1 1 1 ( " II I" ) is the numera l which represents 3 -- and likewise 11 1 when multiplied by 9 (giving 99 9) and then increased by one (giving 1 0 0 0) equals 1 0 3.

The " variables" of SYSTEM C are strings of one or more n's ("n," " nn , " "nnn," and so forth ) and have I D numbers which consist in all 8 's ( 8 8 8, 8 88, and so forth). S o formula # n i s a vari� able if and only if n consists in a l l B's. so we encode 11 Formul a # x is a variable" as fol lows: [" Formula # x is a variable" }

V(x )

= df

(Ea ) (N(a ) * x= ( B *a) )

[" x is eight times some number which is is the formula number of a numeral"] or [" x is eight times some number which consists in al l l's"] or [" x consists in a l l B's"]

So we can easily encode " numeral" and "variable : " But to encode "term, " " wff," "axiom, 11 and so forth we must first deve lop some complex machinery.

We cal l the number of digits in the decimal form of a number the "length" of that number · so 5 has length 1 , 2 4 6 has length 3 , and so fort�. we define this notion as follows: L(x, y )

= df

[" x has length y"]

(GR((l O * *y ) , x) * -GR(( l O * *y ) , (x * l O ) ) ) Y [" 1 0 is gre ater than x but not gre ater than 1 0 tim es x"]

73

This defin i\ion gives the right r e s u lts: 5 has le n gth 1 ( 10 being greater tha n 5 �ut not greater than tha n 5 0 ) , 2 4 6 has len gth 3 ( 1 0 being g reater than 2 4 6 but not greater than 2 4 6 0 ) , an d so forth,

The n ext defi nition defines a kin d of decimal notation (here a n y terms or abb reviations for terms may be substituted fo r the subscripted y's except that a ny nume ral o r va riable must first be e nclosed in pa re ntheses before being substituted for on e of the y's) : x

( L(y2 , a2 ) * =

= df

* L(y i, ai) * < Y 1* (l O** (a2+ • • • + a l )))+

(y2* ( 1 0** ( ,,,+ ai))) +

,,,+

yi)

[ " x equals (y 1 times 1 0 to the power of the combin ed le n gths of y 2 a n d • • • a n d y i ) plus ( y2 times 1 0 to the power of the combi ned le n gths

of ... a nd y i) plus ... plus y i"] We a re n ow gettin g la z y about pare ntheses; p resume oute r pare ntheses a n d association to the left a nd so forth, So "x = 3 91 " ["x equals ( 3 times 1 0 to the power of the combined le ngths of 1 and 9 ) plus ( 1 times 1 0 to the power of the le n gth of 9 ) plus 9 "] would be defin ed as follows: x = 3 19 = df ( En) ( En n)(L(l, n) * L9( , n n) * x = ( 3* ( 1 0** ( n+ n n))) + (l* (lO** n n))+ 9 )

Similarly, " x = y 3 z " [ " x equals (y times 1 0 to the powe r of the combin ed le ngths of 3 and z ) plus ( 3 times 1 0 to the power of the length o f z ) plus z "] would be defined as follows: x = y 3 z = df (En )(Enn)(L( 3 , n ) * L( z , n n) * + n n))) + ( 3* ( 1 0** n n)) + z ) x = (y* (l O** ( n

The i ntuitive idea behind the exp ression "x = y 3 z " is that if, for example, y =77 a n d z = 8 88, then we want x to equal 77 3 8 8 8 : 74

If X = y3 z and y 77 and z 8 8 8, then x 7 7 3 8 8 8.

The n ext set of defi nitions makes it . easier to give the ID #s for a formula . Since " n " has ID # 8, we might as well use " n " as alter native n ota­ tion for " 8" -- for then we �a n say that " n" has ID # £• So we give these defi n ition s : f.

±*

= df = df

= df

1 2 3

.l l

= df = df

= df

4 5 6

-n

= df = df = df

7 8 9

Then , i nstead of saying that " / = /" has ID # 1 6 1, we ca n equivale n tly say that it has ID # � (the form The "L::i. " being alter native n otation for " 1 6 1" ) , main effect of this is to make some of the later defi nitions mo re readable, .}...

The last defin ition above defines " " as " " ; so in effec't the alternative n otation fo r 9 comma (", " ) has ID # 9 , The comma is not part of the vocabula ry of SYSTEM C ; rather it will be used A i n writing sequences of SYSTEM C fo rmulas, " sequence" here is a ho rizo n tal string of o ne o r · mo re SYSTEM C fo rmulas with a comma at the begin­ n ing a n d e n d a nd between ad jace nt formulas, For example, " , n, n n, n nn, " is a sequence; it con tains the fo rmulas " n " a nd " n n " a nd " n nn " with appropri­ ate commas, This sequence has ID # , n , nn , n n n , ( ID # 9 98 8 98 8 8 98 ) , S o, just as individual formulas have ID #s, so also sequences of individual formulas now have ID #s, Formula " nnn " occurs just after sequent!e " , n, nn, " which begins sequen ce " , n, nn, nnn, ." Using the cor respon din g ID #s, we ca n say that formula # 9 8 8 8 (" nnn " ) occurs just after sequence # 9 98 88 (" , n, nn , " ) 8 which begins sequence # 9 98 8 98 8 8 9 (" , n , nn , nnn, " ) , We define the ge n eral n otion here as follows: [" Formula # x occurs just after sequence (o r comma) # y which begins sequence # z "]

S(x, y, z ) = df (Ea)( z = yxa * (y= J.. v ( Eb)y= b ) * (a = "" v (Eb)a = b ) * -x = * -(Eb)x �S- * -(Eb ) x =b.,_ * -(Ebl (tc)x =b.&.c ) .L 75

["For some a : the decimal form of z is yxa, where y is or starts-and-ends-in .J. ' a is o r starts-and-ends-in L ' and x c ontains no i" ]

The next definition defines an analo gous notion :

["Formula # x o c curs just after some sequence (or comma) # whic h is less than Y and which begins sequenc e # z " ] P(x,y, z)

= df

(Ea)(GR(y, a) * S(x,a, z) )

["For some a less than y, formula # x o c curs just after sequence (or comma) # a which begins sequence # z " ]

Here " P" is used because this notion i n effect refers to previous members of the s equence.

A " term" of SYSTEM C is whatever is specified t o be such by its grammar rules -- the relevant rules being ( 1) and ( 2 ) of Chapter II, ( 4 ) of Chap­ ter III, and (7) of Chapter IV. We could c ombine these rules in this definition : Every numeral is term; if x and y cations of this (x*y) , and (x* *y)

a term; every va riable is a are terms (by previous appli­ rule) then so also (x+y) , are terms.

A n equivalent definition could be given which makes use of the notion of a sequenc e of formulas: A term is a member of some sequence of formulas satisfying these conditions : eac h member of the sequence is either a numeral or a variable or else of the form (x+y) , (x*y) , or (x* *y) where x and y are previous members of the sequence.

We encode "Formula # x is a term" to mirror this latter definition: [ " Fo rmula # x is a ter>m"]

T( X ) = df (Ea) ( (Eb) s( X J b J a) * (b) ( C ) ( s(b J C J a) p( e J C J a) > N( b) V V(b) V (Ed) (Ee) ( p( d ' C J a) (b= (d+e) v b=(d * e) v b= (d!!el) ) ) )

---

---

-

*

*

[ " Fo r> some sequence # in which x oc curs : every formula # in any place is either> a numeral # or a va r>iable # o r>, where d and e are p r>evious 76

members

o f the f orm (d+e ) o r (d *e ) or (d * *e ) " ]

A "wff" of SYSTEM C is whatever is specified t o be such by its grammar rules -- the relevant rules being (3) of Chapte r> II and (5) , ( 6) , and ( 8) of Chapte r> IV. We could comb ine them in thi3 defi­ nition : J

-

-

-

-

-

-

- -

-

If x and y are terms, then x = y is a wff; if p is a wff (by previous applications of this rule) then -p is a wff; if p and q ar>e wffs (by previous applications of this rule) then (p * q) is a wff; if u is a variable and p is a wff (by previous applications of t his rule) , then (u) p is a wff.

If we translate this into a definition using the notion of a sequen ce of formulas, then we get: A wff is a membe r> of some sequen ce of formulas satisfying these c onditions : eac n member of the sequen ce is of the form x = y, where x and y are terms ; or of the form -p, where p is a p r>evious member of the sequen ce ; o r> of the fo r>m (p * q) , where p and q are previous membe r's of the sequen ce ; or of the form (u) p, wher>e u is a variable and p is a previous member of the sequence.

We encode "Formula # x is a wff" to mir>ror this latte r> definition : ["Formula # x is a wff"]

W(x) = df (Ea) ((Eb) S(x, b, a) * (b) (c) (S(b, c, a) > (Ed) (Ee) (T(d) * T(e) * b = d= e) v (Ed) (P(d, c , a) * b = -d) v (Ed) (Ee) (P(d, c ;'a) * P(e, c, a) * . b =(d *e} ) v (Ed) (Ee) (V (d) * P(e, c ,a) * b= (d) e) ) )

-- -

--

[ " F or some sequence # in which x occurs: every formula # in any place is either> of t he form d= e ' where d and e are term # s ; or of the fo rm =.d, wher>e d is a previous membe r>; o r> of the form id ¥rel, where d and e are p r>evious members; or of the form �d2 e, where d is a variable # and e is a p r>evious member"]

V ariable "n" occurs in wff "(n */)= /" at a place which ends initial segment "(n" of wff " (n */) = /." Using the corresponding ID # s, we can say that variable # 8 (" n") o c curs in wff # 4 8 3 1 5 6 1 77

[ "( n* / ) =/" ] i n a place which ends 4 8 [ " (n" ] of wff # 4 83 1 5 6 1 [ " (n*/ the gene ral notion (the notion of ring in a particul a r place in the

initia l segment # ) =/ " ] . We en c ode a va riable occur­ w ff ) a s follows :

[ " Va riable # x oc c urs in wff # Y in the place which ends initial segment # z of wff # y" ]

a )(y = z a * O(x, y, z ) =df V( x ) * W( =y ) * (y=)z( zv= a(E * -a =.!2. * x a = (E v x z ( * = ) ) -a n * -( Eb ) a nb -(Eb) a = b� )

[ " x is a va riable # and y is a wff # and eithe r y = z o r else y = z a (fo r some a whi c h isn't .!l and doesn't begin with ,u) a nd either z = x o � else z = ax ( for some a which isn't .!1 and doesn t end with .!! ) " ]

Then we en c ode the n otion of a " free " occur­ rence of a va riable: [ " Vari able # x o c curs f reely in wff # Y in the pla c e whi c h ends initial segment # z of wff # y" ]

= df O(x, y, z ) -(Ea ) y = i_xla F(x y z ) ( a ) ( b ) (c ) ((W(a ) * b = c ,Lxla * ( y = b v (Ed ) y = bd ) ) (GR( c , z ) v GR( z , b ) ) )

*

*

>

[ " Variable # x o c curs in wff # y a t a place whi c h ends initial segment # z of wff # Y , Y doesn 't begin with Del , and : for eve ry w f f # a p receeded by ,lxl which oc c urs in y after some initial segment c, z is eithe r less �f an c or else greater than b (where b is c Lxla ) J

The next five ve ry dif ficult def initions deal with substitution. The first two enable us to mi r­ ror the arithmetic inferen c e rules [ ARl + AR2 + BRl + BR2 + BR3 + BR4 ] -- mi rroring ARl, for example, by whi c h terms "(/// +/ ) " and "//// " may be inte r­ changed. The first definition here prepares for the second: [ " Wff # w results from substituting term # x for ONE instance of term # y in wff # z " ]

78

SUB l T(w, x, y, z ) = df W( w ) * ( Ea )( Eb ) (( w = x b * z = y b ) ( w = axb * z = ayb ) )

*

W( z ) T( x ) * T( y ) v (w = ax * z = ay ) v

*

[ " w and z a re wff #s, x a nd y are term #s, and , for some a and b, ( 1 ) w = xb and z = yb, or (2 ) w= a x and z = ay, o r ( 3 ) w = axb and z = ayb" ] [ " Wff # w results f rom inte rchanging ONE OR MORE instan c es of te rm # x and term # y in wff # z " ]

= df ( Ea ) (Eb ) ( S( w, b, a ) SUBT(w, x , y, z ) ( b ) (c ) ( S(b, c , a ) > (b = z v (Ed ) ( P(d, c, a ) ( SUBlT(b, x , y, d ) v SUBl T(b, y, x, d ) ) ) ) ) )

* *

[ " Wff # w is a membe r of some sequence in which e ach membe r either is wf f # z o r else results from an ea rlie r membe r by substit u ting term # x for ONE inst a n c e of term # y o r by substituting term # y fo r ONE instan c e of term # x " ]

The ne x t two def initions enable us to mi r ro r axiom · Cl, which says " Equals may be substituted for equals." Again the first def inition prepa res for the second (here " f reely" means " whe re term # x substitutes fo r f ree instan c es of va riable # y and eve ry instance of a va riable in term # x is free in the c o r responding instan c es in wff # z " ) : [ " Wff # w results term fo r ONE instan c e of in wff

f rom substituting # x va riable # y freely # z"]

SUBl VF(w, x , y, z ) = df W( w ) * W(z ) * T( x ) * V( y ) (( Eb ) (w = xb z = yb ) v ( Ea )( Eb )((( w = ax z= ay ) v ( w = axb ( E c )( c= ay * F( y, z, c ) ) z = ayb ) ) ( d ) (e )( f )(( V( d ) f = ax GR(e, a ) -GR( e, f ) ) > F( d, w, e ) ) ) )

*

*

*

*

*

*

*

*

*

["w and z a re w f f #s, x is a term # , y is a va riable #, and, for some a and b, ( 1 ) w = xb and z = yb, or ( 2 ) w= ax and z=ay, o r (3 ) w= axb and z = ayb and, in cases (2 ) and (3 ) , va riab le # y i s f ree

79

in the place end i ng ini t i a l segment # c ( wh e re c =ay ) of wff # z, and eve ry va riable # d in term # x is free in wff # w in any place ending some initial segment # e of wff # w ( where e is between a and f, where f = ax ) " ] ["Wff # w results from interchanging ONE OR MORE instances of term # x and te rm # y freely in wff # z " ]

SUBTF( w,x , y, z ) = df ( Ea )( Ee)( Ef )( (Eb)S( w , b , a ) * ( Eb )S ( z, b,a ) * W ( e ) * V ( f) * -( Eb ) F( f,w, b ) * -( Eb ) F( f, z , b ) * ( b) ( c)( S (b,c,a ) > ( b = e v v ( SUB1 VF( b , x,f, d ) * ( Ed ) (P( d,c,a ) SUB1 VB(b,y, f,d ) ) ) ) ) )

[" Wff # w and wff # z are members of some sequence in which there is some wff # e and some variable # f ( not occurring freely in wf f # w or wff # z ) such that each membe r eit her is wff # e or else results from an earlier member by subs tituting term # x or term # y for ONE instance of va riable # f freely"]

The last one here enables us to mirror axioms CQl (" If eve ry thing is such and such then x is such and such" ) and CM3 ( the principle of mathematical induction ) -- as wel l as the "SON" no tion: ["Wff # w results from substituting term # x for ALL instances of variable # y freely in wff # z " ]

SUBALLVF( w, x,y, z) = df (Ea)(Eb )( W(a) * V(b ) * * -(Ec) F(b, w, c) * * -(Ec)F(y, a, c ) V ( y) (SU BTF(w, x, b , a) * -(Ec)F(b, z , c ) * SUBTF(z, y, b, a) ) ) )

[" For some wff # a and variable # b, variable # y is not free in wff # a, variable # b is not free in wff # w or wff # z, wff # w results f'rom interchanging one or more instances of term # x and term # b freely in wff # a, and wff # z results from interchanging one o r more instances of term # y and term # b freely in wff # a" ]

If you are still with me, you will be pleased t o 80

hear that the hardest part is now ove r.

We now encode "Formula # x is an axiom, " mir­ r oring the eleven axioms of the SYSTEM C proof mechanism; I give the name of the axiom ( "A l 11 " C Pl , " and so forth) to the left of the corresponJ­ ing part of the definit ion: ["Formula # x is an axiom" ]

A (x) = df W (x ) * ( ( Ea )( T( a) * Al x = a;:,ii) V ( Ea) ( Eb) (Ec) ( W (a ) * W( b ) * W( c) * CPl ( x = :J.a *-(a!all CP2 v x = �!b)*-al CP3 v x = - - a* -b ) * --(-(b*c )* --( c *aill ) ) V ( Ea ) (Eb )( Ec ) ( EdT(SUBALLVF( a, c, d: b ) * CQl x =.::..U..d.2.b.!:al ) V ( Ea)( Eb) ( Ec ) ( V( a ) * W( b ) * W(c ) * CQ2 x = filal=.,lb!.,::c l* --C (alb* - C a.2, cl.2, ) V ( Ea ) ( Eb ) V( a ) * W (b ) * -(Ec) F ( a, b, c ) * CQ3 x = .::J.b *- ( a ) b ) ) V ( Ea ) ( Eb )( Ec ) ( Ed ) ( SUBTF( a, b, c, d ) * Cl ) ) ) x =.=.{_b=c *--(a!.=dV ( Ea ) ( T(a ) * CMl x = -( a+/ ) =/ ) V ( Ea ) ( Eb)(T (a)* T( b ) * CM2 x = .::.Ua+/ ) = (b+/ ) *-a=bl ) V (Ea ) ( Eb ) ( Ec)( Ed ) (Ee )( SUBALLVF(b, /, a , c ) * e = ta:tL.l * SUBALLVF( d, e, a, c ) * CM3 x = -((b!J.al=ic!.::d))* - ( alcJ ) ) ["x is o f the form a = a

(i.e. a6a ) , where a is a t erm # ; or • • • " ]

Now we mirror the eight inference rules:

ARl AR2 BRl BR2 BR 3 BR4 CPR

["Formula # x foll ows by an inference rule from formula # y and formula # z " ]

R ( x, y , z ) = df (Ea ) ( Eb ) (Ec)(Ed )( SUBT(x a b y) * ' ' ' (( a = i_c±Ll * b =cL) v ( a = .lc±Ldl * b = {_cftdl ) V ( a = rc!L}.. * b =c Y v ( a = Ic!£dl * b = ilc!dl:!:.cJ ) v ( a = ,Lc � * b = c) v ( a = (c !.!Ldl * b =((c** d) * c-) ) ) v z = -( y* -x )

---

81

CQR

v ( Ea ) ( V(a ) * x = taly )

[" Wff # x results from interchanging one or more instances of some terms # a and # b in wff # y, where, for some c, a = i_c+/) and b = cL ; or

•••"J

In Chapter IV we defined a " proof" of SY STEM C as a vertical sequence of wffs , each of which e ither is an instance of one of t'he e l even axioms or e lse fol lows from previous wffs of the sequence by one of the eight inference rules. We define d a " theorem" of SYSTEM C as any wff of a proof. We could equivalently have define d "theorem" in terms of a sequence of formulas more directly, as fol­ lows: A theorem is a member of some sequence of for­ mulas satisfying these conditions: each member of the sequence is eithe r an ax iom or e lse fol ­ l ows from previous members o f the sequence by an inference rule.

We encode " Formula # x is a theorem" to mirror this latter definition: [" Formula # x is a theorem"]

TH(x) = df (Ea) ( ( Eb )S(x,b,a) * (b) ( c)(S(b,c,a) > A(b) V ( Ed)( Ee ) ( p(d ' C ' a ) * p( e ' C ' a) * R(b,d,e) ) ) )

[" For some sequence # in which x occurs : every formula # in any place is either an axiom # or is such that it fol lows by an inference rule from formula # d and formula # e, where d and e are previous members" ]

We define the notion of the SON of a formula (see C hapter V) as follows: [" Formula # x is the SON of formula # y"] SON(x ,y ) = df (Ea ) (NUM ( a,y ) SUBALLV F(x,a , 8 ,y ) )

*

[" Formula # x is the result of substituting the numeral for y in formula # y for variable # 8 (i.e. 'n' )"] or

82

["x is the I D # of the wff which results from taking the wff with ID # y, erasing the variable 'n' in it, and instead writing in y number of /'s" ]

And then the easy:

definition of formula F is . rather

[" The SON of formula # n is a not a theorem" ] F = df

(a) (SON ( a,n ) > -TH(a) )

[" For all a, if formula # a is the SON of formula # n, then formula # a is not a theorem"]

Now to e xpress F in strict SYSTEM C notation we would take F and, in a step by step way, e limi�ate each defined term in favor of its definitional equivalent (using the 44 definitions of this chap­ ter ) ; the result would be the unabbreviated version of F which uses just our eight basic symbols [" / " " +," " * , " "(," ")," "= ," "-," and " n"]. This wouid be a long and thankless job, but it could be done. F is an abbreviation for a very very long SYSTEM C wff in strict notation. So the end result is that we can e xpress F (the FATHER of the Gode l formula G ) as a SYSTEM C formula: F

The SON of formula # n is not a theorem

If you have forgotten what we wanted to do with this formula, then ple ase go back to Chapter v .

83

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