1
i
UDIES IN LOGIC
1
AND THE
FOUNDATIONS OF MATHEMATICS L. E. J. B R O U W E R / E. W. B E T H / A. H E Y T I N G EDITORS
Recursive Number
R. L. GOODSTEIN NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM
STUDIES IN LOGIC AND
R E C U R S I V E NUBIBER THEORY
THE FOUNDATIONS OF MATHEMATICS
: IDEVELOPhIENT OF RECURSIVE ARITHJIETIC
IN A LOGIC-FREE EQC.4TION CALCULCS
R. L. GOODSTEIN L. E. J. B R O U W E R E. W. B E T M A. H E Y T I N G
N O R T H - H O L L A N D P U B L I S H I N G COMPANY AMSTERDAM
Professor of .tlcthema&ics Linioersi
The nature of niimbers. .%rithinetic- antl tlie gRme of chess. I>t,finition of counting. Eiolution o f the concept of 5 forriiul system. C H A P T E R.I . . . . . . . . . . . . . . . . . . . . . . . . . The arithmet,ical operations. Deíinition b - itertition und recursion. Contracted notatiori. Single and niultiple rrc.ur.~ii>~is.
D e h i t i o n of proof. Commutative and associativil properties of addition. The equations z j- (y-z) = y ( 2 - y ) and (1 , r , y l ) j ( x )= ( 1 jz,yj)!(y). Tlie f~mctipns Z,, L ' , and p,. Inequalities. Verifiabdity and freedom frorn contradiction.
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C-R
111
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The propositional calculus. Propositional functions. The limited universal, existential and minimal operators A:, and LE.Mathemetical induction. Tlie counting operator S:.
e,
The fundamental theorems of arithmetic. Prime numbers. Uniqueness of the resolution into prime factors. The greatest common factor. EXAVPLESI V .
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Formalisations of recursive arithmetic. Tlie Deduction Theorem. Reduction of the uniqueness schema. CHAPTERVI . . . . . . . . . . . . . . . . . . . . . . . . . Reductions to primitive recursion. Course-of-values recursion. Recursion with parameter substitution. Simultaneous recursio~w. Generalised induction schemata. Permutstion.
W
COSTENTS
CHAPTER VI1
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134
Elirnination of parameters. The initial functions alid tlie generat ion of al1 prirnitiie recursive functions by the single scliema F z = -4'0. The enumerating function. A doubly recursive fuxiction which cannot be defined by primitive recursion and substitution alone.
CHA~ER VI11
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Godel numberirig and tlie incompleteness of aritlimetic. Tlie aritlimetisation of syntax. The constriiction of a verifiable unprovable eqiiation. Skolem's non-sta~idardmodrl for aiitliinetic.
INTRODUCTION 113
The Nature of Numbers The questioii "lJTllat is t!ie natiire of a iiiathematical entitg?" 1s oiie which has interested thinkers for over two tliolisand yearc and has proved to be very difficult to anslver. Even the first and forernost of these entities, tlie natural number. Iias the elusiveness of a wiil-of-the-wisp when \ve try to,define it. One of the sources of the difficulty in saving what numbers are is that tliere is nothing to wliich we can point in the world around us when me are looking for a clefinition of number. The number seren, for iiistance, is not aiiy particular collectioii of seven objects, siiice if it were, then no other collectioii could be said to have seven members; for if me identify the property of beiiig seven mitli tlie property of being a particular collection, then beirig seven is a property which no other collection can llave. A more reasonable attempt a t defining the number seven would be to say tliat the property of being seven is the property mhich al1 collections of seven objects have in common. The difficulty about this definition, however, is t o say just mhat it is that all collections of seven objects really do have in common (even if we pretend that me can ever become acquainted with al1 collections of seven objects). Certainly the number of a collectioii is not a property of it in the sense that the colour of a door is a property of the door, for we can change the colour of a door but we camot change the niimber of a collection without changing the collection itself. It makes perfectly good sense t o say that a door w-liich mas formerly red, and is now green, is the same door, but it is nonsense to say of some collection of seven beads that it is tlie same collection as a collection of eight beads. If the number of a collection is a property of a collection then it is a defininr~property of the collection, an essential characteristic. This, however, brings us no nearer to an answer to our question
2
ISTRODCCTION
"What is it that all collectioiis of seven objects llave in common?" -4 good may of makiii? proreas with a question of this kind is t o nsk oiwselves "Hom do \\-e know tliat a collection has severi ilienibers?" because tlie aiiswer t o thi.; cjiiestion shoiild certainly briiig to liglit sometliiiig nliich collcctions of seven objects sliare iii coinmoii. An obrious iiiisxer is that \ve find out tlie number of a collection by co?t~ztin!/tlie collectioii biit tliis ansmer does iiot seeni to lielp us becaiise, \rlieii \ve courit a collection, me appear to cio no more thaii '1:tl)í-1' c:rcli ;iieriibt>r of tlie collection mitli ;L i:iini;>er. (Think of a l i ~ i l \o l Uoltiitrsriui::Scririg off.) I t Flcnrly dnes i:ot provide a definition of iiuiiiber to say tliat iiurnber is a property of a collection which is found by assigning numbers t o the members of tlie collection.
The Frege-Russell Definition To label each meniber of a collection with a number, as we seem to do in counting, is in effect to set up a correspondence between the members of two r~111ectioiis.the objects to be counted and tlie natural niimbers. Iii counting, for esample, a collection of seven objects, we set up a correspondence between the objects counted and the numbers from one to seven. Each object is assigned a unique number and eacli number (from one t o seven) is assigned t o some object of the collection. If we say that two collections are simikr when each has a unique associate in the other, then counting a collectioii may be said t o determine a collection of numbers similar t o the collection counted. Since similarity is a transitive property, that is to say, two collections are similar if each of them is similar t o a third, it follows that in similarity we may have found the property, common t o all collections of the same number, for which we have been looking, and since similarity itself is deñned without referente t o number it is certainly eligible t o serve in a definition of nirmber. To complete the definition we need only t o specify certain standard collections of numbers one, two, three, and so on; a collection is then said t o have a certain number o d y if i t is similar to the standard collection of that number. The numbers thernselves may be made to provide the requíred standards
iii the follo~~ing way. We define the property of beirig ail empty collectioii as the property of not being identical with oneself. and then the number zero is defined as the poperty of beiiig similar to the einpcu collection. S e s t me clefine the standard unit coUection as the collection mhose only member is the niiniber zero, and tlie riumber one is defiiied as the property of being similar to the unit collectioii. Then the standard pnit is talcen to be tlie collection m-hose merribers are the numbers zero nnd uriity and the number trro is defined as the property of bniri~similar to tlie standard pair, rintl so oii. This is. iii effect. tlie de!i~ii:ion of niirnber which mas discovered by Frege iii 1SS-k aiid. indepeiiciently, by Russell in 1904. It cnnnot, homever, be acccnted as a coniplete aiismer to tlie problelii of the iiatnre of number.~-4ccordiiiq to the definition, iiumber is a similarity relatioii betwecn collc.ctions in mhich each element of orie collection is made to correspont7 to a certain elernent of tlie other. and vice-versa. Tiie ~realíriessin tlie definition lies in this notioii of correspondence. Hov; do \\-e know vhen tn-o elements correspond? Tlie cups and saucers in a collectio:i of cups standing iii their saucers have an obvious correspondence, but mhat is the correspondence betmeen. say, the planets and the JIuses? It is no use saying that eTen if there is no patent correspondence bet~veenthe planets and the Iluses, \re can easily establish one, for how do we h o w this, and, ~ h a ist more important, what sort of correspondence do me allow? I n deñning number in terms of similarity me have merely replaced the elusive concept of number by the equally elusive concept of correspondence.
Xumber and numeral Some matliematicians have attempted to escape the difficulty in defining numbers, by identifying numbers n i t h numerals. The number one is identified with the numeral 1, the number two with the numeral 11, the number three with 11 1, and so on. But this attempt fails as soon as one perceives that the properties of numerals are not the properties of numbers. Xumerals may be blue or red, printed or handwritten, lost and found, but it makes no sense t o ascribe these properties t o numbers, and, conversely,
numbers may be even or odd. prime or composite but these are iiot properties of numerals. A more sophisticated version of this attempt t o d e h e numbers iii terms of numerals, makes niimbers, not the same thing as, but the nnmes of the numerals ; tliis escapes thc absurdities mhich arise in attempting to ide~ltifynumber and numeral but it leads t o the eqiially absurd conclusion that some one notation is the quintessence of number. For if numbers are tlie names of numerals then we must decide whicli tiumerals theg name; we cannot accept the number ten for iiistance as botli the name of the roman numeral and the arabic .numeral. , h d if it is said tliat tlie number ten is the name of all the iiunierala ten then we reach the absurd concliision that tlie meaning of a number word changes with each notational innovation. The antithesis of "number" and "numeral" is oiie which is bommon in language, and perhaps its most familiar iiistance is to be found in the pair of terms "proposition" and "sentence". The sentence is some pliysical representation of the proposition, but cannot be identified ~ v i t hthe proposition sirice di£ferent sentences (in different languages, for instance) may express the same proposition. If, however, \ve attempt to say just what it is that the sentences express we find that the concept of proposition is just as difficult to characterise as the concept of number. It is sometimes held that the proposition is something in our minds, by contrast with the sentence, which belongs t o the externa1 morld, but if this means that a proposition is some sort of mental image then it is just anotlier instance of the confusion of a proposition with a sentence, for whatever may be in our minds, whether it be a thought in words, or a picture, or even some more or less amorphous sensation, is a representation of the proposition, differing from the written or spoken word only because it is not a communication. I n the same way we see that the view that number is indefmable, being something which we know by our intuition, again confuses number with numeral, that is confuses number with one of its representations.
Anthmetic and the Game of Chess Tlie game of chess, as has often beeii ohserved, affords an escelleiit paralle] with niathematics (or. for ~ l i a tmatter. witli language itself). To the numerals correspond the cliess pieces, and to the operations of arithmetic, the riiox-es of the game. But tlie paralle1 is even closer thari this, for to tlie problem of defiiung number corresponds tlie proble~nof defining the entities of tlie game. If \Te ask oiirselves the questiori "TVliat is the king of cliess?" me finct precisely tlie same difficiilties nrise in trying to h d an ansxver whicli we met in oiir consideration of tlie problem of clehing the concept of iiiimber. Certainly the king of chess, mhose moves tlie rules of tlie game prescribe, is not the piece of characteristic shape n-hich we cal2 the kiiig, just as a numeral is not a number, since aiiu other object. a matchsticlr or n piece of coal. mould serve as me11 to play the king in any gane. Instead of the question "Wliat is tlie king of chess?" let us ask "TVhat makes a particular piece in the game the king piece?" Clearly it is not the shape of the piece or its size, since either of these can be changed a t will. What constitute a piece king are its mores. That piece is king which has the king's moves. And the king of chess itself? The king of chess is simply one of the parts which the pieces play in a game of chess, just as King L a r is a part in a drama of Shakespeare's; tlie actor who plays the King is King in rirtue of the part which he takes, the sentences he speaks and the actioiis he makes, (and not simply because he is dressed as b g ) and the piece on the chess board which plays the king-role in the game is the piece which makes the king's moves. Here a t last we find the answer to the problem of the nature of numbers. \Ve see, first, that for an understanding of the meaning of numbers xve inust look to the 'game' nrhich numbers play, that is to arithmetic. The numbers. one, two, tliree, and so on. are characters in tlie game of arithmetic, tlie pieces which play these characters are tlie numerals and what makes a sign the numeral of a particular number is the part which it plays, or as me niay say iii a form of ~vordsmore suitable t o the conteirt. what constitute
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a sigil t!ie sign of a particular niimber are tlie trrr?asformation mtles of tlie sisil. I t fouo~vs,therefure, that tiie OiJECT OF OUR STODY IS SOT NOSIBEP. ITSELF BCT TIIE TI¿-tSSI:OCJIATIOX RrLES
OF THE
aiid in the cliapters \1 liicli foilo\i* me sliall have iio fi,rtlicr occasion to refer to the iiiinihcr c o i ~ c ~ pBut t . jiist as the rii1c.- of clieus are cwreritly forniulated i ~ terrus i of the entities of CIICLL~S. SO tliat we say, for iristtziice, the kiiig of chess inoves only o n e scluare a t a time (escept iii castli~ig).iristr;xrl of tlie completely tqi~ivalentformulatioii '%he piece p1;iyiri~tl:c part of king (or >iliiply tlie kiiig-piece) is inoved oiily orie arii:ti.re : ~ ta tinie (exce@ iii c,is¿liiig)'' so we shall conti~iiie,iii piirely descriptil-e passages, to fjrniiilate the operations of aritlimetic in terms of aritlimetical riititivs iristead of arithmetical signs. For instance, \ve niay speak *of "ihe sii~iiof the iiumbers two and tlu-ee" ratlier than confine 'oi!rselres to tlie object formulation "2+3", mhere + is the sign ltiiose role in arithmetic is what is called addition, and ''2" and "::" are niimerals ~vhoseroles are t h o h ~of tlie numbers two and t!ree. To put it another way. in defiiiing the part played by a si-ii 1iPe g ,in aritlimetic, we shali say that what we are defining is tlic suiu function, but the dehition itself will refer only t o operations for transforming expressions which contain the sign f. r r - i \ i m n SIGXS,
1
Sumber Variables Tlie paiallel between chess and arithmetic breaks down when wc contrnst the predetermined set of pieces in the game of chess n-ith the licence granted to arithmetic t o construct numerals a t :vill. I n tliis respect arithmetic more closely resembles a l a n g u q e liich places no h i t , in principie, upon the length of its words. -1familiar notation for numerals expresses them as ~vordsspelt v.-itii the 'alphabet7 "O", "1" and "+"; each 'word' has an initial '.O" followed by a siiccession of pairs "+ 1". Thus, for instance, we forrii in tufn '-O", "0 + l", '
+ ".
" x + 1 S 1 - 1", ''O 11 7 1 1 the last of whicll is a numeral. This new sign \ve cal1 a 'numeral variable'. Tlie rules permitting the siibstitution of " x l l " or '.Oe' for "x" iii effect allow the substitution of nny nxmernl for x ; the object of tlie formidation n-e have adopted is that it scrres to define tlie concept of nny numeral and the coricept of a nzo?1erul t.nri«ble siinidtaneously. I n the sequel, not oidy the lcttci. .c. hut otlier letters, too. will be used as numeral variables. Tlie numeral formed by n~itiiigsome iiiinieral for -.z" iii ..x-L1" is called the svcces.sor of that iiuii~er;il.For iiistance, n ~ i t i n g "O+ 1 ; 1" for "x" in "x; 1" n-e obtain ..O - 1 4 1 - l " , tlie successor of "O+ l L 1". For this reason .'.zL1" is called the (sign of the) . successor function. The definite article is somewhat niisleading, however, since we maj- mite. in place of .t., a11y other letter mhich is being used as a numeral variable : in a system iii mhicli x, y and z are all numeral variables, each of "x- 1". "y- 1". .-z+ 1" is a sign of the successor fiinction. Nevertheless we shall tallí of the successor function, the uniqueness ir1 qiiestion heiiig tlie uniqueness of tlie sign which results when we m i t e some definite numeral for trhe rariable, be it denoted by z. y or :. For purposes of standardisatioii of notatioii we sliaii have occasion to introduce, insteaci of the 'alphabet' "O", "1" and " for writing numerals, the 'alphabet' "O". '.S" iii which the numerals become "O", "SO". "SSO", "SSSO" a i ~ dso 011. 111 this notation the sign of the successor function is ..SxY'and the transformation rules for a numeral variable x are (i) S r may be mitten for x , (ii) O may be written for s. Another notation in current use employs 'cx'" for the successor fiinctíon. so that the numerals are vrrítten "O", ..Of", "O"", "0'"" and so on.
"+
Definition of Counting Il'o theory of tlie natural numbers is complete mhich cloes not also take into account the part which niimbers play outside arithmetic. I t is not only a propertj- of the number nine tliat it is a square but also that it is the number of tlie planets. and this
latter property is not a consequence simply of the laws of arithmetic. According t o the Frege-Russell dehition of number, tlie number of a collection is found by testing i t for similarity with the standard unit, pair, trio, and so on, in turn, this testing being carried out by the process of counting, biit as me llave proposed a definition of number mhich does iiot rest upon tlie iindeñned concept of a similarity correspondencc we cannot nccept counting, in the Frege-Russell sense, as a nieans of fincling tlie nurnber of a class, without readmitting this undefined concept. Tliere is, however, an entirely different interpretatioii of the pi-ocess of cóunting, which makes counting arailable to us as a means of recoraing the.. number of a collection. without transcending tlie limitatioii we imposed iipon ourselves of expressing tlie properties of numbers in ternis of tlie transformation riiles of the iiiimber signs. We Jtart by separating tivo distiiict stages in the process of eo~inting. The k s t of tliese is m-liat we shall cal1 "using a collection as a numeral" mhich consists in overlookiilg the individual 'idiosyncrasies' of the elements of the collection and regarding them as being all alike (biit not identical) for the purpose in liand. This is simply a (perhaps rather extreme) form of a treatment of signs familiar in all acts of reading, writing or speaking; the letters "a" on a printed page, for instante, have their several differences and, subject t o sufficiently close scrutiny, are as different as say the soldiers in a platoon, but for the purposes of reading me ignore these differences aild treat the vnrious a's as being the same sign. And SO too, in speaking, we treat as the same a variety of slightly differeiit sounds. I n a differerit contest, signs which we would accept as the same for reading purposes, are carefully distingiiished, as, for esample, when we test the quality of printing. The process of overlooking some differences, but not others, is fundamental in language; it is tlie process by mhich ive subsume objects with a 'family likeness' under a generic name and tlie process ~ ~ 1 i i c h makes possible the use in language of universal \trords. Without it, the concept of tlie number of a class could never have arisen. The second stage ili the process of counting consists in a transformation from one number notation to another by means of the rules "one
and one is two". "two and one is three", "tliree and one is fourW and so on. It is the recitation of these rules (in an abbreviated form in which each 'and one' is omitted, or replaced by pointing to, or touching, the object counted) mhich gives rise t o the illnsion that in counting we are associating a number witli each of the elements counted, mhereas me are in fact making a trarlslation from the notation in which tlie number signs are "one", "one and one", "one and one aiid one", and so on, to the notation in mhich the signs are "one". "two". "three", and so on. The true nature of counting is perhaps most clearly brought out if me re-introduce the older process of making a tally. 3Iaking a tally of a collection consists in some formalised representation of the elements of the collection, say by means of dashes on a sheet of paper, so that in making a tally we are copging a number sign in some standard riotatiori - finding the number of the collection, by treating it as a number sigii and copying this sign. Thus a tally of the planets consists in the row of dashes
If we now proceed to traiisform this sign by means of the transformation r d e s 11=2, 11=3, 31=4, 41=5, 51=6, 6 1 = 7 , 7 1 ~ 8 , 81=9weobtaininturn111111111=21111111=3111111=411111= = 51111 = 6111= 71 1= S1 = 9, which completes the transformation.
I n counting as we teach it today, the processes of tally making and sign tzansformation are carried out simdtaneously, thus avoiding the repeated copying of the 'tail' of the number sign in transforming to an arabic numeral. It is important t o realise that counting does not discover the number of a collection but transfo~ms the numeral which the collecticun itself instances from one notation to another. To say that any collection has a number is just to say that any collection may be used as a number sign. Formalisation of Counting Counting may be formalised in a system of signs by formulating the transformation rules of a counting operator "X". JVe represent t,he objects in the collections t o be counted by letters a: b: c, ....
10
ISTRODvCTION
INTRODUCTIOS
and collections by conju~lctionslike a S: b, n S: ZJ LP: C ; a single object being regarded also a collection. The letter 1 we use as a variable for an object. that is. ii letter for which any object may be n ~ i t t e nthe ; capital letter L berves as n variable for a collection 2nd maj-. in any contest. be rcl~lncc(1by a alletiiiite coilection or by "1; S: 1". The nunier~lsof tlie systcin are tlic signs (witliout X) obtairied from l. x ariti the succrssor furiction n:-- 1 hy sribstitution. Thcn Tre define Y(l)=l, S(L&l)=S(L)--l. 'Lliebe equations sufice to determiiic tlie iiiiiiiber of any c~llectionl For instance, substituting "a" for tlic variable-sign "l", in the k s t . \ve obtain N ( a ) = 1, and tlieii, substitiiting "o" for "L" and "h" for "l" in tlie secoiid. \ve obtaiii
N ( a S: b) = N ( n )+ 1 ! and so. X ( n & b) = 1 -L 1. Y P S ~ siibstituting , "a & h" for "L" and "c" for "1" me find N(a & O &c)=X(a & b)+1=1-~-1+1, and so on. ' I T c observe that the definition of X(L) is by recursion, that is to say. X ( L ) is not simply an abbreviation for some other expression, as, fbr instante, when we define 2 = 1+ 1, the sign "2" may be replaced by "1 + 1" for which it is merely an abbreviation, but X ( L ) is determined only step by step, by introducing the members of tlie class t o be counted one a t a time (or shedding them one a t a time). R'e may express this by saying t l ~ a tfor the variable L, S ( L ) itself is undefind, only the result of substituting a definite class (lilíe a & b & c) for L being defined by the reciirsive definition. Tfie recursive definition is, so to speak, a schema or mould from v.-hicii tlie definition (value) of X(a & b & ... & k) may be found by siibstitution for any particular class a & b & ... & k.
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Evolution of the Concept of a Formal System I n the following chapters we shall set up arithnietic as a formal S Y . S ~ P - B I .Tlie idea of a formal system is one which derives from Euclid's presentation of geometry, but the notion has ~indergone
1
i
considera1)le developmerit during tlie past centiiry. 3uclid7s interition iri the "Ele~nents" was to decliice the 1vliole bod- of geometrical lmowledge of his time from tt few self crideiit trutlls (calied asioms) by purely logical reasoning. Euclid did not. Iiowerer. specify the naturc of 'logical rea?oiiiriq' nrid tlie tirst attenipt to do so was made by Ckorge BooIe. iii Is17. iii Iiis JInfho)inficnl dnnlysis o/ Lqic. Boole coristriicted a ;:.-rnbolíc lurigiinge. in ~vliicli the 'laws of tliought'. formiilater! iis axioriis, niny be sti~cliedbu mathematical tecliniqiies. Iii thc cnri~;lletctlcx-clo1)nicnt of the notion a forinal systcni is ari a-icrilhl:icr. of slgnc qcbpa:.nted into various categories. tlieir usage it~nilr! 1 ; ~\:irious con\-eiitioiis (tlie asionis and transformatioii rules) tiie objvct of the systeni being to arraiige sequences of formiiIne (rhicii are tliemselres svqiiences of signs mith certuin >pecified forniation r.iilcs) iii certniri relatioriships to one anothcr to forin a particular pattern callec1 proof. A formal systeni ninu contaiii l,otli !ii,tthc.rnatical nrici logical s i p s (the distinctiori is an arbitrar'- ori:?). aiitl nintheiiintical and logical asionis; its essential featurc, q ~ 1 foirnal , systeni. is that its operation does not presuppose any !íiiu:r letige of the significance of the signs of the system thali is givcii hy the nsiomq aiicl transformation rules. Tiie mathematical a~ioilisare no loiiger "self evident truths" biit arbitrary initinl ;io.;itions in a ,rran;c. and tlie logical aasioms espress. not the "la-\\-=of tlioiig!it" but arbitrary conrentions for the use of the logical .igns. I n the formal s ~ s t e n mitii i ~rhichn-e :iiall first be coiicerned in this book, the equation calculus. the or:!y :igils are sigris for functions aiid numeral variables. and tlie equality sigii. Tiirie are iio asioms escept tlie introductory equations for function si;ns. and there is no appeal t o 'logic7. the operation of tlie sj-steni being specified siniply by the transformation rules for the mathec?,ztical signs. It is sho~r-nthat a certain branch of losic is defi?iable in !he equation calcl~lusand logical signs. and theor?:i?s, are introduced as conrenient nbbr~z.icttionsfor certain f1inctio:is ancl formulae. This brancli of logic is characterised by the fact that it caii assert the ~xistence of a number mith a giren propcrty uiily miien tlie niimber iir question can be found by a specifiahle riomber of trials.
tri
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CHAPTER 1 DEFINITIOX BY RECURSIOK 1. Variables
We have already had occasion, in the definition of a numeral. t o refer to the use of a letter as a sign for a variable. In terms of two operations (1) replacing x by x i 1, (2) replacing x by O, \ve defined numerals to be tlie signs constructed from x by the repeated application of tlie operation (1) folioíved by an applicatio~i of the operation (2). Starting mith the sign x the process of constructing numerals may be regarded as a process of eliminating x using only tlie operations (1) and (2). For instance the numeral O + 1 + 1 + 1 is constructed from x by three applications of operation (1) follomed by an application of operation (2). The defining property of a numeral variable x is that it may be replaced by zero or x + l. Any letter may of course serve as a numeral variable, but in this chapter only the lett-ers x, y, z and w will be used. By means of variables we can make general statements about numbers, statements which hold true when any particular numeral is substituted for the rariable. -
1.1 ADDITION The fundamental operation of arithmetic is ctdclition. Addition is the operation of joining together two numerals by tlie addition sign ' + '. For instance, joining the two niimerals O + 1 + 1 + 1 + 1 and O + 1 + 1 we obtain (omitting the introductory part O + in the second numeral) the numeral O 1+ 1 + 1 + 1 + 1 1 which is calied the sum of 0 + 1 + 1 + 1 + 1 and O + l + l . To say tliat addition is the operation of joining together two numerals is not, homever, a mathematical definition of the operation
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DEFISTTIOX BIT RECI-RSION
i.,,,y,xIT[!>>-r%y~ : : < ' : - x s I o N
14
previous result ; for instarice we pass from ( S + 2 ) + 1 to 7 + means of the earlier equatioii 5 - 2 = 7.
for \ye llar-e simpli. r<.!;l;l::ctl tlie \\-oi.tlrrtldifion by t'li(' iilidefiiled tcrm joining. There ar:. ohrio~islyi:ia:iy "aya ill i\-liicll we i~ligllt joili togetller t r o nul::~:i.;;]s. !,lii oril? <,11eof t l i e ~ eazy5 gires x h a t wc. iiieaii b ~additioIl. -$ clnlIie~iiatic~il tlefinitioii ~f nddition nlust bt. foriiiiiltitc~Jin ter?ns of :iiimernI variables. niid tlic acldiiion si(rll 2.. ;ilorle: 110 estr:iiicoiic: elemeiits niay b t introcliiced. A 4 cllild hrrt !cai.:;-: :O add by ti:? siriiple coiijiiiictiorl 0f ~iilllierals, for instnrlce t,o fincl the ~u1l-iof 3 nnd -t lir combines . . . 3 r d . . . . to fOrm . . . . . . 3lld 'col!:!ts th(x (l(:is'. i.c. tra~iforiiis. . . . . .: into 7 . Lnt,er11. l:J:tTlis to procFi.tl fToriiC, to 7 b;; t,he rcpentecl nddition of 1. *' ;ind this fiir :nore eco!iomic:il procpcliirc is the basi:: of the followirig fornial defiriition :
1 . 1 1 Addition of tliree or niore nurnerlils is defined iil terms of tlie repeated addition of pairs. Tlius f1)r instance n7e define
x+ytz=(x-y)-z w=((x-y)--z)-w
X+Y+Z+
,
I
A
!
xiO=x
,
x - ; ( y + - I ) = ( x + y ) t l.
4 A,
The sign ' = ' here rignifies thnt tlie expressions mliich stand oii eitlier side of it are cjuirnlent $0 thnt either niay replace, or be replaced by, tIie othei-; that is t o say, A, and A, espress t r a w forntation rules by x-?iich one sign pattern may be transformed into another. (Tliere is of course another entirely different use of the equality sign ' = ' in mathematics t o which we shall have occasiori t o refer later). The ~ a r i a b l ex in equation A, may, by definition, be replaced by O or by x i 1 ; it follows that x may be replaced by a n y numeral for numerals are constructed from x by these same substitutions. And s i d a r l y both x and y in A, may be replaced by any numeral. SOillustrate t'he application of the transformations A me evaluate the sum of 5 and 4. Substituting 5 for x in A, and A,, and 0 , 1, 2, 3, in turn for y in A, we obtain
i
i
and so ori. Tliese clefinitioiis. lilíe d~fi:iitioii9.treat the numerals added unsymmetricnlly. Iii dcfiiiitiori -4. x and y are not on the s a n e footing arid i t is by no menii.; obvious that x+y=y+x. I n A' the three terms x, y, c are addc(1 in that order, and so too in 9". We shall prove in the iiest chapter that the sum of two numerds is iridependent of tlie order ir1 wliicli they are added; from this it follows tliat, for instaiice. y+x+z=(y+x)+z=(x--9)-r=x'y-z.
The equality of z + y + z and y , z x ho~vevwrequires iiidependent proof and the general proof that adclitioli of any number of terms is completely symmetrical is much more difficult. The definition of the sum of a variable number of ternis is a probIem to which me shall return later; there is however a ~pecialcase which we shall consider here, and that is the repeated uddition of a single number. 1.3
i I
~~ULTIPLICATION
Repeated addition of the same n u m b ~ ris called multiplication; the sums x + x , x + x + x , x + x + x + x , and so on, are denoted respectively by 2 - x , 3 . x , 4 . 2 , and so on. the fbst term of the pair denoting the number of repetitions of the second term. The formal definition of multiplication is:
3%
O.x=O
(y+ I).x=(y.x)+x 5+2=(5+1)+1=6+1=7 5+3=(5+3)+1=7+1=S 5+4=(5+3)+1=8+1=9.
Each partial result after the h t is obtained by means of the
by
Substituting O, 1, 2 and so on in turn for y in we verify in turn l.x=O+x=x,
1
and so on.
2.x=l.x+z=x+-x,
M,
& (and using MI)
3.z=2-z+x=x+x+x
w w w iCí
16
DEFMITION
!
BY RECCRSION
17
DEFINITION BY RECTRSION
so that. taking y = O , 1 , 2 in turn \$-e rerify tliat lx=z, -x = 3 x = x ( 2 2 ) = 8and so on. Similarly if x, 5 , %x, ..., are denoted by ,z, ,x, ,x, ... respectively, we may define
x.y is called the product of x and y ; wheri the cont,est renders the sign free from ambiguity we shall denote the prodiict of x and y by xy, omitting the dot betmeeii tlie terms. Products of three or more numbers are defiiied in ternis of prodiicts of pairs. Thus 1.21
"X =
P
3Iultiplication like additioil is a syinmetricsl operation, but the proof of this must also be postponed iintil %e have set up tlle necessary proof processes. . .
1
PI p,
cv+i)~=(fl)X
There remains to be defined one more fundamental operatioil of the arithmetic of natural numbers, namely subtraction, the converse of addition. \Ve introduce first t%e converse of the operation + 1 , which we denote by - 1 , m-ith the formal definition
-
EXPONE'TTIATION Repeated multiplication of tlie same number is cailed exponentiation ; the products x .x, x . x. x; x.s x. x and so on, are denoted respectively by xl, 23, x3, and so on. the 'indes' denoting the number of repetitions of the base. The formal definition of xv is 1.3
,
and theil the difference x L y is defined by
-
Substituting O, 1 , 2 and so 011, in turn for y in E, (and using E,) we find in turn j
and so on. Exponentiation is not a symmetrical operation, that is to say, xv and y" are, in general, unequal. Just as me passed from addition t o multiplication and from multiplication to esponentiation by the iteration of tlie same number, so we can define further operations by iteration. These operations have no established names so that we shall call them simply tetration, peiitation, hexation, heptation and so on. Denoting the esponentiations a?, 8 , 8 and so on, by %, 3x, and so on, respectively me define 'x by
instead of the familiar The use of the modified subtraction sign - marks an important difference between the foregoing definition of subtraction and that in elementary arithmetic. If x is not less than y then x-y has its familar meaning, and, as we shall subsequently show, y + (x y) =x ; but if x is less than y. definition S gives x-y the value zero, so that y + ( x ~ y=)y. Finally we define lx, y], the positive digerence between x and y by the equation
1.o
Fm-CTIONS
Expressions constructed from variables, like x3, x + y or x -y .z are called functiom; more exactly a function is an operation on numbers d e h e d by means of variables. The severa1 notations in which we have expressed functions so far are of considerable hktorical and technical importance, but their diversity tends to
DEFIXTIOS B Y ILECURSION
conceal their fundamental structure, and 1-i-e shall introduce a staritfardised notatioii. \Ve shnll denote a function by prefising to the variables concerned the nnme of the operation (abbreviated wherc necessary). Thus the successor functioii x + 1 will be denoted t q - S(x) (and so tlie iiiiniernls by O. S(()), S(S(0)).S(S(S(0)))and $0 oii). the sum functioii s - ! I b ~ Snin (x, y), the product x. y by PLaíl (x. y) and tlie cspuiiciitinl by E s p (x. 9); the brackets iii r::c..-e signs, like tlie coniriia bctxvr>entlie variables, are not esseiitial i)arts of the sign, biit are iiicliidc.rl only for typopaphical conve;:icnce and will bp ulliitt~cl7~11e11this tines not lead to nnibiguiti-. TIL standardised iiotatioii the definitions A, 31, E, T a ~ i dS talde tlie forms
is a particular case of definition by scheme is
RECURSIOS
19
of which the
(The difference between iteration and recursion is that iil the lntter, function b depends upon t~ as well as upon x and F(x, y).) ,hesample of a function m-hicii may be defined iinder the scheme R is the sum of the geometric series
-
1 -. 1
11
Sum(x. O) = x sum(x, Sy) = S(Sum(x, y)) . Prod(x: 0) = 0 Prod(x, Sy) = Surn(x, Prod(x, y))
!1
Exp(x. O) = 1 Es
T
Tet(x, 0) = 1 Tet(x, Sy) = Exp(x, Tet(x, y))
6
Dif(x, O) =x Dif(x, Sy) = P(Dif(x, y))
¡vilere P(x) is x- 1 defined by
Tiie common feature of all the definitions 9 t o S is apparent; e:ic!~definition takes the form F(x, O) = a(x) F(x, Sy)= b(x, F(x, y)) ~vhereF(x, y) is the functioii defined and a(x), b(x, y) are functions previously defined or are variables or definite numerals. Definition scheme 1 is ca,lled iteration; deñnition by iteratioil
1
~vliichsatisfies t,he equat,ioiis
where B(x, y, z)=Sum(Esp(x. y- 1). z). -\lore specifically thc scheme R is called recursion in y ; the secoiid variable x ici called a parameter of the scheme. Xo limit is placed on the nunibvr of parameters, so t!lat for instance the sclienie
is also called recursion (in x). The variables in a function sign. like x, y, z in F(x, y, z ) are sometimes called the arguments of the function. If a (unique) numeral f can be fou:id such that F(a, b, ...) = f where a, b, ... denote certain definite numerals then f is called the calue of the function F(x, y, ...) for the values a, b, ... of the arguments x, y, ... of the function P(x, y, ...). The characteristic property of a function F(x, y, ...) defined by recursion is that the value of F(x, y, ...) is determinate for any set of values of the arguments provided that the auxiliary fuiictions in the definition scheme have this property, for P(0, y, ...) is a known function aild the ralue of P(Sx, y, ...) is determined when the value of F(x, y, ...) is k n o m , since F(Sx, y, ...) is espressed in terms of F(x. y? ...) by means of a function with knon-n values.
20
DEFINITIOS B Y RECURSION
1.6 It \vil1 be seen: in a later cliapter that the introductioii of additional parameters, as in the scheme R*, is in fact superfluous and that it suffices to consider only tlie simplest recursive scheme
SUBSTIT~~TIOS The simplest operation by which a new function may be constr~ictedfrom giren functions is substitittion. Giren a function of any number of variables, say a functioii of four variables P(a, b, c, d ) , and any foilr functions, say A(x. y, i ) , B(x, y. z ) . C(x. y, z), D(x, 9. z) then a new function F(s.y, s) niay be defined by the equation 1.7
in whicli there are no parameters a t all. We collect here for futiire reference three functions mhich hare an ituportant part to play in this reduction. These are illt x, Hf x = [x/2] arid R t x='[xlly
-
defined by the recursions
.
S i ~ b s t F ( x , y, z ) = P(A(x, y, 2). B(x, y, z ) , C(x, y, z), D(x. y, z)). The function F so clefined is said to be defined from P, A, B, C. and D by siibstitutioil in P. For instance in the definition of addition, x+Sy is defined from x l y and S by substitution in Sx. A definition of the kind Subst is also said to be an explicit definition of the function F, by contrast with a recursive deñnition. I n an esplicit definition e v e y variable which occurs on the right hand side of the equation must occur also in the nem function oii the left, but there may be additional variables oii tlie left-hand side.
(-
(i) A l l ; O = O , A l t S ~ = 1 ~ ~ 4 l t x po that A l t l = l , A l t 2 = 0 , A l t 3 = 1 and so on; (ii) H f 0 = 0 , H f S x = H f x + A l t ~ so that H f l = O , H f 2 = 1 , H f 3 = 1 , a n d so on, that is H f 2 x = H f ( 2 x + l ) = x ; , t x)) (iii) R t O=O, R t S x = R t x + { l ~ p ( x R where p(x, y)= ( S y ) 2 ~ S x;
1 .S
It is often convenient t o be able to express any variable, or definite numeral, in the form of a function. For this purpose we introduce a number of special functions. The first of these is the zero function Z(x) defined by the equation
thus R t Sx is obtained from R t 1: by adding 1 if Sx is the next square after (Rt x)2, and adding O otherwise, i.e. R t x is the greatest number whose square does not exceed X. 1.61 Contracted Notation When a function is d e h e d by the simplest recursion .
This definition could be given recursively in the form
.
but the explicit definition is indispensible in the sequel. I n addition to the zero f~inctionwe define also the identity function 9 ( x ) by the equation 9 ( x )= 5.
its successive values F(1), F(2), F(3), ... , are given by b(O), bb(O), bbb(O), ... , and it is natural t o denote these by bl(0), b2(0), b3(0), ... , so that F(x) may be expressed in the form bz(0), where bO(0)stands for 0. Thus, since x + y is defined by
From the zero function we can obtain any constant function by substitution in the successor function Sx. For esalnple the fiinctioii C(x). wliose value is 2 for any values of x, is d e h e d by
x+O=O , x+Sy=S(x+y) i t follows that z+ 1 =Sx , x + 2 =Sz(x) ,x + 3 =S3x , ... , SO that z + y may be expressed in the form SY(x)where SO(x)stands for x.
THE INITIALFUNCTIONS
I
l
1.9
SJXGLYRECL~RSXVE FCXCTIONS
A fuiictioii is said to be si7lqly rccztrsi?~z(or primiti~ereczcrsire) if it niay be defined from the zero fiinctioii, the siiccessor function and tlie identity function by substitutioiis and recursions. We 1iia.y also cspress this definition by aiiying th::..t tlie zero fiinction, tlie siiccessor fiinction ar,d the icleiitity fii1;ctiori arc siiigly recursive, ~ i i dfiirtlier that a furictiori is singly recursi\-c if it is defined by substitutiiig siiigl-j recursive fiiiictioria ir1 n. sin& rrciirsire firrictioii or is defiiied by a singlj- recursive sc!ienir cmploying orily siiigly recursive fitiictioiis. Thiis al1 the fiiiictioris cicfincd iip to t%i? pointd are siiig!y recursive functions and any fiirictioii fvrriied from thern by siibztitution mil1 be singly recursire. 111 ii sciise which mil1 be made sliarper iii tlie nest chapter a feeiinire fiiiictioii is totally eliminable, that is to say, when al1 its arpu:iiarit places are filled by definite niimernls the function rnay be tt~nrisformedinto a single iiiinieral. For the initial functions are certcinly cliniuiable, and substitution of eliininable functions in an eliminable function residts in aii eliminable f~iriction;finally a function F ( x ) . wliich may contain paranieters, defined in terms of eliniiiiable functions by a singly recursive scheme, is eliminable aincc F(O)is eliminable and F ( S x ) is eliminable if F ( z ) is eliminable.
I
H ( 4 . n. a). - - - define in turn addition. multipIicatioii, esponentiation, tetration, ... ; for H ( 1 , O .
l
so that H ( 1 . n. a ) = a t n . and tliereforp
H ( 2 , O.
tlieii t,hc valiie of H ( p , n , (1) is determined for any values of p. ' 1 . ancl ( 7 : for H(O, n , a ) is given, and if H ( p , n, a ) is given for sci::ie 11 aiicl any n and a then H ( p + 1, n , a ) is definecl by single recursion iii 71. The functions H ( 1 , n , a ) , H(2?n, a ) , H ( 3 , n , a ) ,
r l ) - rr
so that H ( 3 , n , a ) = a"
so that H(4, n. a ) = "a arid so on. The ecjuations H do not fa11 iinder tlie siiigly ~ ~ c i i r a i vscheme e R since the successors of both n and p are iiivolved. H is an example of definition by double r~cur,:ion. The general forin of a definition by tlou ble reci~rsionis aa follows : A function G(p. n ) . TI-hich may de;)end also ii~torione or more parameters, is said to be clefined by doiiblc recur.~ionfroin certain given functions if
Uoltl)ly Recursiee Punctions Tlie totslity of arithmetical operations, additioii. multiplication, esl~oiiriitiation,tetration, pentation and so on, may be expressed by inenris of a single function. If we define a function H ( p , n, a ) by the eqii at'lons
H
. H ( 2 , n - l . a )= f i ( 2 . n ,
so that H ( 2 , 71. a)=na. aiid helice n-e filid ir] tiirn
l .1
E ( ( )n, . n ) = S n , H ( 1 , O, a ) = a , H ( 2 , O , a)=O , H ( p + 3 . 0, a ) = 1 (so that H ( p + l , O, a ) = a ( l - p ) + { l ' - ( - - p ) ) ) aricl H ( p t 1, n t l . a ) = H ( p , H ( p f 1, n, a ) , a )
u )= O
i2
I
i
(?(O, n) is one of the giveii fiiiictions G(P+-1. O ) = ~ ( P , Q(p. Q))) G(p+ 1 ' 7 1~)=c(p, ~ n, G ( p ,d[p. n. G ( p - 1. n)]). G ( p T 1, n)),
mhere a(x. y&) . L(x) . c(x, y, z. zc) and cl(x. y. z ) are given functions. \Te observe that O(p+ 1, O ) is obtained by siihstitiitiiig a given function for n in G ( p ,n) aiid substitutiiig the result in a gicen fiinction: G ( p- 1. n 1 ) is also obtauled b-j- substituting G ( p 1. n) in given functions and substituting for n in (;(p. n ) . V e shnll show i n Exanzple 1.9 that the first trro equatio,is tnny be replctcecl by G(O,n ) = G ( p - 1. O ) = 1. If the given functions are siiigly reciirsive tlieii tlie equations R2 s h o ~thnt for any given valfce of p. the function G(p, n ) is singly
reciirsive in n, for G(0, n ) is given to be singly recursive, aiid if for some p, G(p, n ) is singly resursive in n, then G(p+ 1, n+ 1 ) is expressed by singly recursive functions in terms of G(p+ 1, n). It does not necessarily follow, hon-ever, that G ( p , n ) is a singly recursive function of n nritli parameter p and we shall subsequently show that there are doubly recursive functions which are demonstrably different from any singly recursive function; in particular this is true of the doiibly recursive function H ( p , n, a ) defined above (see Péter [ l ] )* By analogy witli the equations R2 one may readillv define recursions in three or more variables, but me shall have no occasion8 to employ more than a double recursion.
Examples 1 1. Formulate the reciirsive dehitions of t,he arithmetical processes of hesation and heptation. 1.1 Evaluate Tet ( 2 . 3 ) aiid Pent ( 2 , 3 ) and express Hex (-7. 3) as a repeated esponential. 1.2
If f ( O , n ) = i , f ( p , O ) = p + l a n d , f ( p + 1: n+ l ) = f ( p , p n ) . f ( p t 1, n),
find the valnes of f(2, n),f(3, n) and f(4, 4).
1.92 From single and multiple recursions we derive the general aotion of a recursive fiinction. A function sajd to be recursive if it is singly recursive, or d e h e d by a recursion in any number of variables with recursive given functioils, or defiiled by substitution from recursive functions. We shall show that al1 the processes of clmsical arithmetic m u y be described by means of recursive functions.
prove that f(x) is recursive.
*
prove that f(x) is recursive.
The number in square brackets refers to the bibliograplphy.
1.3
1.31
1.4
If y(%) is recursive and f ( O ) = O , f ( x + l ) = ~ ( ~ i l ) .
If g ( x ) and h ( x ) are both recursive and if f(x)=g(x) for x g 2 . = h ( x ) for x > 2 ,
Prove that the following functions are recursive
1.41 The remainder when x is divided by 2, and the quotient; the highest common factor of x and 2, and their least common multiple.
1.42 1.5
As in 1.41 with '3' in place of '2'.
If g ( x ) , h ( x ) are recursive and f ( x )=g ( x ) if x is even, = h ( x ) if x is odd,
show that f(x) is recursive. 1.G Show that the coefficient of xr in the expansion of ( 1l z ) " is a doubly recursive function.
1
1.7 Assuming that a+b=b-a aiid ( a + b ) + - c - a - ( D t c ) prove that tlie sum a+ b ~ c - d is iridepeiideiit of tlie ortler { ~its f factors. 1.8 Define primitive reciirsive fiinct;oiis i , x ) . j( r. y). ?(n.y ) riich that i ( O ) = l , i ( x + I ) = O : j(O.y)=O, j ( ~ -1. 0 ) - i . j(x-1. !/-- ] ) = O ; k(0,y)=O. k ( x + l , 0 ) = 0 , k ( r - l , y - p l ) ; i. 1.81
CHAPTER 11 THE ECJUATIOS CALCGLUS
If n ( x ) . b ( x ) and c(x. y) are priiiiitit-e rcciirsive and
2. $Ve linve described the two proccsscs. siibktitiitioii aiid rrcursion, by which iicw functions m a l br definc(i from fiinctions previously introcliiced. aiid have show~ilion. al1 rt~c~i:~ive fuiictions are built up from tlie zero and siiccessor fiincrIulis. \:e' tiirii nom from the cqiiations. Tiie eqiiations construction of functions to the proof.5 we consider express the fact tliat ccrtni:i tiinction signs may be substituted for others ; for inst~iicetiir eciir~tioiix -- y = y x says that x + g may be snbstitute(i for ?J - J . aiiti z.ic~,-rersa.Tlie general forni of siicli equations is F-!7 niiprt. I; ;iiici (: are recursive
prove that f(x, y) is primitive recrirsive. 1.9
3 ) f
If A(p. n , PL, u )= i ( p ) .~ ( nA j(p, ) n ).O(?)- 1. u ) + k ( p , n ) . c ( p - 1 , n'1.
ZL, u ) t anti p ( p , n. w ) = i ( p ) - j(p, n ) . S d ( p - l ) ' k ( p . n ).Se(p- 1, n- 1, w )
antl if y ( p . n) is defined by the double recursioii
1
pro\-e that, if G ( p , n )=y ( p + 1, n + l ) , tlieii
l 1.91 Formulate the introductory equat,ions of a function f(x, y, z ) defined by triple recursion. I
I
I
lI ¡ l
functions. 2.1 DEFT'U-ITION OF -1 Pnoov -4 proo/ is a table of equatioris enrh of n-hich is citlier (part of) the definition of a functiori, or a11 rquatioii of the form F = F, or is a proced equation. If F =G is or:e lif tlio cqtintioiis of a proof, then a proi-ed equation is obtained hy rcplacinq the function F by the fnnction G a t one or more of tlie places nt which F occtu-s in some equation of a proof. Furtliermore, the eqiiation forrueri 1))- ;*eplacin~ n 1-ariable a t all the points a t which it occiu-s in some equntioii of a proof, by another \-nriable, or by a definite niiineral or furiction. is a proved eqiiation. Finally. F = G is a proved ecluatioii if equations of a proof are obtahed by siibstituting the fiiriction F for a fiinction H , and by substituting the function G for H. in the eqiiations which define the fiinctiori H. If the results of substitutirig F and C2 ui tlie tlefining equations of sonie fuiiction are equations of a proof. we shall say that F and C= satisfy ~JZC sctme introductory eq~intion.9. The riile that F=G
THE EQUATIOX CALCCLUS
if F alid G satisfy the same introductory equations is just another may of saying that the processes of recursion and substitution define unique functions. For instance, the equations x + O =x , x + S y = S ( x + y ) , which introduce the sum function, define this fuiiction completely, so that any f (z, y ) mhich satisfies the same equatioiis, namely f ( x , 0 ) = x , f ( x , S!/)=Sf ( x , y ) , iu jiist another notation for the same function. 2.11 To illustrate the proof techniqiie, and ir1 preparatiori for siibsequent applications, me collect together here proofs of the principal properties of tlie functions we introdiiced iii the previous . ' chapter.
1
We observe that ( 2 . 2 1 ) , ( 2 . 2 5 ) are defining equations of the sum function; ( 2 . 2 2 ) is derived from (2.21) by substituting S x for x ; ( 2 . 2 4 ) derives from ( 2 . 2 3 ) by substituting x + 0 for x (on the left hand side) as is permitted by ( 2 . 2 1 ) : ( 2 . 2 6 ) derires from (2.25) by substituting S x for x ; ( 2 . 2 8 ) is obtained from (-7.20)and ( 2 . 2 7 ) . Tlieequations ( 2 . 2 2 ) ,( 2 . 2 4 ) , ( 2 . 2 6 ) and ( 2 . 2 8 ) shom that both S x l y and S ( x + y) satisfy the introductory equatioiis
so that (2.29) is a proved equation. Substituting x for y and y for x i ~ i(2.29) we obtain
I
Properties of the S u m Fz~nction 1 As the first esample of a proof we consider the commiitative property of addition which is espressed by -the equat,ion 9 3 -.-
39
The nest step is the proof of the equ at'ion 1
x+y=y+x
the pair of equations
To make it easier to follow we divide the proof into two part,s, starting witli the proof of the equation
O+Sx=S(O+x)
O+O=O
which follow by substitution in (2.21) and ( 2 . 2 5 ) , show tliat the function O + x satisfies the introductory equations
Sx+ y=S(x+y)
The proof consists of the nine equations
which are also satisfied by the identity function 9 ( s ) ,proving 2.3. The four equations 2.21, 2.3, 2.25 and 2.291 show that sf y and y + x both satisfy the introductory equations
which completes the proof that el
(2.")
S(x+Sy)=S(x+Sy)
(2.25)
S ( x + # y ) = S ~ ( +Xy )
(2.29)
Sx+y=S(x+ y).
*
l
x+y=y+x.
We take as the next example the important equation
which espresses the associative property of addition.
TRE EQI--4TIOS CiLCTLUS
Sirice ( x - y) -i- S .
and
+- y) z] aiid
=& ((z'i'-
(where \ve are using aii abbreviation of the forrn a= b = c to denote the pair of equations a = b, b = c ) it follo\v-; tiiat both the functioiis (r- y) -'- z and x k (y+ z) satisfy thc c~i1u:it;l)iiy(%. y. S:) =Srp(x. y, 2 ) : frirtheriilorc (.r + y) -L bo
-x
y = r (y - O) J
t i i ~ tthese fuiictioris also satisfy the ecl~itition
-
and so 2.4 is proved. 1
2.5
TRODUCTS
The commutative propertj- of miiltiplication, which is expressed by tiic equatioii
Ss.yiSx=(x.y+y)-Sx
=(z.y-rSx)-y by2.4 =S(.t-.y+x)+ y -Sf(x.yTxi- y ) by 2.29 =(~.ytx)-A
g>(0,2/) =0 (Asd.. :/) = Y(X,y) 89. proving 2.59. The equatioiis 2.51. 2.53 show that the functions y - x and x - y satisfj- the sanie iiitrotiuctory equations, which completes the proof. 2.53 If for some f (..L.). f (O)= O a:id f (S;:)= O are provable then f (x)= O is provable. For froiii f : . i r )= o and O.f (x)= O we derive f (Sx)= O .f (x) aiid since ZSx = i ) . Z x t iierefore f (x) and Z x botli satisfy tlie introductory eqiiation.;
y(0)= o . y ( S r )= o.'~(x)
xy ='yx is readily prored ~ v i t hthe help of equations 2.2 and 2.4. The defining eqiiations of the product yx are
as in tlie proof of 2.2 me start b y establishing the compaiiioii equations
31
so that f (x)= Z x aiid f (s) = O i+ provable. 2.6 PROPERTIES O F THE DIFFERENCE FUNCTION We commence by pro"tig tlie important equation
x-y=sx-fy.
5.61 From the equations
2-o=x, S x - S 0 = ( . ? 7 ~ ~ 0 ) ~ 1 = s x ~ 1 = x , The first of t,hese is proved b y the two equations O -O = 0,SZ. O =X. O (iristances of the introductory equations of the product), which in conjunction mith Z(O)=O,ZSx=Zx (mhere Z x is the zero function), show that
x.O=Zx=O.
and
x=Sy=(x'y)-
1 . Sx~SSg=(Sx-ay)- 1
(mhich are instances of tlie introtiuctory equations of the difference function) i t follows that .c2y and Sx=Sy both satisfy the introd uctory equations
-
rp(x.O)= x , ~ ( x.Sr/) , = p(x, y) 1
Foi t.he secoiid we observe that so that
x-.-y=,Sx~Sy.
It follows that for (x+Sy)-Sy=S(y+x)-Sy=(xf2/)'?/ and therefore if Il(x,y) iS defined explicitly' by the equation I,(x, y) = x both (x+y) - y and I,(x, y) satisfy
y ( x , O ) = x , cp(x. Sy)=cp(x, y ) . From the equations 0 1- O= 0 , 0:Sx = (0:s) 1 and ZO = 0, ZSx=Zx 2 1 folloms O 2 x= O and similarly using Z ( x , y) defined explicitly by the equation Z ( x , y) = O me may prove 9 (x+ y) = O, ' and thence y- y = 0. We define the positive difference of x ancl ?J (2, by the equation
-
I
~ h i c hcompletes tlie proof. The common value of x + ( y - ~ - zand ) y + ( x- y ) is the greater of x and y, denoted by max (x,y). Iri preparation for a theorem connecting equality with zero positive difference we prove nest that for any recursive function
f (4 The special case of 2.63 obtained by siibstitutiiig x + y for x, riamely 2.831
The only applications of the equalisiilg rules for functions which we have made so far have been confined to single recursions. However, the following proof of the equation
requires an application of the rule to a double recursion. \Ve observe that
x+(O=x)=x=O+(x~0)
is evident, since ( 1 -x)f ( x + y ) and (l'2x)f( y ) both satisfy
d o . y ) = f (Y), cp(Sx.y)=Zrp(x, y)
Sx+(Sy=-Sx)=Sx+(y:x)=S(x+(y-x))
+
-
(x-y)=S(y+(x:y))
SO that x (y - x ) and y + ( x y) both satisfy the doubly recursive equations
1
substituting x L y for x in 2.631 n-e find
and after multiplying by 1 -Ir, y1 and using the provable equation ( 1 - ( x - y ) ) { l A J E , y))= ( 1 -Ix, y)), (see Example 2.241), this becomes
(~'Ix,
Y ] ) /( Y ) = ( ~ ~~ Il x) (f, y j ( x L y ) )Substituting x for y and y for x in 2.65: since lx, y1 = ] y ,21, we obtain 2.65
2.66
(lL1x, 91)f( x ) = ( l-Ix, y])/( x + ( y - x ) ) and so 2.63 folloms by 2.62. Taking f ( x )t o be the identity function Y ( x )= x in 2.63 we have 2.67
Sy+(Sx'Sy)=Syf
( 1 :x)f @ + y )= ( 1 ' x ) f ( y )
d
) x ,y1 = (2-1-y)+ (y'x) . 1 It follows that Ix?y1 =1y, xl, and iising 2.61, that
and
( l L l x ?~ I ) f ( x ) = ( l ~y Il )xf (, y ) .
2.63
(1+,
Y ~ ) x ( =~ A I
x,
!/))Y
and so if 1 F , GI =O is a proved equation for certain functions F , G then substituting F , G for x, y in 2.67 we see that F=G is a proved equation. Conversely from F = G we derive, by xneans of F&.F=O, both F-G=O and G-F=O and so lF?G(=O.
Thus we have shown that each of the equntions IF,GI=O, F=G muy be deduced from the other.
34
TIIE E Q r A T I O S CALCLZCS
-
The equation 2.63 may be espressed in another forni mliicli is often more convenierit to use; if rve multiply 2.63 by 1 f ( x ) .and iise the equ at'ion
'i'hus the foregoing theorem may be representecl by the schema
f ( O ) = g(0) f ( S r )= g(&x)
f (x)= g(.r) .
S-hicli is obtained b ~ .-ltOst,itiitirig f (.t.) for x ir1 the provable c q ~ ~ a t i ozn( l z ) 0 ('.;w Ysaniple 9.1) we obtain
2.i;
-4s a first estiiii;>i of the use of the equivalence of the 1.qiiatioris 1 F, GI = O niitl F =C: \ve prove tlie equalby of , two
are ~n-ovable.tiicri tlie eqii.ztioii f (.t.)= (1 is provable; in schematic form
-
2.7
fii:lctioiis f and g whicli are k n o m to satisfy the equatioiis
If for
so^:!^
fiiriction f ( x ) the equatioris
!(U) = O ( 1-'f(x))f(,Cx)=O
f ( O ) = g(0) , f (8s)= g(Sx) . \Ve detiiie p?(x)= 1f ( x ). y(.r)l so that ~ ( 0=)0 , ~ ( S X =0 ) are proved equati6ns, and therefore g(Px)=Zq(x). Since 2.r also satisfies tlie introductory equatioiis
-
!
tlicrcfore p(x) = Z z , aiid so ~ ( x=)0 , are proved equations, whence, ;)y the previous theorem, f(x)=g(x) is a proved equation. A gmeralisation of this result is given in example 2.73. To record the fact that a proof of some equation F = G may be ('Rtnined by combining tlie proofs of one or more equations Ji qi. f2= g2, ... , f k =gk, me m i t e simply
By esaniple 2 . 2 3 2 . ( 1 :ri)b n-e dediice
f)
L r ~ bnnd so from ( 1 f ( x ) ) (f S r )= 0
and so if g(x) is defined bj7 the recursion ~ ( 0=) 1 , ~ ( S X =g ) ( x ) ( 1 2f ( x ) )
niid if h ( x )= g(6.r) tlien
-
biit h(0)=g(O) (since f(0) = 0 ) and so h(x)=g(x) ; Thus g(0)= 1, g(Sx)= g(x) and so (by example 2.7304), g(x)= 1, and therefore 1 = f ( x )= l . Hence f ( x )= f ( x ) ( l - f ( x ) )= O? using esample 2.1. \Te shall show later that theorem 2.8 establishes the validity of the method of proof known as mathematical induction. nnd cal1 this a proof schema. --
* .~íxltiplyiiiga n equation by a factor is a shorthand way of clescribing the
,: .ri\-ation
of an equation ca=cb from the equation a = b ; tliis derivation is vf couise effected by substitciting .'bu for .'a" on tlie right-liand side uf t h e equntion ca = cn.
2.9
THE FCXCTI~SS .Zj, 4 and
u,,.
For any function f ( x ) me define
.
37
THE EQO.lTIOS CALCULUS
so that (1-A(?n))A(Sm)=O, and therefore by 2.5, A(m)=O. I n particular A ( n )= O. that is. ( 1 -g(n))L',(n)=O.
so that for any numeral p
2.92
If for some f (x), g(x)
lT(p)=f(o).f(l).....f(~) ;
and
( S n - a ) ( l f ( a ) )g(n)= O 2
for a function f ( x ,a ) depending nlso iipo~ia 13ararneter a \ve w ~ i t e
with an analogous formulation for thc case of i:iore tliaii -oile parameter. We take for granted tlie coni~aliioil defiiiitions;for the function For any assigned numeral p r e have
n.
and
then ( 1 17,(n))g(n)= O. 1 ~ At( m ) . B(m).C ( m ) denote respectively
aiid ( S n l S m ) ( l -f ( S m ) )g(n) . By hypothesis C ( m )= O and so ( 1 -T- B ( m ) )C(m)= O ; moreover
-
A ( m ) )B ( m )= ( 1 A ( m ) }{ A ( m )= ( 1 117,(m))g(n))= O nnd so (by eñample 2.47, with IT,(m) for a, f ( S m ) for b and (Sn~ S m ) g ( nfor ) c), (1
~T(P, a)=f(O,a ) . f ( l ,n)...:f(p, n )
In using the fiinctions Z,, 17, we take tlie first vnrisbIe be the operative variable; to study suclr a siirn as
iii
f to
we introduce ~ ( xa ), by the explicit definition s. a ) = f (a,z ) and express f (a, O ) f (a, 1 ) ... + f (a, p ) as Z&p, a).
+
+
2.91 If for some functions f(z), g(z) the equtttion
is provable, then
is provable. Denote ( S n - m ) ( l - g ( n ) ) Z l ( m ) bi- A ( m ) ; then by (i) A(O)=O, and A ( S m )= ( S n-Sm)(l = g(n))(.Z,(,m)+ f ( S m ) )
and therefore A ( m )= O , whence A ( n )= O, that is
(1 -27,(n)} g(n)= O . ( 1 - f ( x ) )17,(x)=0.
For and
{l'f (O))n,(o>={l2f(o)}f(o)=o ( 1 f ( S x ) }17,(Sx)= (1 f ( S x ) )f ( S x ).q ( x )= 0.
-
2.93 I n the following section me require a number of the elementary properties of the function u ( x )= 1 2 ( 1 x ) which are proved in Exumples 2.26. I n particular we shall use the equations
and
a(xp)= .(x) .&(y) . The function n ( x ) obviously satisfies the equations For some given f ( x ) we define ~ ( x ) = a ( xf ) ) .It folloms that
d d x ) )= a ( 4 f( S ) ) )= d f(4) = a(x) and
( ~ ( 1~2( x = ) )1
% ( Q ( X ) )= 1 2
e(x).
t l
w
/
3s
EQC.\TIOx
&loreover
CALCULES
4 q . r ) )= fle(x)
for
1 1
Oe
THE E Q U A T I O S C'ILCKLUS
3n
11% note the following immediate consequerices of the definitions:
Y
( l L f l r ( n ) ),u,(Sn)= ( 1 -O,(n)) ,uj(n),
.(q,(t))) =?(O)
3(17,(Sx))= L 4 & ( .~ @)()S 4 so tlist x(U,,(x)) satisfies the saitie introdiictory eqiiatioiis 17,(z) it.self.
L)b
C~Y
If for some a? [<(a)= 1 aiid ? (.Sa ) =O then 8,(a)= 1 and so ,u,(Sa)= S o : if either T I c ( @= ) 0 or then pj(Sn)=lc,(a).
For Y ( X ) 1 = 0 and so ~ ( x'-)1 = 0 wheiice f ( x ) ( e ( x ) 1 ) = O ; but f(-t')(l:?(x)) = O and so. by addit,ion, f ( x ) .( ~ ( x 11) ,= 0. 2.!):34
For
--l(O)=Q(O) ( 1
-4(Sx)= e ( O ) ( 1 =~
I j ~ ( x= )x(f ( 2 ) )then %(x) = cu(lir,(x))
whilst
cuLT,(O) = af ( O ) =?(O)= %(O)
so that q ( x ) and a n ; ( x )satisfy the same introductoiy equations. 2.935
for 2.94
If cu(f(x))-g(x)=Othen f ( x ) - g ( x ) = O .
o=f(~)'~(f(x)).g(x)=f(~).g(~). For any function f(x) we d e h e O,(n)= Zi,,(n) :Tí,(Sn)
and where
{/,(O) =O
. ~i,(Sn)=Sn.Bl(n)+pf(n).(l -O,(n))
e@)= 4 f(4).
( 0( 1) I ~ , ( S S X ) ) I I R ,((1X ) 17,(S.r))
so that ( 1A A ( x ) ) A ( S x= ) O and t herefore d(s)= O . Since e(0)( 1=17,(Sx))=af ( O ) ( 1 ~JT,f,qx)) = .x(f ( O ) ( 1
= nZ7,(x).?(Sx)
n e ( S x )= q ( x ) .~ ( S X )
-
U , ( l ) )( 1 -O,(O)) = ? ( O ) ( 1 A ? ( ( ) )( 1 U0(1)) = 0. and 17,(SSx))Z7Rt(~){i(17,(Sx) U,,(SSx)))
= (1 l & ( S S x ) ) A ( x )
x ~ , ( s x=) LYLT,(X). ~ Y ~ ( s x )
and
i
/ ( o )( 1 - ~ , ( s x ) ) ~ , , ( =x )o .
Denote e(0)( 1 IIp(Sz))Z7E,(x)by --l(x). theii :
l
U,(Sx))).
2.941 now folloms by 1.933.
Since
e m-
O,(n),u,(Sn)= B,(n). S I L
then
q
{ l :O,(x)) {p,(4Yx) ~ p , ( x ) j . O=
arid
O,(x)(p,(Sx) - S X ] = O therefore, by example 2.36, (P,(sx)- S X ) ( ! l , ( ~ ~, (~x ) )= o ; hence. by esample 2.420, {Sx-,ui(x)) {p,(Sxj-Sx\ =O. %ice (by esample 2.41) ,Sx~p,(x)= [ x L , u , ( x ) ;]1~L ( , u , ( . r ) ~ x ) ;
40
41
THE E Q C . ~ T I O X CALCCLCS
THE EQUATION C A L ~ L X C S
and therefore
therefore (1 2- ( p , ( x )' x ) ) {p,(Sx)- S x ) = o ;
but , ~ , ( 0 ) -O= 0 , and so b y tlieorem 2.5. p,(x) ~ x O.=
Since ( b y esample 2.74)
As in the previous proof
therefore
-
{{L,(x) ',Ll,(S.r)){ S X ' { L ~ ( ~= XO) }
whence, since b y 2.942,
/L,(X)
2.944
(Sn
X)
-
IP,(x+~~),
{ l - (l'1T7p(x)).1p1(x+r), p,(x)1) ( 1 ' n e ( x ) )
which together mith (1 - U O ( x ) )I,ui(x),P ! ( x )=o ~ , proves
(/L!(x) A pi(a))= O
By example 2.461, from p,(x+r) - p , ( x + S r )
1
Hence b y example 2.451
x = O, therefore (using example
2.423), p l ( x )'-p,(Sx) =O.
=O
( 1 2 1 7 , ( x ) )I,u,(x-Sr), p,(x+r)l = O
we derive
L n , ( Ip!(x+r), ~ ) ) pj(x)l =o -
-
Since 1 2 Z7,(x) = a ( 1 ZI,(x)), it folloms b y 2.935, tliat
{''4(")) I{l,(x t.;r ) , p,(x)l
{ i2 - ( p f ( x ) ' { ~ 1 ( x i - r ){),)L L , ( X ) ~ ~ ~ ~ ( X + S ~ ) ) = O
whence, since p,(x) -p,(x + O ) = 0, it foll0-a~that l u t ( ~- ~) t ( x+ r ) =o> and therefore, b y example 2.71, ( S n L x ) (,rl1(x) :p!(a)) = O -
(1
! ~ 1 ( 4=1O
=0
.
'x )f ( S x )= O .
( { L, ( S x )
2.947
(,u,(sx)' x ) ~ ( s x=o ) .
2.9471
From p,(x)-x=O we deduce, b y esample 2.46, We have 17,(0)p1(0) = 0 and
-
(1 L n e ( ~ ) ~ , ( ~ ) ) ~ ( S ~ ) ~ t ( S ~ )
-
ne(z)p,(x))ne(x)~(Sx){Sx~e1(~) +p,(x) ( 1 e l ( x ) ) ) -17,(x)p1(x))17,(x)e(Sx)pl(x),since ~ ( S X01(x) ) . =0 , = O , which proves that i 7 e ( x ) ~ l=( O~ .)
=( 1
=( 1
But n , ( x ) = a ( n , ( z ) ) , and so b y 2.935
{/¿,(SX) - 5 ) {Sp,(x)- p , ( S x ) )
=0
and since ( b y esample 2.41) Sp,(x) -p,(Sx) = {p,(x)' , ~ , ( S X + ) )( 1 '(icl,(S~) -ic~t(x)))
multiplication b y 1 -8,(x) gives
-e,(x)) = o
( p l ( ~ ~ ) -( x1 )
-
and therefore ~ ( S X ( p), ( S x ) x ) ( 1 O1(x)) =O ;
From
( 1 e,(x)) ( p , ( s x )- P , ( X ) ) = 0
and
( 1-e,(x)) ( P ~ ( X ') p j ( S x ) )= O
( l L e i ( x ) ) Ipt(z) . P~(SX =o) ~ which in conjunction with ( 1 ~ 1 7 , ( x )Oj(x) ) =0 ,
we deduce
yields ( b y example 2.36)
but o(Sx)B,(x)= O so that
-
p(Sx) ( p , ( S x ) xP,(x) = 0
9
vhence by addition { / L , ( S X' ) x)g(Sx)= 0
from which 2.947 follows b y theorem 2.935.
43
T H E E Q t J A T I O S CALCL-LCvS
T l i E E Q L 7 A T I C ) S C \LCL*r.L-?;
Theoreni 2.9471 follows from "947 by niearis of esample 2.741.
f ( O ) ( 1 f f , ( S x ) )f ( u , ( S x ) )= O . . . . . . . . . . . . . . .
.
{ , u ~ ( L ~ xX )) f ({!,(SX T r ) )=
2.948
lieiice by tlieorem 2.941 (and esample 2.36) Since. b ~ 2.63. .
From 2.946 arid 2.9471 \ve obtain (as ir1 esaiiiple 2.36)
-
{ p ,(Sx'; 2 .ci l i ~ , ( St~r ):. !6,(ASz)l O
....... ..
and so, by example 2.47.". {1 2-
I{L,(~X).
.Sx/} I,t,(S.r
and by 2.946 (taking z = 0 ) . ( 1 f ( O ) ) p , ( r )= O n e havq. o11 miiltiplying (ii) by 1 f ( O ) ,
r ) , , / ~ / ( S=Xo) ~
-
ancl similarly from 2.9-F7
aiid so ( i
[ l ' I / ~ , ( S X8x1) ) . f (,u,(A\'x))= { l
ljl1(8x).9x1) f ( S X )
&iridso Ip,(Sx), 8x1) f (,u,(Sx))= O
{l
-
. . . . . . . . . . . . - (u)
( { l i ( S X ) )= O .
B y ( i ) , (ii) and theorem 2.63 we obtaui 2.948, which may also be written in the form (1
+
I,uf(Sz),S x l } f ( p , ( S x r ) )= 0 ( 1 - q x ) }f (P!(x))= 0
2.949
Finally we observe that ( 1 17,(0))f ( p , ( O ) = ) ( 1 = j ( O ) ) !(O) completes tlie proof. 2.941
-
=O
mliich
f ( 0 ) ( l ~ f ( x () l)~ ! l , ( x ) ) = = O .
( 1 '-1 ( x ) )f ( p r ( x ) = ) " :
however ( 1 ~ / r ! ( x f) () p ! ( x ) )= ( 1
/c!(x))f (0)
and so 2.9491 follows on multiplicatioii by 1 = f ( x ) . From 2.0491 and 2.944 we deduce. by esnrnple 2.162.
-
Since nOi(x)= it%(x). ~ (I-@ l S x =17,(x) ) ( 1 2 e S x ) = 6,(x)
-
.
For ( 1 f ( x ) ) U , ( x= ) O and so, by the prcvioiis theorerii.
which. by example 2.-1.81. traiisforms into
{{,,,(SX) '-x) f
f ( O ) ) ( 1 M,(Sx))f (,u,(Sx))= 0, nliich added :o (i) yields ( 1 - - n , ( S x ) )f (,u,(Sx))= o
[l-Ip,(ICx),Sxl}f(Sx)=o ;
but
-
,
aiid so, by 2.92, we have
therefore 1 2 R , ( x ) = O,(x) and O,(z) O,(x) = O,(x) and
YO
{l
R , ( x ) )Ip,(Sx), 8x1 = O ; hence by the last theorem (1
-
R , ( x ) ) f ( p , ( S x+ r ) ) = 0
.
It follows froin example 2.711 that ( S x - a ) ( 1- R , ( a ) ) f ( p , S x )= O
atid therpforc by theorem 2.92 (l
L n ~ ff (,+(Sx)) ( ~ ) =)
;
First we prove tliat { l 2 f ( a ) )( p , ( a- n )' - a } = 0 . Denote ( 1 2 f ( a ) ),,,(a v.) by g(n): then since. by esample 2.74, (1 '- f ( a ) )O,(n -- n ) = o . we liave g ( S n )= y(n) and so g ( n )= g(O) tliat is :i f ( a ) ) t t l ( u n)= { l =f (tr))!r,(n).
C' or -
* e-
Qr
e-
6i
m
C e,
44
h h
But p,(a) - a = O and therefore (1 f (a)) {p,(a n) - a ) may be expressed in the form
b
(Sz-a) ( 1 - f ( n ) ) ( / ~ , ( x ) - a ) = O . . Honrever. by 3.944 and 2.042
b b h
TAE EQL7ATION CALCULUS
-
=O
wliich
. . . . . . . . . . . (i)
(Sa -2) (,L,,(x) -/¿!(a)) = O and so
. p,(a) - a
=O
. . . . . . . . . . . . . . . . . . . (ii);
h
(Sa-x) (p,(x)-n)=O
h
multiplying (ii) by 1-f (a) aild adding (i) we find, since
h h
(by examples 2.145 and 2.41),
h
1 We shall show in the next chaptey that theorem 2.945, 2.949 and 2.9493 prove that when f (x)is zero for sonte x from O to n then ,Lr,(n) is the least value of x, between O and n, for which f (x) is zero, but if none of f ( O ) , f ( l ) , .... f(n) is zero then p?(n)= O.
rr h B
-
-
~ ( 0 )these ) equations still hold if v e replace 1. by p. The equatioiis obtained from (i) and (ii) by replacing i. by ~íare proved by means of the relations p(Sn)- 0(n)= O and p(n).U(n)= O previoue1~established; instead of (iii) me have e(O){l2Q(Sn))(1 ~ , u ( n ) ) p ( S n ) = ?(O){l-e(Sn)) (1 ~ ~ ( nU )( n) ).S77 =?(0)(1-o(Sn)) (1 ~ , u ( n ) ) . S nsince . 17(n)(1~
From (i) nnd t,he companioii eqiiatiori in !L nre llave
-
~ ( S n ) ( l I?.(n),p(n)l) Ii.(Sn), /c(Sn)l = (1 I;l(n),,U(TL)I))?(S~)A(S:L), o(~>~)t(~71)/ = p(Sn){l
-
(1 f (a)) (,LL,(x) a ) = O
.
P
+
A,(Sn)= R,(n)+Sn. (1 (A,(n) f (Sn)))
and
~ r ( n=e@) ) .Un)
then
I;.(n), p(n)l) I?.(n).p(n)j= O ,
thnt is
211d
o(Sn)(l li.(la), ,u(n)l}Ii(Sn).Ic(A'ia)l froin (ii) and its coinprinion {l
-
= {1
i.e.
h
~ ( 7 1= ) T;r(n\. )
= (i
. . . . . . . . . . . . . .(iv)
14n)>P(n)ll P(n)l1.(Sn)> {l(W\
= (1A
IA(n),,u(n)l)Ip(n)j.(Sn),,u@)p(S?z)I IA(n),~ ( n ) lIA(n)i(Sn), ) p(n),u(Sn)l
= {l
li.(n), ,u(n)l}lJ.(n)j.(n),p(n),u(n)I= O
i r ( % ) (1 AIA(n),p(n)l} /?.(Siz),p(Sn)J= O .
. . . . . . . . . . . . . . (r).
Finally, i~sing(iii) and the corresgonding equation for p,
e
I n the following proof, t o simplify the typography, we shall drop the suffix and write simply A(n) and p(n) for A,(n) and ,ul@). The proof turns on the following properties of A(x), obtained from the defining equations by inultiplication by an appropiate factor. e(Sn) A(Sn)= Q(Sn).A(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . (i) e
2(n)-f.(Sn)= R ( n ) . ~ ( n.............................. ) (ii) (1 lA(n)) (1 -g(Sn)) A(Sn)= (1 -A(n)) (1 -g(Sn)) S n . . . . . (iii)
It is readily seen that (apart from the introduction of the factor
and so ( l L p ( n ) ) ~ ( 0(1)-e(Sn)) (1 -lJl(n), ,u(n)l)l?.(Sn),,u(Sn)l= O ; . . . . .(vi) addiilg ~ ( 0(1):p(Sn)) times (r)to (vi),since ,L + (1 .-,u) = 1 t (,u-L 1). n-e Iiave e(O) (1 -o(Siz)) (1 -l?.(n), ,u(n)l)IA(Sn),p(Sn)l = O
46
THE: I.:Q L-.ATIOS
(-A I.CI-I,US
2.062
to which we add p(0) tirii~s(iv) aiid fiiid
Vsing esaniple 2.471 it ft,ll:tn-s tiiat
The ineqiiality relatioiis are traiisitive, that is to say
2.9621
a: c follows from
(1
2.9622
n -, c follo~vsfrom
ci
b and b> c -. b nnd b
'c .
ivliicli coiiibined witti I N ( ~ ) ; . ( oc)((l)p(0)l ). = O proves
Sirice n : b is equiralent to 0 - ci (that is. eacli is derirable from the other) the transitivity of tlie relations and < follows from 2.9621 and 2.9622. Tlie proof of 2.9621 i.s coiitained in esample 2.161 : n-e observe nest that siiice ( 9: S x ) = (y - 2 ) l . tlierefore S x y follo~vsfrorn .r > y aiid tlicb:.efoi.e 2.!)622 follows from 2.9621. In fnct >.!J622 is a coiisccliieiice of
by theoreni .>.!)403.
2.!)63
[ 1 Ip(O)A(,t).
I ) ! , ( , L;)/?(o);.: ~ S I L )~, ( 0 ) , l ~ (=S 0~ ¿ ) ~
(1 .-?(O))
cc(.i:)
;S
-
=U
a =. c follows from (r - b ancl b z c
which follows from 2.9621 n11d 2.964
?vliich is the desiretl resiilt.
if h>c then S h >Sc
mhicli is yroved 1)'. obeerving tiitit fro!:i
2.961
\Te tlcfine the ine?:irrliftJ cr ;b to stand for the equatiori aiid so
The inequality a>2, is read as ' a is greater than or equal to b' aiid u ~b as ' a is less than or equal to b'. IT-e intrc'cluce also two further inequalities
a t b ( a is less than b), a> b ( a is greater than b ) which are tlofined to stand for S a g b, a >Sb respectively. It follo~~rs from example 2.5 that the ineqiiality a ~ may b be derired from the equation a l - b = O, nnd con.versely, and that simiiarly the inequalities a ~ b b>a , (and therefore alao the inequalities a t b , b > a ) may each be derived from the other. The follon-ing simple inequalities derive immediakly from the detinitions axi:l the properties of the difference function:
81) = S'c
(Sb S e ) .
2.97 The inequalities tr > 6. t r - c > ó - e iiiity each be derived from the other. 2nd so too may etlcli of tlie inequalities a > b, a.Sc> b.Sc. F o r ( b ~ c ) ~ ( n + c ) = Oh.Sc~n.Sc=(b-a).Scand(b~-a)Sc=O ~n, if and only if b=n=O, since (b-ci)Sc=(h-a)c+(b-a).
and t,he iri~ql.:ality ci g b to stand for
l
2.95 Recursive aritlimetic ns it has been constituted in 2.1 is demonstrably free from coiitradiction, in the sense that we can shom that if p = g is a prorable equation, where p and q are deñnite numerals, then p aild q are tlie sanle numeral. I n other words it is impossible to prove tlie cquation 0 = 1 iil recursive arithmetic. If me say that aii equatioii F = G is cerifiable oidy if F and G are the same numeral or tlie substitution of numerab for the variables in F aiid G al~vaysreduces F and G to the same numeral, then nTeniay espress freetlon~from coiitradiction by saying that only rerifinble ~qlicctionstrre procable. -4s me observed in S 1.9, when the variables in its argument
respectivel?, then 17, and TVo each eqiial n(d) mhence
i aiid so 011. up to V,- TI-, xliicli shows that the equatiori F = G is vcrifiable. is doubly recursire. aiict if its introductory eqiiatioiis are If
-
!,-'-l. n L 1 ) = c ( a , p, n. gjn. p, d(a. p. n, ~ ( ( 1 P+ . 1, n ) ) )d . a ,P+
Examples 11 '1 -.
2.01 2.02
then F ( d , O , N ) = G ( d , O , S ) for any numeral N , and
1.03 2.1 2.2
F(-.!, 1. O ) =CT(A, 1, O ) = a,,. say, d ( A , O, 0 , vio) =aw, say, ) ~ ( - 41,, l ) = c ( A , O, 0 , t7,,(l,, v,,)=G(A, 1, l)=vll, say,
2.201 2.21
~ ( ( 1 .
l 7 n))l
Iff(O,y)=yandf(Sx,y)=f(x,y)provethatf(x~y)=~. Proce the equationu 2.01 - 2.251 a(b + c ) = nb iac (ab)c= n(bc) (nb) (cd)= (nc). (bd) x(1-x)=O llx=OZ
u.--
9 99
~ 1 ( 1 L X ) + { l ( l :x)}= aL(b+c)=(u-b)-c (a,b):c=(a~~)~b
and so on up t o F ( A , 2, N) = @ ( A , 2, N ) .
2.23 2.231 2.232 2.233 2.234 2.24
(a+x)-(b*x)=n-b a(x- l ) = n x l n a(b-c)=ab-ctc 2'-x2= 0 ( 1 - x ) ( 1 ~ x ) 1=~x ( 1 2 la, 61) ( b - u ) = O
Thns step by step me reach
2.241
( l ~ l abj) , ( 1 I - ( b - a ) ) = 1-10. bl
2.242
( b l a )( S n - b )= O
2.213
( 1 = ( S u - b ) } ( 1 -Y ( b ~ a = ) )0
2.214
( S u - b ) (Sb-.-n)la, bI
d ( A , O, 1, v11)=d,,.
F ( d : 1 , 2 ) = c(A,O, 1, t.,, do,, ,t.,*)= G ( A , 1, 2 ) and so on, up to F ( A , 1, N) = G ( A , 1, N ) =a,,, numeral N.
say, for any
Then F ( A , 2, O)=G(A, 2, 0)=v2,, say, d ( A , 1, 0 , vm)=dlO, say,
for a:iy set of numerals ( A ,P, N ) and so F = G is verifiable. -1 similar argument applies if g, is recursive of higher order. Thk completes the proof that every provable equation is verifiable. Tl'e sita11 show szibsequently that the converse does not hold, since t1ier.e are rerifiable equntions which are not provable.
=0
2.2441 { l - ( S n ~ b ) ] - { l i l a ,b J } = I = ( n ~ b ) 2.215
(b-a)i(A'a-b)=l+(a-b)-(h-Sa)
2.246
Sn-b=(n-b)~{l-(bla))
2.25
l : ( p ~ q ) = f l lp ) ( l L 9 )
2.251
(I 1 ~ ( q r-) ] - ~ ( 1 - y ) r = ( l ~ g ) - ( l (~r q- 1) )
2.26
If n ( x ) = l - ( 1 : ~ ) prove that a ( x ) l - l = O ,
THE EQCBTION C . 4 L C a U S
TXE E Q C A T I O S C A L C L Z C S
a ( x ) + ( l ~ c u ( x ) ) =. l(u(x)'x=O, x.cr(x)=x, (Y(1'-x)= 1 - a ( x ) . (Y((Y(x))=(Y(x) , ~ ' ( Y ( x )1='I , u ( x ) a ( x )= (Y(x), cu(xy)= ~ ( b &(y), ) . and that E ( / ) - g = O follows from f .g==O. I f ¿u(x?y) = "(15, y'l)) prove that ( 1 '(Y(x,y))x= ( 1 a ( x , y))y and (b-b.a(c, b))+c.cr(~? b) = C . Prove the equations 2.27 1- 2.274
S x - (5-(z)=S(x= (x'a)) Prove the index laws ?.?SI - 2.2%
( x.y)" z x n .?/',
Prove that 3 . 2 = 6 , 4 . 2 = 8 , 3 . 7 = 2 1 , 4 . 7 = 2 8 and 34.27=918 Establish the following proof schemata
{l~(g'f)}h=O ( S f-g)h=O
3
Ib.a+(b-~-a)l=nib
2.44
( a ,b ~ ( b - a ) ( = a - b
34
THE EQCATIOX C.\LCCLT7S
2.5201 Prore the equations { l ~ ( b ~ {b'-(a-(a'-b)))=O a ) } 2.73
Prove that, for anv definite nuineral p f ( p '- r)=O 2.7301 f(p+Sr)=O f(z)=O
2.5
Prove that each of the equations (a)b=a+(b.-a), ( p ) a = b = ( b - a ) , ( y ) a l b = O follows from any other
-
2.6
Prove the equation {1 (a.-b))+ (1 ~ ( 8 b ' - a ) ) =1
5.7
Establish the schemata 4.701 - 2.72
2.74
If p(x, O ) =f ( S x ) and y ( x , S r ) = ~ ( xr), .f ( x+SSr) prore that n,(x+Sr)=17,(x).pl(x, r ) and deduce that (1 - 1 7 , ( ~ ) ) 1 7 ~ ( x + r ) = O .
2.741
Prove that (1 = f ( 4 1f l i ( 4 = 0 With the notation of
2.3
1.6 prove
2.9
If the function n! is defined by the equations O! = 1 , ( S n ) != ( n ! ) S n prove that 1 n!= O and ( S n )! =%(n) .
2.91
Prove the equation ( l ~ ( 1 - a () l ~ b )( )l ~ a c( )1 - ( b + ( l ~ c ) ) ) = O .
-
5. 7
THE LOGICAL CONSTANTS
of a true proposition is a false one. and the negation of one that is false is a true proposition.
CHAPTER 111 THE LOGICAL CONSTANTS If a and b are natural iiiiinbers a.nd if la. bl =O is a provable equation then \ve call the equation a = b a true yroposition, and if the inequality la. bl > O is provable then the equation a = b is called a fcrlse proposition. Since the equations a=?, la, bl =O may each be derived from the other it folloxvs that a t'rue proposition is a provable equation and conversely. Moreover. as Ix, y1 is a reciirsive functioii, for any natural piimbers a. b there is a unique natural number c such that 3.
la, b l = c '
3.12 We denote propositions by single letters p, q, r xvith or without numerical subscripts. The negation of the proposition p is denoted by p, which is read 'not-p'. When the contest renders the usage free from ambiguity we shall write simply 'p' for 'p is true' and ' - p' for ' y is false'.
-
3.13 If p, q denote the propositions a = bl c = d respectively then the proposit,ion [a, bl
-
/C.
d( = O
is denoted by p & q whicli is read 'p and q', the proposition (a,b(. (c,d[= O
is denoted by p v y xvhich is read ' p or q'!
ia provable; if this riumber c is zero then a = b is a true propositioii,
and
and if c is not zero then 1-la, b[ = O is provable, so that a = b ia a false proposition. Bccordingly, a n y proposition i s necessarily either true or false, and n o proposition i s both true and false.
is denoted by 2) -t q read ' p implies q'. p + q is thus the same proposition as Finally, we denote the equation
3.1 We call a(la, bl) the number of the proposition a = b , so that a true proposition has the number zero and the number of a false proposition is unity. Conversely, if &(\a,bl) = O then la, b) = O SO that the proposition a = b is true, and if &(la,bl)= 1 then la, bl > O and a = b is false, and therefore the number of a proposition is zero if and only if the proposition is true and the number is unity if and only if the proposition is false.
by p -eq which is read ' p is equivalent to q', so that p the same proposition as (p + q) & ( q -+ p). We observe that, since 1 la, bl, 01 = la, bl, therefore
3.11 The propositioii 1 :la. bl = O is called the negation of the proposition a = b. Since x(1
/x. Y / ) = 1A Ix. y(
it follows that if the proposition a = b is true, so that la, bl = O , then the riumber of the negation of a = b is uiiity, and if a = b is false then the number of its negation is zero. Thus the negation
and since and therefore
( 1 :la, b [ ) . [ c dl= , O
N
p v q.
-S-
q is
(i~x)a(x)=x(i~u(~))=O nc(l*(x)) = a(x)
(x= O)++(a(x)= O)
\Ve shall justify the suggested readings of the signs a, and ct in the next section.
V!
-.
-t
58
TIIE LOGICAL CO?~ST.\STS
THE LOG1C.a CONST.\ZTTS
3.14 From the equation la, bJ+ Jc.dJ= O it follon-s that la, b/ = O and Ic, dl = O , and conversely, and tlierefore p & q is true, if, and only if, p and q are both true.
50
n-hich is summed up by saying that '+' is a transitise relatioii. Siniilarly from the schema
3.141 The product la, b/ . [ c .dl x-aiiishes if either la, bl = O or dJ=O; aiid if both la, b/ > O and jc, dJ> 0 then la, bl . J c ,d J>O,
x=O (l=x)y=0
Ic,
and so: p v q is triie if and only if either p is true or q is true. 3.112 It folloms that p -> q is true if nnd onlj- if either 2 is false or q is trtie. which n-e t ~ k cto be tlle ser1se of the espressirm p inzplies p.
3.15 The definition of eqiiivalence makes y and q equivalent if and only if p aiid q are both true or botli false. From tlie proposition p I A q and a proof of q we derive a proof of p. for if
and
(1-la, bl).Ic, d l + { l - ] C . dl].la, bl=0 dl = 0
Ic,
are proved equations. then la, bl = O is a proved equation; accordingly, t o prove some proposition p it suffices to prove a proposition equivalent to p. 3.16 The signs a, v, N, -+ and tt are known as logical constants; their introduction effects a considerable economy in proof technique by revealing a variety of easily recognisable patterns of procedure.
xve derive the fundamental schema for implication (1-~ n o ~ \ -as ti modzis ponens) P P Y Y -\
I
3.22 Since addition and multiplication are commiitatire the following proof schemat.a are ralid:
Tliese shom that ' a ' . . v ' and
3.23 From the equation x ( l =x)= O u-e deduce botlr p -t p and p ct p mhich show that '-+' and 'ct' are reflexire relations. 3.24 Since f = O and g = O follow froni f -!- g = O. nnd concersely,
P P+q 5' !I+P P & Q P"Y
and
It folloms that and therefore, if p, q, r are any propositions al= %, bl= b,, c,=c,, say, taking lai, +I for x, lb,, b,] for y and Icl, c,] for z respectively, we obtain the proof schema
are s!j»lnietrical relations.
' t . '
'e' like '+'
is transitive.
3.3 We consider next some of the important relations which hold between the logical constants. Since 1 .- (1 ( 1 ~ x ) = ) 1L X it follows that
From tlie equation I
-
(x + y) = ( 12 x) ( 1 .-y) we deduce
60
TIIE LOCICAL COSSTASTS
THE LOGICAL COXST.AXTS
3.32
I t remains only to prove t,he scliema
-(Paq)++-Pv-q
-
aiid from 1 -xy = ~ ( ( 'T-x) 1 + ( 1 y ) ) follon~s
+
Fiirthermore. since x ( y + z ) = x y XZ, p v ( q a r ) is the same proposition as ( p v q ) a ( p v r ) so that r ) -'( P
P v (P
3.33
(11 & ( P
aiid this foilows from the eqiiation {l
r)
++
q2 then
1 We observe first that silice the suffixes l,:! are interchangeable in the given conditions pl tt p2 , ql tf q2 therefore it suffices to prove the foregoing propositions with implication in place of the equivalente relation. Since p -t q is the same proposition as N p v q , the truth of the fea-tli proposition follows from tliat of the first and third, and the fifth then follows by means of the second. That p2 -+ p, follows from pl + p2 is shown by the equation N
{ 1 2 ( l : x 2 ) } (1~x1)=(1:x1):{(1~x,):x2(1~x1))
-
by means of which ( 1 -1- ( 11x 2 ) ) (1: x,)
=O
follo\vs from
( 1 x1)x2= o.
Since 3.341
r
++
r me deduce from 3.321 that pvq--(-p&-q)
so that the truth of the third propositiori follows fi-om tliat of the first and second.
y1)!/22 xl!jz)
+
P & ( Q ~ ~ ) + + ( P & ' I ) ~ ( P ~ ~ )
If pl up2 and qi
[ ( l x1)x2 y1x2) k {(L
{l (xl + g1)}(x2 y2)= O from the equations ( 1 '-rt.,)x2 = 0 , ( 1 l y l ) y 2= O.
For equivalent propositions we cari prove interchangeability; @is is given by following theorern: 3.34
=
by nieaiis of mliicli \ve derive
and since { l ~ ( z + y z ) ) ( x - y () z - z ) = { l ~ ( x + y () r ~ z ()x)l y z ) = O therefore 3.331
(q+ g1))(x2+?/J
i1
3.342 Tlie importante of the resiilts contained in theorem 3.34 lies in the fact tliat i t alloms iis to replace any proposition, ir1 u r i esp~essioiiriiade up of logical constants, by an equivalent; t h t resulting espression will be equivalent to tlie original one and so by 3.15 a proof of the transfornied expression suffices to prove the original. I n particular, since the propositions la. bl = O and a = b are equivalent, we m u y zoithozit loss of generality suppose any proposition to huz+e the form c= O .
As an example of tlie use of this principie nre consider in detail a proof of the proposition 3.343
If p', q', r' are equivalent to p, q, r respectively then p' -+ q' , r' v p' , r' v q' are equivalent t o p -t q , v p , r v q respectively and so P is eqiiivaleiit to
P'
:
( p r -+ q') -+ {(Y' v p') 4 (r' v q')) Whatever the propositions p, q, r me may take p', q', r' to liave the forms a = O, b = O, c = O respectively, and so P is equivalent to
P* :
( 1 ~ ( 1 - 1 - a ) b( 1) ~ c a ) c b = O
where a, b , c are certain numerals, which may be derired from the provable equation (emmple 2.48)
6"
(where x, y, z are numeral variables) bu substituting a. b, aiid c for x. ?/ and z respecti\-el?. If me allon- the lettcrs p. q. 1- t o hnre thc clual roles of ii:inies for propositions and iiiinier:d variables tlien rnny 1;~iteG directly in tlie forni {l -L (1I- p)q) (1 I- r p ) r q = 0 : that is to say, \ve may formiilate ari eqiiivnlcnt of P simply bu \ ~ i t i n g(1 p)q for p y. rp for r v p arid so o11 (thoiigli of coiirse )ve cliaiige the signi!ic:iiice of tlie 1ctter.s when wé rnake tlie trniiscriptiori). -\
then n-e denote l z I F ( x ) , f (x)J= O by
and
Froni the provable equatioiis
y c derive the truth of the propositioiis pv-p
-
Q(Y) by
242)
-
p(x),
Q(Y)
-+
-
~ (v 4P(X) from y(1-y) = 0 by takiiig IF(n),f (z)l for y. JVe use tlie term formula to cover both propositions and propositional functions. I n tlie notation of tliis cliapter tlie fundamental theorem 2.68 takes the form
and - ( p & - p )
which are líilomii respective11 as the priiiciple of exclzcded middle (or terfiurn non dattr,~)and tiie principie of noncontrndiction. 3.4
P(X)
63
{P(x)+ dY)} {q(?/) ~ ( 2 ) by ) P(X) ++ q($!). 3.42 The relations betweeri tlie logical coiistnnts mhich we llave established for propositioiis hold also for propositional functions, the proofs proceeding on tlie sariie liiies; for instance me derive
e
3.35
-
THE LOCICAL COXSTANTS
THE LOGIC.41, COSST-ISTS
PROPOSITIONAL FUXCTIO-*TS
If x is a variable and f (x), g(x) are two given recursive functions then the equatioii
f (x)=S (4 is calied a propositional fzcnetion; if for some value a of x, f (a)=g(a) is provable then the propositional function is said to be true for the value a ; if 1f (a), g(a)l > O is demonstrable then the propositional function is said to be false for the value a. More precisely, the equation f ( x ) = g ( x ) is called a one-variable propositional function, ""'1 f ( x , y) = g(x, y) is a tmo-variable propositional function, and SO on. Propositional functions are denoted by p(x), p(x, y) etc., accorciing t o tlie niimher of variables. 3.41 As in tlie case of propositions, if p(x), q(x) denote the propo.citioiin1 functions
3.5 The logical co~istaritseriable us to introduce the conditional eqiiations of elementar>- algebra. Unlike the variable x wliich is characterised by the propertj- that it may be replaced by zero or by Sx wherever it occurs, the x in a conditional equation is a missing number sign which m$!- only be replaced by some def%ite numeral. For instance, wheii r e say that x = 3 is a soliition of the eqiiation x2= O, we mean that a true equation results from replacing x by 3 in the second equation. but neither in x = 3 nor in x2=9 may we replace x by zero or by Sx. I n fact x = 3 and x2= 9 are not equatiom but propositional functions; and the fact that 3 is a value of x which satisfies ; c L 9 is espressed by the implication
F:
( 5 = 3 ) -+ (x2=0).
Formula F holds for any ralue of x as rnay readily be verified, for , 31 = 0 (by example 2.4'7) and 12 13+Sr , 31 = 0 and 1= 1;: 35 91 = O , so that, by esample 2.7303, (11-[.t.. 3 j ) . ( 9 . ,[=O which completes tlie proof of formula F.
64
LOGICAL COSSTANTS
T H E LOGIC.1L C O N S T A h T S
Similarly. the fact that the 'equation' x2 + 6 = 5 s has only the twvo solutions x = 2 and x = 3 is espressed by the implicat.ion
-
i.e. ( 1 Ixz+ 6. 5x1). Ix, 4. lx, 331 = 0 ; denot,iiig tlie left-hand side of this equation by f ( x ) we liave
and f ( 0 )=f ( 1 )=f ( 2 )= 0 mhich proves t>liatf ( x )= O.
+
3.6 I n additiori to the logical constarits \%-eintroduce &e linaitd universal, existential and nainininl operators A:, E: and L: as follows : A:(/ ( x )= O ) stands for the propositional f~inctionL',(n)= O ; E:(/ ( x )= O ) for the propositional functioii I7,(n) = 0. and I LE(f (x)=O) for tlie ftinction p,(n). The operators 'A:', 'Ej', 'L:' are read as 'for al1 x from O to n'. .for some x froni O to n' and 'the least x from O to n ' ; we proceed to justify these suggested readings. (We use tlie term 'justify' to cover an informal discussion - there is no question of a formal proof, since only the sign itself and not the interpretation finds a place in tlie formal work.) 3.61 The logical constants and the operators may be regarded simply as abbreviations for the expressions by means of which they were introduced, in which case in any formal proof the logical constants and operators must be eliminated and replaced by the expressions which they denote. Alternatively we may regard these signs as an additional part of the formal system satisfying, by definiticm, the equivalences { ( a = b ) v ( c = d ) } tt {la, bl-Ic, dl = O } {(a=b ) a ( c = d ) } tt {la, bl Ic, dl = O } {(a=b) -+( c = d ) } tt { ( l ~ l abl)lc, , dl=0} {(n=b) * (c=d)} tt { ( l = l a , bl)lc, dl (1-Ic, dl)la, bl = O }
+
+
65
which must be added t o the list of permitted formulae in a proof; in this case the new signs cannot be totally eliminated but espressions containing them may be transformed into equiz+alent equations in which they do not appear. The sign x in the operators d:(f(x)=O). E:(/ ( x )=O) and L",f ( x )= O ) is not a true variable but an auxiliary sign k n o m as a bozind variable. JITe could readiiy reserve a special class of signs for bound variables but since there is ver- little risk of confusion it is customary to use the sanie sigris as for variables. The fact that tlie x in AL(f(x)=O) is a bound variable rnay be expressed formally by the rule that d i ( f ( x ) = O ) may be replaced by A ; ( f ( y )= O ) , or by the espression obtained by replacing x by any other variable, but substitution for the bound variable is not a l l o ~ e d with , a corresponding rule for the other operators. ,Uternatively we may formulate the rule as an equivalente
in which x and y may be replaced by any other variables. with corresponding equivalences for tlie other operators. 3.62 A:(/ ( x )= O ) is the proposit,ion Z,(O)= 0, that is f ( O ) = 0 , and if for some p, A:(/ ( x )= O ) is equivalent to
+
+
then since I=,(n 1)= Z,(n) f (n+ l ) , so that A:+ ' ( f ( x )= O ) is the proposition A!(/ ( x )= O ) a f ( p+- 1 )= O , it follows that A+"(/ ( x )= 0 ) is equivalent t o
so that for any assigned p, A!(f(x)=O) is equivalent to Similarly, since
q(f( x )= O ) is equivalent
to
66
THE LOGICAL C'ONSTASTS
THE LOGIr \ L l'OXST.\STCI
For the iiiterpretatiori of tlle operator LZ we recall tlie cliaracteristic properties of the f~inctionp , ( n ) proved in the lmt cliapter. From the proved equntioiis (2.942 and 3.949) we have
Tlie companion formula
may be callecl the principle of matlienirtici~lind~ct~ion. JTe derive formula 3.5 from whcrlce we obtain the foriiiiila
wiiic!i says that if tlierr is a 1-alue of x betveen O aml n for ~vhicli f \.i) vanishes, then p,(n) is one of these values; and from tlie eqiiation (2.9493) {l'f
Iwe Iisre
(4{ P , ( x ) - n ) = O
f ( n )= O + p t ( x ) G n nhivh is equivalent to
mhich says that f (n) does not vanish for aii n les8 than ,ui(x), so that if f (x) vanishes for some value of x from O t o n , then ,u,(n) is the least of these values. Tf j(s)>O for all x from O to n , i t follows from the equation (2.!145) n i ( n ) P , ( n )= 0
LE(f ( x )= O ) =p , ( n ) = O .
¿hzt U .7
~IATHKYATICAL I'YDUCTION
I t folioms from theorem 2.8 that the proof schema
iu \-%lid. Siis schenla is kno\n. as tlie scl~emsof ~~zathematical
indt!di~n.
I
I
any p ( x ) has an eqiiivaleiit propositional f::nction. f ( x )= U . say; by esaiiiples 3.01, 3.03 snd 3.322 n-r +e ttiiit formula 3.51 is eqiiivaleiit to ~ ( 1 1 = ) 0 wheir
~ p ( n ) = ( l - f ( O ) ] J 7 0 ( 1 ~ )rf (1 n) . O(.T)= f ( x ) - ~ ( l z f ( x1 + )). Since ~ p ( O ) = { l i f ( O ){ )/ ( O ) - ( l i f ( l ) ) ] j i l ) = @ and c p ( ? z ~ 1 ) = ( 1 - f ( O ) ) Z 7 , ( n ) ( f ( n - ' - 1 ) . - ~ 1 ~ f ( n - 2 ) ) ) f ( n + 2 ) =~ ( n f (n ) 2)
-
+
( 1 p;(n))~ ( I i Z 1 ) = O , wlieiice i i - ( i z ) = 0 so that follows by the induction schema. If \ve denote the propositional functioii :.S by P ( n ) then P ( 0 ) is
wliich holds by esample 3.031, and P(?L-1 ) is eqiiivaleiit t o p ( 0 ) & A ; { p ( z ) -t p ( a f 1 ) ) & ( p ( n + 1 ) -+ p ( n + 2)) -+p ( n + 1) mliich follo\vs from 3.81 (aild exaniple 3.031). Since P ( 0 ) and P(n+ 1 ) are proved t,hen P ( n ) folloms by 6.7. We collect here for refcrencc tlie principle properties of the operators A, E ancI L;the fractioiial part of the numbers of the follov-ing formulae are the same as in tlie numbers of the theorems in the previoiis chnpter of n-hicli tliese formulae are transcriptions. 3.9
TEE LOGICAL C O S S T A S T S
68
THE LOGICAL (:ONSTANTS
1
3.95 We consider nest some furtlier theorems ori t,hese operators which are needed in the sequel. I n the following proofs we systematically employ the corresponding lower case letter as the representing function of a predicate expressed by a capital letter. The product I?,(n) and the sum Z,(n) will also be deqoted by 17 f and ,Y j ( x ) respectively.
(X)
2471
69
A similar proof shows also that
is valid. In either of these schemata. by taking 0 = 0 for H we see that the schemata are valid also if H is suppressed. 3.053
E;(x
For
a F ( x ) )+ E ; F ( x )
x g n & F ( x )+ F ( x )
whence t,lie result follows by 3.951.
e
zS n
By esaniple 3.Y31. Jn terms of representing functions we have to derive the equation
3.950
E:F(x)
AEG(x) -t E : { F ( x ) a G'(x)).
The result is evident for n=O. Non( l ~ h( 1) - f ( x ) ) g ( z ) = O .
from
For n=O the derivation is obvious. By 2.7, ( 1 - p ) = p q = l l p , and thence by 2.7 (over q )
[ l - ( 1 2 0 ) ( 1 - b ) ~ ]( 1 2 ( a + q ) ) (1-Op)c(p+q) = [ l - ( 1 - a ) ( 1 - b ) c ] ( 1 - ( a + q ) ) (1-b)cp, since (1-q)q=O and ( 1 - b p ) p = ( l - b ) p ,
1 - p q = ( l ~ p )( 1 - ( 1 - q ) } + ( l : q ) .
and therefore
It follows that
whence 3.955 follows by 2.8. which completes a proof of (i) by 2.7.
-
For
-
A l ( x á n ) + {E,"F(x)-+E; ( x ,< n a F ( x ) ) }.
For [ l ( ( 1- h ) ( 1 Z;(n))Zlg(n))] ( 1~ h( ( 1) Z;(n))-f(n+ 1 ) ) = [1
( ( 1~
-
(.qn)+g(n+ l ) ) = h( 1 ) Zj(n))Zg(n)}] ( 1 h ) ( ( 1~f (n+1 ) )-Zj(n)> g(n+ 1 ) = 0 ,
-
nsing the hypothesis ( 1 h ) ( 1 2 f ( x ) ) g ( x= ) O.
By 3.052 and 3.956. 3.955
F ( n ) + EEF(x)
For (1-f(O))f(O)=O and ( 1 - f ( n + l ) ) { ~ f ( x ) ) f ( ~ n + l ) = O . S
l
l
THE LOGICAL C O N S T A S T S
3.959 If G does not contaiii tlie variable x anrl not coritained iii F or C the:i
12
is a variable
By 3.96 aiid 3.961. 'rlie case n = O is obvioiis. Sirice ( p -> q ) a ( r (I) -,(1, v r -.- y), as is rcadily proved. therefore ( E : F ( x ) -+ G ) ( I ( n- 1) C ) -.- (E:'"F(x) U )
-
-&
ri:id so, from F ( n ; 1) - C: follorvs
(E,"F(x)-t G ) -t (E;."'F j x ) -r (:) aiid tlie proof is completed by 3.7. It Iollows that tlie schema
-
3.964
AzG(x)++E",-G(x)
The case n = O is obvious. To prove -~47G(r)-+f?:-G(x) we tise the equation
proof is completed by an application of scheiiia 2.8. Similarly to prove
is valid, arid so in particular is the schema
we use the equation The validity of the schema for n = O is obvious. Since
C
= O,
xán
g ( x ) )( 1 - f ) {
C g(x)+g(n+l))
XBn
-
(which is also proved by applying 2.8 t o the variable c) with the same siibstitution for a, b and c. 3.9641
hare
Taking
-
G ( x ) for F ( x ) in 3.957 and using 3.964 r e
using the hypothesis ( 1 f )g(x)= 0 ,
tlie proof is completed by an application of 3.7. 8.961
For G(O)i G(O) and
@912 7
which is proved by applying 2.8 to tlie rariable c ; taking 1( 1 - g ( x ) ) for h and g(n+ 1) for c, the g(x) for a, X1
(1-(1-f)
e-
A ; G ( x ) + E",@)
From the propo~it~ional schema
, and 3.964 we derive immediatelj
v w
@-
TKE LOGICAL CONSTh\TS
THE LOGICAL CONSTAIVTS
52
3.966
xgn
b y 3.951 1 we derive A:(F
Denote this implication by P ( n ) . P ( 0 ) is evident, and ( z ~ &nf(x)=O)
-+
[ P ( n )-+ EY(y=x a f ( y ) = 0 ) 1 ;
so that
{ P ( n )-t E: G ( y ) )-> {P(n)4 E:"
whence. taking y = x
&
-?
t so that
EY+'{x
=
&
y
f(y)=O)]
. -
1 {f(n+l)=O,
&
a f ( x ) = O -t [ P ( n )-+
f ( y )= O) , b y 3.958,
E;+l{y=x
&
f(y)=O)].
Since xan+1
a f(x)=O),
f(x)=O -+ { x ~ &nf(x)=O) v { x = n + l
it follows that
{z
a f(y)=O)]
P(n)--+ [{z
i.e. P(n)-+ P(n+ 1 ) , mhich completes the proof.
If F does not depend upon x then: 3.9661
A = ( F a G ( x ) )tt F
3.9662
A:(F v G ( x ) )-eF v AZG(x)
3.9663
E:(F & G ( x ) )tt F
3.9664
EZ(F v G ( x ) )tt F v E:G(x)
&
&
G ( z ) )+- F
A,"(F a Q ( x ) )-+ F A','G(y)-t { x < n
follows
F a A;G(y)
whence
F
A;G(x) follows.
&
-+
G(x))
a {z
-t
a A;G(y) -+ { x ~n ->
( F a G ( x ) ) ),
arid so, by 3.96, 3.9641 F
x=n+l
&
Conversely from
But f(x)=O) -> ( z = n + l )
A;G(x)
F & G ( x )-t F
whence
G(y))
f ( y )= O for G ( y ) .
{ x g n & f(x)=O) -t [P(n)-+ E;"{y=x
{z=n+l
and from
& G ( x ) )-t
by 3.959, 3.961 we derive AZ(F
E; G(Y)-+ E,:+' @(Y
But
F a G ( x )-+ G ( x )
From
f(x)=O -+ EY{y=x a f ( y ) = o )
&
,4,"G(y) -+ A l ( F
& G(z))
which completes the proof of 3.9661. Again, by 3.9641, .
and so
F v A;G(y) + F
whence
F
v
A;G(y)
-t
((2
and so
F
v A:G(y)
-t
A + ( F v G ( x ) ), as above.
Conversely
A,"(F v @(Y)) + { X 9 n -+(Y G ( x ) ) )
and so
AV(F v G ( y ) )a
and therefore
AY(F
v
v
{x
-+
G(x))
< n) + ( F v G ( x ) ) )
G ( y ) )&
-
F + { x a n -+ G ( x ) ) F + A;G(y) ,
whence 3.9662 follows. Theorems 3.9663 and 3.9664 follow from 3.9662, 3.9661 by taking F for F and G ( x ) for G(x).
-
AEG(x)
3.97
EZG(x)
.
N
The Counting Operator X ; F ( x )
To express the number of roots of an equation f (x)=O in the
1
!
THE LOGICAL CONST.\'
74
range O<x g n . or what is tlie snme tlii~ig,tlie riumber of values of x in this range for whicli tlie propositioiial functioii f ( x )= 0 is true me introduce the ftiriction S': (f íxj = O ) clefined by the recursion
I
l
[A:
-
I f f ( x ) is the representing fiinctioii of tiic ~)i'ol)ositionalfunction F ( z ) then \ve lvrite S : F ( x ) S ; ; i ( x )= 0 . The first theorem n-e 1,rovt. f»r tlie coiiiitiiip o p cv.itoi' .~ h
Tf D(a. b ) denotes the positive difference bctn-eeii 1
l
l l
D(a. O)= O , D(0. Sb) = D(8o.
=0
,
that D(a. Sb) = O and tlierefore D(ri. 1)) = 0 and finally
n-lience
A:;
-
-
[-4:L1 F ( x )
-\
-)
{S:'' F ( x )= O ) ]
F(x)a [ S I P ( x= ) 01 .
Tlie eqiiivalerice 3.972 iiiay also 11e espress:~i
iii
tlic forni
-
Let S ( n ) denote the positil-e difference I~ctweentlie two sides of tlie equation. The proved equation 1 - p = : l l ( p - q ) ] - [ l : ( p - ( 1 - q ) ) ] suffices both to prove # ( O ) = O and S ( n )= . 4 ; ! i ~+ i ).
[ l ~ al ~, b \ = l ~ { l ~ l ( l (~l ~a b) )ba,.l ] Thus from ( 1 2 f ( x ) ) ~ (=x {) l g(z)}/( 2 )= O we derive
and
3.9i5
NZF(x)=x G ( x )
I X ~ + ' F ( x )N;+'G(x)l= , (X"(x), N;G(x)l
whence, by -3.8, IN;F(x), N;G(x)l= O , mhich completes the proof.
{ e F ( x )= O } tt A:
3.072
N
F(x).
Por n = O the equivalence becomes simply ( 1 : f ( O ) )=O e(1-f(O))=O. Silice X;'-'(f therefore
( 5 )=
Proof as for above, usii~gthe equatioii. 1~~=(11-p)-[l~{q--(l:p))]
1-f(x)=l'-g(2).
It follows that
~
F ( x ) -> {X''F(x)=O)]
bl and
l ~ { l ~ [ ( l ~ (l:b)al] f l ) b .
then SO
11 : a , 1
-
o) = N;(f ( x )= O ) + (1 / ( n+ 1 ) )
N:(F(x) a G ( x ) )G S t F ( x )
I n the equation { 1 2 (a: b)} {[rz - ( 1 (1)~ q ) ) ] [b = -- ( 1 ~ p ) ]=)O which is prored by applying schema 2.7 to tlie variable q, \Te take NE(F(x)& G(x))for a,NEF(x)for h. F(n-c 1 ) for p and G ( n L 1 ) for q. 3.0í6
x>n
[S;(y = x)= O)
-t
Let P ( n ) denote the implication: then P(0)is equirnlent to
{l:(l:x)) Since x > n L 1 + x > n and tlierefore
.i,
P ( n ) -+ [ x > n - 1
[l:r)=O. >n f l
->
(1-Ix,nLll)=O
- [ X y ( p = x ) = o]>
and so P ( n ) &rc>n;-1 -+ { S ~ ( y = x ) + ( l - l xw. i - l [ ) ) = O
76
THE LOGICAL C O N S T A ~ S
whence
P ( n ) -t ( x > n + 1 -> XY+'(y=x)=O)
A-;+'(xgn
+ 1)= N S A 1 ( x g n +N:-'(x=n+ )
P ( n ) -t P(n + 1 ) .
i.e. 3.977
-
AT,"+r+l(y=r)=NCAr(y=r)+ ( 1 L [ n + r + 1 , rl}=iV~+"~=r). -t
by 3.977.
We consider first the case n =O.
For n= 0 this reduces to the previous theorem: furthermore
n>x
1) , by 3.980,
= X ; ( x g n ) +1 ,
( 0 0 )= 1 ; and Nn+l(y=n+ 1)= N,"(y=n + 1 ) + ( 1 ~ 1 n + l , n + l I ) = l by , 3.976.
It follows that
-
1 1
whence the result follows.
N i ( y =n )= 1
For n = O this is simply 1
"
THE LOGICAL C O S S T A X T S
iV;(y=x)= 1 ,
for
Since
x
therefore
x.0
a /(%)=O
and so
zs.0
a f(x)=O+x+f(O)=O
whence
x
a f(z)=O+
4
-t
f(O)=O}
f(0)=0
{ 1 1 ( x + f(O))}=l
which proves tlie case n = O.
I
By 3.972 [ N ; { ~ = & x f ( 1 / ) = 0 ) = 0+] A y N { ~ = & X f(y)=o) From
and so E;(y=x
follows
x > n -+ { q ( y = x ) <11
and from
x
follows
x g n + {N:(y=x)
which proves
~
n{NY(y=x)= 1 )
-+
&
f(y)=O) ,iV;(y=x & f(y)=o)>O. by 3.965,
whence x g n
f(x)=O -+ N:{y=x & f ( y ) = O ) > O ,
But NY{y= x
f ( y )=O) < 1 , by 3.0i9 and 3.973
and therefore x á n a f(x)=O +iV;{y=x
&
by 3.966.
f(y)=0)=1.
( x> n ) v ( xg n) -t N,"(y= x ) g 1
and since ( x> n) v ( xg n ) is provable by double recursion, we conclude
{ N Y ( f ( y ) = o ) = k + l )-+E~{f(x)=O&NC(f(y)=O& y z x ) = k ) .
From N",j(y)=O)=N;(f(y)=O &y+x)+N;(f(y)=O ay=^) follows x g n Since and
whence x < n
x g n + l t t ( x < n ) v ( x = n f 1) x=n+l
tt
(x=n+l) &
-
a
f(x)=O + N;(f(y)=O)=N;(f(y)=O & y # x ) + 1 & f(x)=O+
(N;(f(y)=O& y + x ) = k -+N~(f(y)=O)=k+l)
(xgn)
the result follows by 3.971 and 3.974. 3.981
Nt(xgn)=n+1
Por n = O this is evident ; and
whence
q ( f ( z ) = Oa {N;(f(y)=O & y # x ) = k ) ) -t ( N ; ( f ( ~ ~ ) = o ) = k -+ l )
Con ccrsely from
follon-s .r
; :
IL &
. = 0 & !j 7;X) 1;) +- 1 - .- xr'(f(y) {(.t..:: n & f ( x ) = O )
= O) = , ;'
f (x)= O ->
u-licncc {St(f(y)=O)=X:+ 1: -+
!I
;-
;tllci SO
{q(f ( y )= 0)
-..
:/
11
i-
L
ir.
1, &
&
/(.t.)
sg(f(;l/)=O& y+x)=l;)í-t
:~-&
{,zp~:n-& f ( x )=O
I Y ~ ~ ( ? / )a=?/+I;:c)=L)] O
whe!ice. by 3.951,
J
:Xt(! (y)= '1) = k + 11 -> :E:(.c ;, IZ & f (x)= O) -- EZ(n:< n & f ( x )=O & A7;(f( y )= 0
&
!/
+ X ) = li)]
arid therefore, by 3.957,
Bi;? SYff ( y )= O ) = k+ 1 -> E;(/ ( x )=O), by 3.951, and so X ; ( ; ( y ) = O ) = k ~ - 1- > q { f ( x ) = O & N ; ( f ( y ) = O& ? j t x ) = k ) wi:ich completes the proof. 'ft hllows from 3.971 aiid 3.983 tliat
{,\írc/ ( t )= 0 )= 1 ) -e+E;{/
( 2 )= O &
++E:(f(z)=O
&
N:(f (t)= O & t # X ) = 0 ) A:(f(t)=O -t t = x ) ;
nnd so 011. Tliese equivalentes show tliat tlie operator .Y:{/( x )= O ) Iias tlie desired properties. Thus S:(f(.r)=O) has tlie value unitv if and only if there is a uniqzlr r froiu O to n for which f (x)=O. Xnd ,Vz(f ( x )= O ) has tlie valuc 2 if aiid only if theic is an x and y from O t o n, x different froni y. siich that. f ( x )= O and f (y)= O and f ( t ) does not vanish for nny otlier V R ~ U Vof t from O to n, and so on.
p
ciP
w
e@
w v
THE LOGICAL C O S P T A 3 T S
3.063
P+cl ( p v r )+ ( q v r )
3.0S3
a-tb (p+a)*(p-+b)
Examples 111 3.
Prove the formulae 3.01 - 3.05
Establish the schemxt,a 3.051 - 3.093
3.061
9 p-i (q+r) P+T
3.0611
P+P * q-tq* p&q-ip*&q*
1
I
TKE LOGICAL COXST-tVTS
3.194 4 , r
-t
{ ( ~ ' r ) - ( q ~ r ) = ~ ~ ~ )
3.2
Prove the schemata 3.31 - 3.3s
3.3
Prove the formiilae
3.31
(l~~~)L~{~(n~z)=O)=0
3.32
-
3.36
8 f f " g ( t , y) = O ) -+g(x, y ) = O
3.37
--l:{f(t, a ) = O )
3-38
E + ( f(4= a ) fk H a ) + R p ( f( 4 ) (use example 3.42 below).
I
-4:p(x) -> E:
p(x)
a A z { f ( x , u )= O ) + { f ( x , a ) = 01
Prove the schemata 3.39 - 3.392
3.4
Prove the formulz 3.41 - 3.42
3.41
E:p(x) +-E;Snp(x)
THT2 LOUIC.\L COSSTA'FTS
Pt-ove the foliolvirlg forrnulae
Establish the schemn
AZQ(X) Q(P +Sr)
q(n)
where p is anj- definite numeral. If 1= L;p(n L X ) ancl g = 71 1 prove (i) E:P(x) + p(g) (u) g
a g n ->
&
N
p(a)
so that g is the greatest x not exceediiig n for mhich p(x) holds. If ab> O prove that tliere is n ,arc%a.test vnlue of n b g x & nn. g y. Prove the scliemata 3.S1- 3.82
wliere x does not occtzr in p. Prove the forrnulae 3.S31- 3.S34
AZ(P q(x))
++
A:(P + q(x))
( P AZq(x))
++
( p -+ L4;q(x)j
Prove the schema
71
for which
THE FUNDAME-VAL TREORE>lS OF AR[TH>l\IETIC
S7
Proof of 4.02 From the introductory equations of R(a. h ) it folloas tliat cs(SR(a.,b), b)= O -+ R(Sn. b) =O and so, by theorem 3.43
THE FUNDXMEXTXL THUORESIS OF =\XITHi)lETIC 1. The notions of qltotient and reniítinríer arc iiitroduced into recursive number theory by rntla:i.; of thr rec.irsive fi:iictions Q(u, b) arid R ( n , b) whicli )ve defii:~a3 fullowa: To simplify the forrniilae \ve v-rite a(c, d) for a(lc, d l ) so thht a(c. d) = 0.1 according as c. d are equal or unequal, and define
4.021
- - h'(8"tr. b ) < /)]
x ( S R ( a ,b). b )= O -+ {O h
-
Since s. a ( x )=x therefore x =. 0 -t a(x)= 1 and since x O t,herefore
By theorem 3.43
{a(SB(a,b ) , b)= 1 )
R ( 0 , b)= o R ( S a , b) = S R ( n . b ) . x ( R R ( a ,b), b ) arid
-
([R(Acr.b ) =SR(a. h)._\(SR(a,b). h ) ] - [R(Sa.i?)=SR(a,b ) ] )
whence
u ( S R ( a .b). b) = 1 + R(Sr1. 6 )= S R ( a , b)
&(O: b) = n
&(Su,b) = &(a.b) + {l-L x(SR(a,b), b ) }
and
YO
S R ( a , b ) < b -t R(Sn. O ) =SR(al b). Tliat t,he.se functions have in fact the required properties is shonm by the following formulae:
But
R(Sa, b)=SR(a: b) +- {SR(u.b):< b i R(Sa. b)< b )
so that
Since x =y -+ a ( x , y ) = O therefore
4.023 Proof of 4.01
f ( a )=bQ(a, b ) + R ( a , b)
If theii
f ( O ) = bQ(0, b) + R ( 0 , b)= O
ant1
/ ( S u )= bQ(Su, b)+ R(Sa, b) = bQ(a, b)
+b ( i
.
Denoting the formula O < h -> {R(a,b) < b ) by P(cc),we derive from 4.02l, 2, 3 (and example 3.085)
P ( a )i P ( S a ). Since P ( 0 ) holds by definition of R(a, h). 4.02 is proved.
x(SR(a. b), 6 ) )+8R(a. 6 ) .x ( S R ( a , b), b)
= hQ(a. b) +SR(a. b).
R ( a , b) < b -+ {SR(a.,b) < b v a ( S R ( a ,b), b) =O)
by- esamples 2 . 2 7 , 2.08,
= $ / ( a ) , so that f(a)=a.
Proof of 4.03 IVriting Q, R, for Q(a. b). R ( a . b) respectirely we have. by theorem 3.43,
SS
{(a=bc+r) -+ (bc+r=bQ+R)}
a=bQ+R+
mhence, by 4.01 a=bc+r -+ (bc+r=bQ+r) ;
4.031
but {Q=Sc+(Q'-Sc)}
.- [ { b c ~ r = b Q + R }-+
{r=b+b(Q-Sc)+ R}]
THE F C T D . \ X E S T A L TíFEOILCXS OF ARITEAíETIC
Y(3
aiid r < b -+ (bc= bQ + Q = c ) , using esample 3.095, and so a=bc+r a r t b -+ Q = c usirig esample 3.0911, which completes the proof of 4.03. 4.1 It follonrs from formula 4.03 that if a= bc tlien R ( a , b) = 0. and conversely if R ( a , b) = O then a = bQ(a,b), so tliat the truth of the equation R ( n , b)= O is botli necessary and sufficient for a to be divisible by b.
and so (using example 3.061) ( a = b c i r ) + (Q>c
-+
r>b).
fjimdarly
(n=bc+r)
-t
( r < b -+ Q g c ) .
(n=bc;r)
-+
( R c b +Q>c),
and so (using examples 2.043, 3.09) (a=bc+r)
-+
{(r
( R < b ) -t Q = c } .
Since r
{(a=bc+r)
PRIME NUJIBERS
then the equation p(n)= O says that ,n exceeds unity, and every namber fram O to n whicl~exceeds isnity bzit is not n itself, does not divide n , i.e. tliat n has no divisors apart from itself and unity, so that is a prime number. Thus p ( n )= O asserts that n is a prime number. 4.21
If q ( n ) is the least divisor of n, after unity, that is, if
a ( r t b ) }-t ( Q = c ) ;
but 4.033
4.2
If p(n) = O denotes the propositional function
whence (by esample 3.06%)
q ( n )= LE{R(n:x ) = O a x > 1 )
then, using esamples 4, 4.1, ( Q = c ) -+ {(bc+r=bQ+R)
-+
(r=R))
R(n,q ( n ) )= 0 , q(n)
and so (by 4.031, 2, 3 and example 3.093)
dir L
TE€E FUNDAMESTAL THEOREMS OF ARITIINETIC
4
6.
G
4.034
Since and
{(a=bc+r) r=R
-+
a ( r < b ) } -+ ( r = R ) .
{(bc+r=bQ+R)
(a=bc+r)
it follows from 4.034 that
-+
-+
(bc= b&)} ,
(bc+r=bQ+R)
n>l
and
-+
q(n)>l ,
a < q ( n ) -+ { R ( n ,a)>O v a á l }
so that (using theorem 2.91 and example 3.063) n > 1 -+ { q ( n ) > l A ~ " ' ( n ~ 1a = ~ (vnR(n, ) a)>O)}, that is
n > 1 + p(q(n))= O
which says that q ( n ) is prime if n > l . From the formula biit
la
90
T R E FLTXDA>CE>T.%LT H E O R E J l S O F .XI:TTt~JlI-TIC
we deduce that ?i
.1 &q(n)=n
-
THE FUXDIDIESTAL THEOREJIS O F ARITHIIETIC
9t
We show first, tliat p(n) is prime for any n. Since p(n)-O.
q ( P ( k ) !'- 1)
From R ( n . n )= O and
aiid
p(q(p(E)!- 1))= 0
therefore we derive, firstl?.
aiid
p(k+-l).,p(k) . p ( p ( k - 1 ) ) = O ~ ( k -1r) < q ( p ( k ) ! - -1) ,
u((')
tliat p ( k - 1 ) is prime: but. by esarnple 4.45, is priiiic. aiid so p(n) is prime for any 71. It foiloms by esaniples 4.5 aiid 4.61 that p ( n - 1) is an odd prime. To sliow that (for n > O ) p(n) is ir1 fact tlie ntli odd prime, so
By means of the proved formula q ( n )';;n
derive
1%--r
v e shall show that every prinie is a p(n) for soriie ri. that is p ( m ) = O + EP{?)i= ~ ( k ).)
I4.E
Th?cs q(,n)= n i s a necessary nnd strficient condition for n to be a prime number.
B'- the properties of the function p u-e llave
1.3 Folloq-ing the idea of Euclid's famoiis proof of the iníinitude of primes we can readily construct a formula giving prime numbers as great as we please. TVe prove in fact that for any n, there ia a prime betmeen n and n!+l . By the characteristic properties of q ( n ) , since n ! + 1 > 1 (by example 2.82)
Let p l ( m ) = L r ( m < p ( x ) ) then since bx example 4.51 p ( m ) > m , n-e have p(pl(m))> na and ?(m) :m ;
+
+ .
p(q(n!+ 1 ) )= O and q ( n ! 1 ) g n ! 1 Next. from example 4.44 we have
( R ( n ! + l a)=O , and
&
DO
+-
whence q ( n ! 1) > n, which proves that q ( n !+ 1 ) is a prime number greater than n and not greater than n! + l .
If
{ m >1
p(x)> m ) +
and so
{m.> 1
p ( x ) > m} -i x > O
n>l)-+a>n
( R ( n ! + l , q(n!+l))=O & q ( n ! + l ) > l } - t (q(n!+l)>n)
4.4
but
n-Iience
m ., 1 + pl(m) 2 1
that is
m>i +q ( m ) = i - ( q ( n r ) ~ l ) .
TVriting ~ ( u L )= q ( n i )-I- l. so tliat m -. i
then (for n 10). p(n) i s fhe nth odd prime number.
-+
: y ( m ) . 15-e hare
~(17))
(since ~ ( m is) the first x for which p(z) exceeds m ) m :1 -- { p ( y ( m ) ): m
THE SEQUEXCE OF PRWES b ( 0 ) = 3 , p(n+ l)=L,P'"'!"{x>@(n) & p(x)=O)
>p(o)
& p(~p(m)
1)
-- n i ]
that is »? 1 + [[p(tp(m)) :m
& p ( y ( m ); 1) -
m ] v ~ ( ~ l ( »=?m) :) .
Iii this and the foilowing section it is ciesirable to rc-iiitroducc t,he coiiveiitioiial sigil by whicli to espress the procluct of a. niimber of f~ctors.For any functioii f ( x ) we defiiio (as in 9 2.95)
it follo~vstliat
Accordingly, by esaniple 3.42
4.5 Iii preparatioii for theorein 4.51 belo~v.we piuve -riext : if bd> O and R(nb,d ) = O. and if
Oiir principal result is that any niiniber inay be espressecl in one wa5- orily as a product of prime factors. JI7e st'art by considering tlie liighest power of a given prime contained in a given number. We define v(O, k )= O "(AS%, k)= L:R(sn, ( p ( k ) Y')f > 0
~,
Siilce p ( k ) :1, tlierefore by esample 4.701, (p(k) > n and so
R(b, 1) = 0 .
thieri
R(Sn, ( p ( k ) J n " )> O
We observe first tliat
ancl therefore (by t!ieoreni 3.940)
R(Sn, (p(k))"'S"."+l) >O and Let
R(ad,d)=O -+ Z g d .
i.e.
denote Q(b, l ) , R(b,1) respectively so that
4.61
d>O
B,
Q
&
+ R(n, (p(~))"'"."+l) >O
and (by 3.9493), v(n,k) < n, and
b=Pl +e aníl e
x < v(n, k ) + 1 + R(n, (p(E))") = O
+
since R(ab, d ) = O tlierefore R(al? ae, d ) = O ; whence, since R(nZ,d) = O , we have R(ap, d ) = O ; but
1.1 > O
so tliat, in particular,
(x<1 & R(ax,d)=O)+x=O
and so, from the inequality pletes the proof. 4.51 If p is prime then
e = O,
Formulae 4.61 and 4.62 shom that v(n,k ) is the exponent of the greatest power of tlie izth odd prime contained in n.
~vhichcoiilt
4.63
m>0 -+ m= 17 (p(x))"'"~") Zdm
S
For by the previous result, since p> O R(up,p) = 0, it follo~vs that R(p, 1)=0 and l ~ p and , therefore, since p is prime, eitiicr 1 = 1 or l = p . But R(a1,p ) = O and R(a. p) > O so that (1 l ) , and tlierefore 1 =p. By 4.5, R(b, 1) = 0 and so R(b, p) = O.
--
By esample 4.86, m is divisible by
17 (p(x))"".", nnd if Q is zdrn
tlie quotieiit, it follows from forniula 4.61 and esample 4.87 tlint ailcl heiice (by esan~ple4.W) that m > 0 + Q = 1.
THE FVNDAXEsTAL
THEOREMS OF . m I T I I E T I C
05
and so, by theorem 4.51 (and esample 4.82 again).
To show that factoristitioii is iinique we isolate an individiial factor iii n product 1 1 ~ -menlis of the followirig theoreiiis: if p (m.r: 0 )= 1 4.64
r < k + { J ( k )-+ J ( S k ) )
and the proof is completecl as before. x* l. -
1 1 ~ ( . rk ) ) [p(I.Ck))"("'.'). q(ni. r, k ) 2-
We have t o prove that if rn admits two resolutions into prime factors. 17 {p(x))"" and 17 {p(x))"".". s a ~ - .theii
"'
zhm
r<;m
<(m,I)= v(m, X ) for .2::.. ni. 1.A42
r -k
-t
H(rc(,n. r. ir). p(r)),O.
iVe have
Denote forrnulae 4.641 and 4.642 by "r i k -> H(k)" and "r < k J(k)" respectirely : theii r < O -t H(O) and r S O -+ J ( 0 ) i r e clenrly triie. If r =S k tllen 17 {p(.c)]'l"'.") = {p(SfY))'"n.s" 1 7{p(x))
-
and
xg
+
R ( m , {p(x))""'.")= ( 1
x
-t
m = { p ( z ) ) ' " ' ~ ~ ' ~x.( nm~).
xs nj
+- R ( q ( m ,x: n2). p(.c)).;
so that. (by esample 1.801)
and so (by esample 4.71 1). v(m, x ) 3 [(m.x) for all X G m. Similarly :(m, x ) > v ( m . x). so that [(m. x ) = v ( m . x). for all x < m . and so {r < k )
- [H(I:)-+ { 17 { p ( ~ ) ) " ~=. {p(r))"rn."q(m, ~) r, S k ) ) ]
( r ~ k-t) { H ( k )-+ H ( S k ) ) . . . . . . . . . . . . ( U ) T t follon-s from (i) and (ii), . . using - esample 2.491, that ( ( r< k ) -+ H ( k ) }-t
<S k ) -+ H ( S k ) )
((7
which completes a proof by induction of 4.641. The proof of 4.642 proceeds on the same lines;
if r=Sk t,lien
~ ( mr,, S k ) =
17 { p ( ~ ) ) " ~ .and ~ ' so (by examples zg k
4.82. 4.55)
1
{ ~ ( mx ,) = O ) -> { [ m> O
-+
R(m. {p(x))""."-'
)>o]
-+
[m > O
e-+
R(m, ~ ( x > ) )O ] )
m > O -+ R ( m , { p ( x ) ) ' ( m . z ) 4 1 ):O
and
therefore, by esa.n~ple3.064, v(m. x ) = O
-
{nz > O + R ( m , ~ ( x > ) )O )
which proves 4.71.
It folioms that
( r = S k ) + R ( q ( m ,r, S k ) , p(Sk))> O
If r *:k then q ( m , r, S k ) = {p(Sk))""'.").q(m, r, k )
e-
v
Since
Z<Sk
i.e.
F
so that. if nt > O , v(nt. x ) > O is a necessary and siifficient condition for m to be divisible by M X ) .
p
l
THE FCSD.IJIEST.\L THE0REJI.S O F ARITUJETTC
I
THE GREATESTC O ~ I O ' FACTOR T Followirig esample 3.36 we can readily define the greatest value of x which divides each of tu-o non-zero numbers a and b. FTe prefer however to proceed differently and to establish tlie greatest comnion factor as the least non-zero value of the function a x A b g . \Te shalí determine fiinctions k(n, b) and l(a, b) such that h(a, b)=ak(n,b)=bl(«. b) has tlie least possible value, not less than iinity. From tlie equatioii
l 4.71 THE GREATESTPRIMEFACTOR If l(n)=L",v(n, n 2 z) > O ) aiid q(n)= n 2 1(n) \re shaU sho~vt hat p ( g ( n ) ) is the greatest prime factor of n. For, by example 4.S3,
whence n >1
- y,
n) > O )
[ ~ l ( n ,
.
I l
4.9
I t follows. by esample 3.36. :Iiat we derive
E~(h.a~l.h=ub)
and
aiid froni this in turn we derive Since n > n -+R ( n , P(a))= 71, t,herefore n > n .+ v(n. a )= O and 4.722 may be rewritten in the forni
From this. and the inequality nb > O, follolvs
EC{c 2 1
&
Ei(EP(kn2 lb =c ) ) )
Formulae 4.721 and 4.723 show tliat p(g(n))is tlie greatest prime factor of n.
and, therefore, if
4.8 The function v(m, z ) makes it possible to assign a unique number t o any given collection of niimbers. For given a collection a,, %, ... , a,,, mith a, > O, we sssign to this collection the number given by
we have, for a, b satisfying ab > O!
E!(EB(k.a-lb= h(a, b ) ) )and h(a, b) > 1 . Hence if
k(a, b)=Li{E;(xa lb = h(a, b ) ) ), ab > O , The number N is completely determined by the given set of numbers a,, a,, ... a,, with a,> O, and conversely the set itsclf iu uniquely deterrnined by the single niimbcr N, for 3 = Q(X) z
x á g ( N ) -t v ( N ,x ) = a,, ;.e. the numbers v ( N , O ) , v ( N , l), v ( X .9,..., v(X, g ( N ) ) reproduce the given collection (where g ( N ) is defined as in 4.72).
then
E;(a. k(a, b) .-lb and so if
= h(a, b)) and
-
k(a, b) < b
l(a, b) = LU(a.k(a, b ) yb = li (a:b ) )
we have
97
98
4.01
THE FUXD IJIESTAL THEOREXS O F A R I T H ~ I E T I C
TI%
When one of u or b is zero me define h(a. b ) = a A b , so that ( l a 0 & b-0 a=O a b-:O
and
1.h-O.q=Ii(n. 6 ) .
1
,-
By example 3.193. if nb .<x, nn y ancl
h(b, n ) G h(a, b ) .
If Y > u . S : b . theii b y esample 2.41. froni the iiiequalities N b < x , N n < TJ \re derive .c > (S 1)O, !/ ( S- 1)a coiitradicting the definition of .Y, nnd so either Y í n or X; h. If S : b and I ' ; n then a X 2 b Y - rtb-bi7-b(n-1')--O aiid so
li(n, b ) 9 h(b. a )
-
1<,, + [ l ; r r - Z b = ( a l - l ) b ~ ( b ~ k ) a ~
wherice, since a ll(rc, b) < n , b L k ( a , b) g b ,
i, bin~ilnrly
n x tlien
~X=bl-=~. I
&
<-
herice if is the greatest n satisfyirig ~i1,q.r n n ~ (given ~ y by example 3.7) ancl if we denote . r 2 S h . !/ L S a by S. 1' re have
Ry esample 3.1s kgb
!/J
-
FSL--
,Y,b
-
Y ; ( I --E!EY(ax-by=c) & c - - u &c-.h
khence, from E y ; ( n x by = c ) & c > 0 +- c J h (theoreiii 4.92) we 4.94 ]?(a, d ) = O & R ( b , d ) = O +- R ( h ( a , b). d ) = O. For if R ( a , d) = O and R ( b , cl) = 0 then, denoting &(a, d ) and Q(b, d ) b y S ir~:d B respectively,
i s not zero then it i s a multiple of h ( a , b ) . lt'e rnay without loss of generality suppose that ab > O. Denoting k(b. a ) , l(b. a ) and h(a, b) b y k, 1 and h for short we have m-*Y',. "* 2 ., kb-la=h. 4.95 I f ra-sb
Let ? he the qiiotient and e the remainder when raLsb>O is dividtd by h, so that ~ < h and ,
conclude
-
(X>b
&
'C
'w'
x«-yh=~.
4 . 9 1;9 b a l < (1 & c -> 1 & í;,!:lb = c -> 1 ~ ( ab) . 5 c. For k q b l s a & k n ~ l b = c Eb,(Ef(lrn~-lb=c)) and Ei(Ef(k(i-= lb = e)) & c 1 h(rr. b ) -,c ( b y theorem 2.0493). 4.93
99
which we shall m i t e for short in tlie forni
-- l . n ~ O . b = h ( n .6 )
-
FUND4'dEYTAL TSEORE>fS OF ARITIl\IETIC
Y
If X t b, since Y b <*Ya ;ve 11nve Y < a (for b> 0 ); but
but
4.95 From the eqiiation k a z lb= h , where k, 1 and h denote k ( a , b), l(a, b ) and h ( a , L), and ab>O it, folloms ( b y esample 4.31) that Ea = h (mod b )
and so
nnd from ( a - l ) b L ( b '- k ) n = h me derive (al-l)b = h ( m o d a ) .
'IP
' íp
'e'
'w 'u' w'
cía
;Yi
w m'
Siiice La = O ( m o d a ) and ( a 1)b=)I ( m o d b ) me find, for aily r and s. (using examples 4.3-3-3, 4.323) ( a 1)br + kas = rh (niod a )
-
( a 1)br+ kas = sh ( m o d b).
-
IJi.o\-e t h e formulae 4 - 4.:;
W h e n c e , b y esample 4.91. if n denotes R ( ( n l)hr + kas. nb). n = 7.h ( m o d a ) n = slr. (inod b ) Accordingly. if a and b are relatively prime, so t h a t h= l,,we hat-e proved : If a and b are relatiaely prime and if r , s nrp rcny t l ~ onlrncbers, there is a number n c ab such thnt n =r ( m o d n )
?
72 = S
-1.
R(,1.71)=0
4.01
R(n. O ) = n . Q(rr. O ) = 0
4.01 1
R(.r. y) = 0
-
(y > O)
v
(.r = o)
(mod b) .
R ( n . bc) = O
-t
R(cr. b ) = O
R ( n . b)=O & b > l + R ( a ; l , b ) = l
-
b> 1 + R(ab+ 1 , b)= 1 If a
cd = b and b > O prove t,hat
R ( n , d ) =R ( b , d ) Show t h a t a = b - d x + a = b ( m o d d ) Prove t h e schematn 4.3.' !- 4.323 a =h (inoti d ) h=c (mod d ) u = c (mod d ) a = h (mod tl) nr =br (rilod (i)
TEiE F C N D A 3 I E S T A L THEOHESlS O F ARITIIMETIC
THE FL7lDAYE'iTAL THEOREJI': OF .\I:ITR(TJIETIC
R(m, a ) = O a R(m,pk)= O a R(u, p) > O -f R(m, apk)= O
al=bl (mod d ) a,=b2 (mod d ) a,+ a,=b,+b,(modd)
4.802
Prot-e that
4.81
R(pkypl)=O+k=l
4.82
N
4.83
m>i+EZR(m.p,)=O
4.84
Show that
Prove the forrnulae
í 7 , ( n ) is divisible hy f (;
j
17,(i.-n) is divisible bv IT,(r) If
a -; n
103
then Tí,(n) is tlivisihle by n , ( u ) .
Prove the forrnulae 4.4s -- 4.44
(k = 1) + R(PEk. pl)> 0
Prove the formulae (where p is a prime number) 4.85
A p ( f ( x ) p) > O
+R
(
17 f (z), p) r- O
r
Prove that 2 is a prime number Prore that 3 is a prime number Establish the schema
Prove the forniulae 4.5 - 4.7 1 l
4.9
If a(1)> O and A:=
17 p(x)"("' prove that, in the notation z á1
of theorem 4.7. g ( N )= l. 4.91
If b> O prove that R(a, bc) = a (mod b)
If 1) is a prime number prot-e that
R ( 2 - l . p) = O R(n. p) -> O
&
-t
R(x, p) = O
R(ab, pk)
=O
+ R(b, pk) = O
4.92
If h is the R.C.F. of a and b. arid H is the H.C.F. of h and c. prove that H is t,he H.C.F. of a, b and c.
4.93
Prove that the L.C.M. of x and y is a primitire recursive function.
FORMALIS-STIONS O F PRIMITI\S: 1WCLTSIFE ARITHMETIC
Sb, we reacii A = A. Since E = A folloas from A \-ve have proved
= B,
A
= A bj7
105
T,
CHAPTER V
FORMALISATIONS OF PRIMITIVE RECURSIVE ARITHMETIC
-4 schenin equivalent. to U is
I n this chapter Ive consider a codification of primitive recursive arithmetic in mhich the oiily asioms are esplicit and recursive function definitions, and the only inference rules are the substitution schemata -
The passage froni U, t,o U is obl-ious;' for the converse we derive
f
where F ( x ) , G(x) are recursive functions and A, B , C are recursive terms, and the primitive recursive uniqueness rule
where the iterative function Hxt is defined by the primitive recursion HOt= t, HSxt=H ( x , Hxt); in the schema U, F may contain additional parameters but H is a function of not more than two variables. I n Sb,, the function G(x)may be replaced by a term G independent of x, provided that G ( A ) is also replaced by C. The novelty in this codification lies in the derivation of the key equation a + (b a ) = b + ( a b) by means of the primitive recursive uniqueness rule. instead of requiring a doubly recursive equalising rule as before. We start by proving a few auxiliary scliemata. From t,he defining equations x - O=x, x+ O= x follows s = x by T, ancl thence by
. g(x)= HZg(0)
( 5 )= HTf( 0 )
from the stated hypotheses. by V, and Hxf(0)=H"g(O) from f ( O ) =g(O) by Sb,. whence f ( x )=g(x) follows by T and K. -Is an illustration of the use of Sb, we derive F(n. b)= F ( A , B ) from the pair of equations a =A , b = B. First ure derive F(a, b)= F(n, B ) from b = E by Sb,, and similarly F(a, B ) = F ( A : B ) from a= A ; hence by using K and T we derive F ( a , b) = F ( A , B ) from a = A , b = B . Two further schemata of importante are
To prove E,, we define Hl(x, t), C(t) explicitly by the asioms
H,(x, t )=t C(t)= F ( 0 ) , wheilce we readily derive C(0)= F(O), C ( S x )= H,(x, C ( x ) ) ,F ( S x )= H,(z, F ( x ) ) which, by El,yields F(x)=C(x), and from this we reach F ( x )= F ( 0 ) by Sb,. T . For E, we define Z(t)=O, so tliat Z ( F ( x )= ) O whence from F ( S x )= O follows F ( S x )= Z ( F ( x ) ) .This equation together with Z ( S x )=Z ( Z ( x ) ) and F ( 0 )=Z ( 0 ) yields F ( x ) = Z ( x ) ,by U1, and E, follows.
106
PORMALISATIOSS
FORYALISATIOSS OF P R I U I T I \ E RECCRSIVE ARITHUETIC
O F PRIJIITIVE R E C G R < I V E ~ R I T H U E T I ~
We establisli nevt some results for additioii, subtraction and multiplication, taking the definin; cqiiatioiis for thesc operations to be: a-O=a . u + S h = S ( a ~ b ): O ~ l = O . s a : l = ( - .~a - O = a . a ' - S h = ( ~ ~ ' - h ) - l ; n - @ = O ,n . S h = a . h - o .
]\-e use a - ~ S o = S a = S a + 0 ,a-';.Sh=S(a+Sb), aiid 1 ;.
10;
Sa-Sb=S(Sni.b)
From a + 0 = 0 +-a , a fSb = - 3 j. iising 5.07. follon-S S b -L n = b +So = S ( h + a). Then 5.l'lS f o l i o ~ ~by s U,.
For ( u - ~ ) ) - l = ( a ~ l ) : ( >( U . - S { I ) : ~ = { ( ( .L. f > ) L l l : l . 1)-Rb= { ( a - 1) 1- hj 1, illl(l the resiilt fnllows by-C;. J?-ith c as variable, apply I;,. For S « - S b = ( S a ~ l ) - b = a - b ,
using (5.01). For Sa.O=a.O+O , S u - S b - S a . : - S a . a.Sb+Sb=(rr.h - cr)tr*ib= = S { ( a . b ; a ) + b } = S { ( a . b t b ) - 1 1 ; . by 5.08. 5.10, a ~ i dso n .Sb ' S b = ( a b + b ) + S n , whence 5.1 1 follows b y t-,.
Proof by E,, using O-&=(O-1)-a.=O-a.
For (a+Sb)-Sb=S(a+b)'-Sb=(a+b)-b by E,.
-
+
so that (a+b)-b=a,
5.13
a(1-a)=O.
For 0 ( 1 ~ 0 ) = Oand Sn(1-Srl)=.Sn(O-o.)=Sn.O=O.
+
+
For ( u t S n ) ( b+ S n ) = S ( a + n)2 S ( b n)= ( a n) ( b m), and (a-c0)2(b+O)=a~b. 5.15
By 5.051 and 5.04.
a(b- c)=lz.b-a.c.
This is a consequence by El of the provable equations a(b~0)=a.b=a.b~a-O.
For O - O = O , O + S u =#(O + a ) , S a = S u , and the result foilows by U,.
n(b-Pc)=a.SíF, - c ) = u ( h 4 c ) ~ n , n.h-ct.Sr=nb-(rlr
-a)=(ah~oc)+a.
For a(b- 0 )= O = ( a .b) . O and a(b.Sc) = a(bc + b ) = n(bc)+ ab, (ab)- S c = (ab)c+ ab. \Ve prore non- aii esteiision of schenia &. so tliat / ( a '- 72, b L n ) f p(n, a, b) = ? ( O . b ) + p(0, u. 6 ) = f
f (a. b) = f ( a - b , O ) + ~ ( bCL., b ) = (a=h ) - - ~ ( ba. , b).
whence $2, follows by U,.
for froni Id,BI = O fol10n.s \ A . BJ1(B'-_4)= O by 5.04, mhelice b y 5.05, A - E = O, and sin~ilarly.B = A = O ; from these we reacli
Plor ( 1 - O ) b = b = b - ~ . b , ( l - S ~ ) b = ( O ~ - c r ) b = Oand , b ~ S a . b = b - ( b f u.b)=O. Next r e prove the key equation
+ (b- a ) ,
and therice A = B, by 5.17. The derivation of [ A .B [ =O from A = B is of course trivial. We come now t o some induction schemata. Let P ( x ) denote the eqtiation f ( x )=g(x). The familiar induction schema is
so that f ( u , O ) = u , f(0, b) = b,
f (SCL,S b ) = S f ( a , b) ; and define g(u, b )= b + (a: b) so that da,O ) = a, g(0, b ) = b, g(Sa.,S b ) =Sg(o, O). JVe start by proving By
6 ) . wheiice
Similarly g(n, b) = ( a-1- 6 )+ y(b. a , b ) whence equation 5.17 follows. 1%-ederive from 5.17 tlie schema
f ( 8 ~= 0) . f ( x )+ f ( S x )= H 2 ( 5 f (x)),
Define f ( a , b ) = a
((1.
or va-iting p ( x ) = 1f (x),g(x)(
E,,
= ( a ) - { ( ) whence / ( a , O ) = f ( a - 1 , O)+ ( 1 ~ (i a l) ) which establishes 5.171 ~vithO for b. 1Vith Sb for b, 5.17 1 becomes
As in the proof of 2.8 we define q(0)= 1, q ( S n )= q(n)( 1 -r-p(n)); then q(SSn)=q(Sn) ( 1 .- p ( S n ) )=q ( n ) (1 2 p ( n ) ) ( 1~ p ( S n ) )
f ( a ,S b ) = S f ( a = 1. O ) which is a corisequeilce of the equations f(0. Sb)=Sb=Sf(O, b),
f (SU,S b ) = S / ( a , b ) , completing the proof of 5.17 l . Kext we define e
/ ( a - n t b 2 n ) - p ( n . a. 6 ) = f ( n - S T Z ,b - S n ) ~ y ( S i i .( 1 . O ) : in fact. by 5.171
where the last equality sign holds according to hYvpothesissince
Iience q ( S n )= q(S0)= 1, that is q ( n ) ( 1=p ( n ) )= 1, and multiplying by p(n), p(n)=O, by 5.13.
is valid. -4s particular cases of 1,. 1, me note that from f ( a , O ) = 0 , f (O. S b ) = O and f ( S n , S b ) = O folloms f ( a , b ) = O ; from f (a. 0 ) = 0 , f (O, S b ) = O, f ( a , b ) =! ( S a . $15) folloms ! ( a , b) = O ; nnd from q ( a . O ) =yi(a. O ) . p(O, S b ) = !?(U.S b ) . ~ ( ( 1 ,b ) = p ( S o . S b ) and y(a. b ) =y ( S a , S b ) follo~vs~ ( r rb. ) = y(a, b ) ; for if n-e denote Ip(ct. b), ~ ( ab)l, by ( a , b) theii f ((1. O ) = O. j ( O . S b ) = 0 : and from q ( a , b ) = p ( S a . S b ) , y ( n . b ) = y ( S a . S b ) follows ! ( u . b ) = f ( S n . S b ) ; mhence f ( t i , b ) = O and so g(cc, b ) = yi(ci. b ) . -4s instaiices of this last sclieina \ve meiitioii
\Ve observe first t,hat fmrii f ( O . O ) = O. f (O. S!))= 0 follo~vsf ( U . h ) = O. The irnplicntion hypotliesis stniitls for the c:iiintioii
Non- { l
f (O, b - 1 ) ) f ( O . h )
=;
(1
aiid fruiii
-
{ l = f ( o , h.- 1); ! ( S u . b) = O .
To complete the constriiction of recursive aritlinictic there reniains oiilj- to prove tlie siibstitution theorciri
tlierefore { l =!(u: 1, b
1): ! ( u . i,)
11
hnti so
This iu :.tbarlily derived fronl tlie equatiuii
( 1 ' - f ( a - ~ n , b ~ S n ) ) f ( a - n h-n)=O. .
( 1 :Ix. y j ) F ( . ~=) ( 1 -1x. y / ) F ( y ).
Esactly as in 2.63, we stnrt froin tlie equntioii
To this end we prove [ l ~ ( a ,fb ) { l = f ( a ~ nb ,~ n ) }f (]a , b ) ( lf ~(a:&,
b2Sn))=O;
1
which is proved by applyiiig E2 with z as variable, and derive
1
u i t h p, q, r standing for f ( a , b), f ( a = n , b =n)and f ( a = S n , b -Sn), respect'ively, the left hand side of this equation has the form
{ l ~ p ( l : q ) ] p ( ~l r ) { = ( 1~ ~r ) ~ p ( l( l~ ~qr ) ) = p { ( l = r ) = p ( l = r ) } , since q(l:r)=O, = p ( l = r ) (l=p)=O whicll completes the proof of (j) by 1, (the validity of (j) with O for n being evident). By writing [ ~ ( ab),, y(a, b)l = f ( a , b ) it follotvs from I2 that tlie sciiema
aiid rnultiplying by 1
lx, y\, me reacli
( 1 :Ix, y l ) F ( y + ( x L y ) ) = ( l : / x , y l ) F ( y )
since
-
similnrly ( 1 1s. y l ) F ( x ( y2 x))= ( 1
d a , 0 )= y @ , O), ~ ( 0S b, ) = d o , S b ) , the required result follon-s by T.
- Ix,
jF(x) aiid since
FORUALISATIONS O F P R I X I T I V E R E C U P . S I \ T .\RlTHUETIC
F O R U I S A T I O N S OF PRIMITIVE RECURSILE ARITHXETIC
We call t,he foregoing formalisation of recursive arithmetic system G?.
this of course is a consequence of Sb, itself. For Sb2 we hare t o prove the validity of the derivation
112
The Deduction Theorem If the equation A = B is deiivnble iti J' fiom aii hjpothesis F = G . (Le. an unproved equation) aiid if tlie deriration does not inrolve substitiition for the rariables in tlle hypothesis, theri
-1
is provable in .%.' We multiply each equatiorl of tlie derivatiori bi. 1 The hypothesis becomes the prored equatiori
iT.GI.
(1-IF,GJ)F={l=IF.GI}G
113
To this end we remark that, since RI A. BI = 1 RL4.RBI, by equations 5.15 and 5.18, therefore
( R d = R B ) -+ ( A = B) v (R=O) , (F(A)= F ( B ) )-t ( R .F ( A )= R.F(B)) ( R = 0) + (R.F(A)= R.F(B)) , and by the substitution theorem whence. using the schemata
land the final equation becomes
from which we may derive
which follow from the provable equations
we prove If P=Q is a proved equation then for any function R,
is a proved equation, and so multiplication by 1 1 F, Gl turns a proved equation into a proved equatioli. We show next that multiplication by a factor does not ixivafidate an application of any of the schemata Sb,, Sb,, T and U. Por Sb, --e have t o prove that
is valid wlien the factor R does not contain the variable s. and
(R.A= R.3) + {R.F(A)=R.F(B)) , and so, (taking 0 = 0 for H in the second of the above sohemata) = R.F(B) folloms from R.A = R.B. we see that R.F(A) Por T we have only to prove the schema
and this follows by T itself. It remains t o prove that an application of U, remains valid under mutiplication by R, i.e. that the derivation R. F(0) =R.G(O) R.F(Sx)=R.H(x,F(x))
is valid. when R cfoes :iot contüiii the variable x. \Te start by proving tlie achema
by niiiltiplyiiig each equation iii :he derivation from these hypot heses by
( l L ] F 1G , , ] ) ( l = l f i . G?l) (l.-/&.
c:,l),
and so on. By Sb,,
Red~tcct.ionof Scherna U TVe consider f i s t a system 2:' n-ith Sb,. Sh,. T. the schema
aiid the asiom niid, in place of tlie familiar introductory eqiiations of tlie predecessor functioii, the axiom
aild the desired derivation follows by the schema
l
P which is also proved by Sb-. From the formula (X.a= 1;b) + { k J ( a )= M ( b ) ) , whicli is pioved above, follows
whence, by the given hyypotheses and the above schemata,
and tliis, ~ i t htlie fist hypothesis, proves R.F(x) =R.G(x), by inclcctioii schema I,, and the dediiction theorem is proved. Tlie decl~ctiontlieorem holds for any number of hypotheses. Fcr instance given a derivation of A = B from two hypotheses Fl=(;r,, F2=G2 me obtain a proof of the implication
Sa=Sb=a-1.b.
Tlie axiom a -i (b a ) = b + ( a b) enables us to deduce a = b from a-b=O and b l a = O . For by Sb,.
and from a i - ( b l a ) = a , b + ( n l b ) = l , and cr+(b.-a)=b+(alb) follows a = b. Derivation of a = b from a l b = 0: b n = O we call schema A. To prove schema &, namely
we define @ ( x )= F ( x )A F(O), then @(O)= 0 aiid
-
@(Sx)= F ( S x ) F ( 0 )= F ( x ) F(O)= @(x)
whence @ ( x )= O by E , that is F ( x ) F ( 0 )=O. Similarly F ( 0 ) - F ( x ) = O and so (by scliema -4)
by multiplying each equation in the derivation by the factor
F(X)= F ( 0 ) Similarly we discharge the hypotlieses F, =G,, F2= G2, F
3=
G3
which coinpletes the proof of schema E,.
116
J!ORSULISATIONS
OF PRIXITICT. RECURSICX ARITHSIETIC
MTeturn now to a reconsideration of the equations and schemata prored in 9. Equat.ion 5.01 we leave to the end. Equation 5.02 is riow an asiom. The proofs of 5.03 ancl 0.05 remain unchanged.
The proofs of 5.051, 5.052 remain unchanged. and from 5.051 it follows that la t n, b - nj = ja. bl
Proof of 5.10
Proof in &'* of 5.07
I C L + ( b+ 0 ) . ( a + 6 ) 01 O. In + ( b -! Sn). ((L + b) i S n l = ltr - ( h - n). ( a -:-b ) -- nl . etc. -
(a+SO)-(Sa+O)=Sa2Sa=O.
+
( a + S S b ) 1( S u +Sb) =S(CL j S b ) - S ( S a 4-b )= (a T S b ) ( ~ Sa b),
proving ( a+ S b ) 2 ( S u+ b ) = O. Similarly ( S u + b ) 1( a t S b ) = O wlience 5.0'7 follows by schema A.
-
=
Proof of 5.11 Since S < r . S b = S a . b ~ S aand
Proof of 5.06
I
.
n.Sb+Sb=(ab+u)$Sb=S{(ab-a)cb) (0fSa)-Sa=S(O+a)-Sa=(0+a)-1-a
=S((ab-6)+a)= (ccb~b)+Sa
so that ( O + a ) ~ a = ( O + o ) - O = O . S i m i l a r l y & ~ ( O t S a ) = a ' - ( O + a ) whence 5.06 folloms by schema A.
therefore ISa.Sb. n .Sb +Sbl= ISn .6. a.b + bl , etc.
Proof of 5.08
The proof of 5.12 remains unchanged. (a+0)-1-(O+a)=a2a=O,
so that ( a + b ) ~ ( b + a ) =whence(b+a)-(a+b)=Oanda+b=b+a ~, follows by A.
For 5.14, 5.15 and 5.151 we note that la.Sb, (Sb)aJ= Iab-a,
bu-a.l = lab, bal,
la(b+Sc), ab+a.Scl =la(b+c)+a, (ab+ac)+al =la(b+c), ab+acl,
+
+
la(b.Sc), ab.Scl= la(bc) ab, (ab)c abl
The proof of equation 5.09 remains unchanged.
= la(bc),(ab)cl : etc.
The schema
Proof of 5.04 a t ( 0 - a ) = O + ( a ~ O ) = a , therefore { a + ( O ~ a ) ) - a = a 2 a = 0 , whence, by 5.09, O-a=O.
is proved by means of the equations 5.09, 5.04 which yield
( F ( x )+ G ( x ) ) ~ G ( = x )F ( x )
The schema and
To prove schema E, follows by two applications of schema E and two of schema T.
0 - G ( x ) =0
.
118
FORJl.\LISATIOSS G F P R I > I I T I F T l:?:C'f~ESI\-E
.\ I:ITHJIIX"TC
we define @(O)=O, O(Bx)= O ( x )4F ( x ) so that rP(SSx)= @(Sx) wheiice @ ( S x )=@(SO)= and so @(x) A- F ( z )= 0 wlietice F ( x )= O. The proof of 5.13 follo~x-Sby E, iii :%'* as ir1 2, and the proof of 5.16 reiiiains iinchanged. So, too. tlie proofs of tlie iiiductioi: schem:tta 1,. 12. 1, aiid the siibstitotioii schema carry straight througfi froni .# to &*. It remniiis oidy to pro\-e schema tT1,
Define (li(z)= 1 F ( x ) .Cr'(x)l. theii # ( O ) = O ztrid by tlie substitution scheina ! ( 1 ' @ ( x ) ) / H ( t f. (s)). H ( t , g(x))l=0 wheiice [l ' r l i ( x ) ) J H ( x .f (x)).H ( x , 9(x))1=0 aiid so ; 1 @ ( x ) ] @ ( S x=)o
.
Course-of-values Recursion There are many kinds of reciirsil-e ílcfiiiition whicli. tliough seemingly very different froni printitive reciirsioii. iiiay in fact be tiansformed into a primitive recursioii. Consider, for instance, the sequeiice f (U). j ( 1 ) . f ( 2 ) . ... in which f(O)=a, f ( l ) = b . f ( 3 ) = f ( l ) - i ( o ) = u ~ b .j ( : I ) = f ( ? ) - f ( l ) = n - 2 b . and so oti. The general law of tiie sequence is f ( n- 1)= f ( 1 1 ) 1f ( u 1) so that f ( n T 1 ) depend.;. iiot jiist on f ( n ) biit also on f ( n i l ) . To show tliat this sequeiice j o l ) may also he obtained by prinlitive recursions alone, we introdiice t he funct ion 6.
-
whence @ ( x )= 0 and therefore f ( x )=g(x).
-
The proof of eqiiation 5.01 in 9 is therefore ralid also in g*. We nlay 013viously take the equation ( a O ) L.1 = ( a 1 ) b as a n axiom in W* in place of S a 2 S b = a l O. The system 9*may be fiirthcr niodified by taking the schema
so that f (n)=v(g(n),n ) , the esponent of tlie greatest power of tlie prime p, which divides g(n). 1). so tliat Let y(n, b) =v(b, n ) i v ( b .
y ( n , g ( n ) ) = f í n ) - / ( n l l ) = f ( . n +1 ) I t follows t,liat
in place of asiom A, provided tliat we add the axiom o-l=O.
For by S and Sb, alb=O,b:c=O cz=b
TO prove O = o = O \lre use 0 - S a = ( O . - n ) - l = ( O ~ l ) ~ r ~ = O ~ u , ar:d 0 - 0 = 0 (iii place of axiom A). Fiiiaiiy tlie eyilibtioti u (b- a ) = b + ( a - b ) is proved exactly as in d.
-
SO that g(n) is primitix-e reciirsive and tlierefore f (n) is primitive recursive. The function I7 prr' is cnlled tlie cou,ae-of-rctlue function of r
f ( n ) . and a recursion in \~rliichj ( n 4 1) depends not jiist on f (n) but also on the ~ a l u e sof f ( r ) for valiies of z < n, is called a course-of-values recursion. The foregoing method of transforming a course-of-valucs recursioii into a primitive recursion is of general applicability. Thus if P(n. b,. tl. ..., b,) is a priniitive reciirsive
1-0
REDUCTIOXS TO PRIXITIFE RECURSION
function of k + 1 variables and if &(n)is primitive recursive and satisfies A,(n)g n for r = l . 2 . ..., k, then the coiirse-of-values recursion
f (n+1)=B(n. f ( l i ( n ) )f, (;lz(n)),... , f ( U n ) ) ) is transformed into a primitive recursion by means of the courseof-values function fin) g(n)=p~'".p:'".... .pn ,
To determine f ( n - 1, a) from the second of these equations \ve need to know the 1-alue of f ( n , z) not just for x = a , but for the value y(n, a ) of x, which of course raries witli n. The method of traiisforming this recursion (due to R. Péter) is of considerable interest apart from the present application. If we calculate in tiirn the values of f (n,a) for n = 0. 1. 2. 3. and so on, we determine the sequence of terms a>y(O, a ) , y(0, ~ ( 1a ,) ) ,./(O,y(1. y(-, a ) ) ) .
Por if y(n, b)=B(.n,v(b, 17,(n)),~ ( b&, ( n ) ) ,... , i ( b , &(m))),80 that y . .. is primitive recursive tlien
+
so that g(n)is defined by primitire recursion and f ( n 1 ) is obtained by substitution from y and g. Similarly, if í.(n) is primitioe recursive and
then we take g(n) as above and
and so on. which are formed by repeated substitutioii for the parameter. The essential idea of the transformation is to disentangle these substitutions by means of a function y ( n , a ) with the following property : for any n we can find p. q and for any p, q we can find n so that
With a suitable initial condition, like y(0, a ) = a , this function transforms any term like y(0, y(1, y(2, a ) ) ) successively into ~ ( 0~ ,( 1 ~9 ( 2Y(O3 , a))))~ , ( 0~ , ( 1Y(h,, , a ) ) ) ,y(0, y(h,, a ) ) . and finauy into v(h3,a ) , for appropiate h,, h,, lb,, so that if y can be defined by primitive recursion, and also the auxiliary function h,, then so can f (n,a). To construct such a function y(n, a ) we observe that, for any n, n+ 1 is expressible in only one way in the form 2*(2q + l ) , where in fact p =~ ( n1, 0 ) and q = [ ( n+ l)/2'+ '1, so that p and q are primitive recursive functions of n (and of course 12 is primitive recursive in p ancl q). IVe define
+
$0
that y(n, g ( n ) )= f (n+ 1) and y(n. o(n))
s(n+ 1)=s(n).P,+, as before. 6.1
RECURSIOX WITH PARAMETER SUBSTITUTION
Another recursion transformable to primitive recursion is recursion with parameter substitution. As an example of such a recursion we consider
(where p, q are the functions of n just introduced); since q t n f 1 this is a course-of-value recursion so that y is primitive recursive. To complete the transformation it remains t o show that there is a primitive recursive k ( n ) such that f(n, a ) = y ( k ( n ) ,a ) : first however we introdiice g(n, a ) such that
TO determine g(n, a ) we consirler the relatioii
yj(g(n- - l . S ) ,u ) = y(n -- 1 , ~,LJ(s. a))
=;'(P. ~'('1, ~ ( 8a. ) ) ) = ; ' ( p , 1/l(!~(q, $1, ~ 1 ) ) =7p(2P(2g(q, S ) -- 1 ) . a ) ~vhiclireveals tlie definitiori by coiirsc-of-1-alues reciirsioii
q(n L 1. ,y) = 2L' ' ' .?(y. .S)
-:- '2" , ,
,.
the definition of g being conipletetl by t'akirig g(O, S ) =s. Similarlj-, to det,ermiiie k ( n ) we considcr 1pt.'(k(71+l), (1) = f
I
( n - l. a)=f
( 7 ~ y. ( ? ) .C I ) )
= ~ ( 1 ¿ ( 7y(%. ~ ) .a ) )
= y ( k ( n ) ,y ( i n .a))= y ~ ( g ( k ( n ).>"), . a)
1vhr.iice k ( n ~ 1=)g ( k ( n ) Y), , which togetlier ~vitlitlie initial condi:ion k(0)= 0 is a primitive reciirsire defiiiitioii of the fiinction k(n).
S~IULT-LNEOUS RECURSIOXS -1s a final illustration of a n iiidirect definition of a primitive rc2ci:r~ive function me consider tlie sirn~~ltuneous recursions
nliicli. togetlier witli tlie iriitial coiiditiori //(O) = 1 . coiiipletcs tlie primitive recursire tlcfiiiition of i r ( n ) . i i i terrii:, ;f tlir $\-en fiiiictioiis P ( u , u ) . Q(u,T). 6.3
GESERALISED INDC'CTIOS SCHEJIATA
6.31 We consider riow sonie generalisatioii~c-if i~idi:ctioii wliicli are provable in .%'. The first of these is tiie sclicnlii
P(f (n. n ) , n ) -> P(r7.S I L ) P(a. n )
62
w $(%reP. Q are primitive recursive. Hr ~igain\ve introduce an auxiliary function
so that
f ( n )=v(h(n),0 ) : g(n)=v(h(n).1 ) .
Tt remrtiiis t o show that h(n) is primitive recursive. 11-riting p(x) = P ( i ~ ( sO), . V ( X , 1 ) ) . q ( x )= Q(v(x,O ) , V ( X . 1 ) )
This sclienia has aii obvious affinity with reciirsioii mitli parameter substitution, and tlie proof of the schenin drperids in essense upon tlie reduction to primitire recursion of a rrciirsion with parameter substitiition. I n the following proof howel-cr this coniiection is iiot brought out, tlie reduction liarin,a bcen made in advance. I n preparation for the proof \Te shall establish the eqiiations (1,) S n á O - + n = h - S ( b 2 S n ) . (1,) S n , h - - + S ( b - - S n ) = b l n , and tlie proof schema
Tlie eqiiat,ioii (1,) iu evident with 0 for b : n-ith Sh for b the eqiiation becomes
n-liich follo\r-3 by tlie substitiitiori tlieoi.eni (2.13) froni
gi
Cs
1 24
b 1- (bA n)= n
-
REDUCTIO'IS TO PRIMiTIVE RECURSION
ItEDUCTIOXS TO PRIXITIVE RECURSION
02.: b), (esample '7.4'71). For (1,) we have 9 n < b-% b = S n + ( b = S n )
-
12.;
6.32 The single premise in hj-pothesis (j) in schema 1, may be replaced by a sequence of premisses. in fact a sequence of variable length, leading to the schema
+b = n + S ( b ~ S n )
- -
-
b 2 n =S(b-Sn) .
For tlie schema we note that the hypotheses are equivalent to p v q and p v r v S and so iniply p v S v (- r & q) which in tiirn is equivalent to ( q -t r ) -t ( p + S ) as required. To prove schema 1, we define g(n, n. b ) by the primitive recursion
d o , a , 6 )=a From hypothesis (i) follows P(g(b,a , b). 0 ) and thence
o < b + P(g(b o. n , b), O )
(k')
p ( a . n)
(The liypothesis (j') is the conjiinction of k(a, n ) terrns
P(f,, n ) a P(f,, n ) a ... a P(f,. n ) n-here f , stands for f (r?n. n ) ) . F(a,n ) =
1 f ( x , a , n) < kío. ni which by esample 3.103, satisfies r
x g k(a. n ) + { f ( x , r l , n )F(tr, ~ n ) ).
From (1,) and the substitution theorem (3.43) we h d
Sn
P(a, 0 ) A:("."'P(f (c.a, 71). n ) -+ P ( a ,Sn)
To prove schema 1: we introduce first the function
g(Sn. a. b) = f (g(n.a. b) , b - S n ) .
e) follows.
1;
(ir) (j')
-+
Nest we define
{ f(g(b1-Sn,a , b). n)=f (g(b'Sn. a. b), b ~ S ( b z S n ) ) )
G(0, n)= F ( 0 , n) G(Sa?n )=G(a, n ) + F ( S a , n)
and so, by (1,) ,
S n g b + {f (g(b- S n , a , b), n)=g(b n , a , b ) ) . But by hypothesis (j) P ( f ( g ( b z S n , a , b), m), n)
-+
P ( g ( b L S n , a , b), S n )
from which it readily follows that
F ( a , n)a G ( a , n ) and G(a, n)á G ( S a , n) From the second inequality, me derive, by induction over m,
whence we derive
S n G b -+ { P ( g ( b z n ,a , b), n ) -+ P ( g ( b ~ S na , b), S n ) )
and hence, taking b L a for m and using the implication
and since S n g b -t n g b, it follows by schema (m) that
a
~ -tb (a+ (b-a)=b)
{ ( n a b ) -+ P ( y ( h ~ na , b), n ) }-+ { ( S n G b ) -+ P ( g ( b 2 S n , a , b ) ,S n ) )
v e derive
which in conjunction with (n) completes an inductive proof of
a G b + {G(a,n )gG(b, n ) ) . (1') Let p(a, n) be the representing function of P ( a , n ) and let &(a, n) denote the equation 2 p(x, .n)= O so that the equivalentes
n á b - + P ( g ( b = n ? a ,b ) , n ) and substitiiting b for n we derive n ( n -t P ( a . n) whence (k)follows, since n g n is provable.
zda
&(O, n )
Q(a,n ) 4 A: P ( x , n) P ( 0 , n) , Q(Sa, n ) tt &(a,n) a P ( S a , n ) hold.
It f0llovv.j that Q(a. n ) + P(u, n ) and Q(&, n ) -- Q(n.n). R r considering in twii the ctises mith O for a and &'afor n. wc readily prove ()(u, :i:itl hciir:c,
n)
\
by i~iductionover
Q(a
l.
il)
Finally, froin the eqiiiraleiices
and hypothesis ( i r )we derive
171
'rt~ki~lg (i .- b for in aiid iising tlic iinplicatiori b i n + ( 1 2 -l ( a A b ) = b}, :\.e dcri vcb
which proves &(a, O ) . From Q(n. 0 ) and (n')\ve cleriv? (k') by 1,. proving the schema 1;. 6.34 S!ie Imt genernlisatioii of iiidiictiori ~vliich\ve consider is the schenla
(11") (i")
J x 9 1:(b,n ) -t { f ( z ,b, n )9C(b, n ) ),
I > ( ( 2 . I?)
.
(j")
P(f(0.7 2 ) , 71) -+ P(0,S n ) P(f(Sn. n,).n ) {P(cl. Sn) - P(S,i.S n ) ] --- . - -
(k")
P((1.n )
-
i T e start 1 ) ~ - definiilg
g(0. v ) = f(O. ,¿) g(Sa, 11) = g(u. )1) + f (Sa. 7 1 )
n ) -> {Q(G(b, n ) , n ) -t Q ( f (x,b, n ) ,n ) )
XG
from wliich me derive iil turn
x 9 W ,n) -+ {Q(G(b,n ) , n) -+ P ( f(2,b, n ) ,n ) )
from \rhicIi it follows that
and
Q(G(b,n), n ) -+ AE'b.n'P(f(x,b, n ) ,n ) and
tlic:li,
by hlvpothesis (j'), Q(G(b,n ) , n ) -+ P(b>Sn)
whencc., by (m'),
b G u -> (Q(G(a,n ) ,n ) ; . P(b,8 n ) ) from wliich follon-S in turn
(nr)
Q(G(n,.12): n ) -> Q(a, Sn)
2nd hence
b
Qc.
71)
P(n, n ) .
The nest -tep in tlle proof is to cstablicli tlie implicatioii
-
Q(g(tr.8 ) . $2) Q(n. S n ) . (m,') The case a = O is hypotliesis (ir'). To complete ail inductive proof of (m") wc use the scliema
+
128
REDCCTIOh-S TO PRI>IITI\E R E C U E S I O S
-
-
which is proved by observing tliat ( A a ( B += C ) )-t (C a D ) is equivalent t,o d v ( Ba C ) v (C a D ) , and tlie equatiori
follows from the equations (1:n)b = O , ( 1 :a)d = O. We take Q(g(Sa,n ) , n) , Q(g(u,7t),n ) for A . B and &(a,Sn), P(f (Su,n), n ) for C, D. The implication A -+ B follo~vsfrom (1") since g(Sa, n )>g(n. n ) and similarly from f (Sa,n )
Q(g(Sn.n). n ) + Q ( f(da, n ) ,n,) ; but
Q ( f (Su,n). n ) + P ( f(Sa,n ) ,n,)
and so
&(g(Sa,n ) , n ) + P ( f(Sajn ) , n )
l i.e.
-
d+D.
From A + B , A -+ D we derive { A &
In this section we define by recursion the processes of t,ransposit~ion and permutation, and apply the results obtained to prove that the sum of a variable number of terms
a ( l ) l a ( 2 ) +... + a ( p + l ) is independent of tlie order of the ternis. K e coinmence by defining iii terms of a ( k ) tlie function t ( k ,n,) by the recursion
it follows by induction orer r thnt 6.41
~ ( pq ), + t ( p + q + l . r ) = t ( ~q + . r+ 1).
Denoting 1 :(l -I-z) by a(x)we define
(&+ C ) )-+ (C a D )
a(p,q)= d p , q L p).n(Sq-I-p ) so that
But
&(a,8%) P ( f(Sa, n ) , n ) -t &(a,S n ) & P(a, S n ) a P(f (Su, n ) , n ) +
P(a, S n ) a P ( f ( S a ,n ) ,n )
Q(a,S n )
-+ P(Sa, S n ) & &(a,S n ) , by hypothesis (j") -+ &(Su,Sn).
Thus we have proved
and
o(p,p+q+l)=a(p, p L q ) - t u ( p f q+ 1).
'CTe observe now that
a(p,p+q)+a(p+q+ 1 , p-q+r)=o(p, p+q+r); for with O for r the equation is evident. and with r +- 1 for r the equation becomes 6.41. Nest we prove
which is equivalent to Gnsider first the equation with O for q, i.e. and this completes an inductive ~ r o o fof implication (m"), From &(a,O ) and (m") we derive &(a,n ) by 1, and from &(a,n) we derive P(a, n ) , proving the schema 18.
this equation is evident with p = O and witli p + i for p it becomes
130
REDUCTIOSS TO PRIXITIIT
RECGRSION
+
which is equivalent to r ( 1 , p+ 1 ) = r ( 1 . P ) U ( P + 2 ) "hich f~llows from the definition of T . With g + 1 inst.ead of q. (atid using 6.121). equation 6,42 is equi-
It follows esactly as in 6.422 above that
.xhicli follows from 6.41. and this completes the proof of 6.42.
We prove nest the three equations
6.422
It follows that
6.47
o*(,.-- 1. q+1.
7
+
rl - 3 . r - q - p - ? ) = t 7 ( r - y + 3 .
Consider first 6.45. Ti~t.rcliiation nith O !or equation with r+ 1 f i ~ rr is equiralent to
r t q + p f 2)
r is evjdent; the
A.43 To define an intercliange of a ( r ) with a ( r + s ) we introduce the fli~iction
To prove 6.452 \ve corisirier the equntion
O(r, S , k ) = [ k + { ( r + s ) - k ) I k , ~ I * ] L ( ~ : T ) ( T +kl* s,
which we denote hy P ( k , r). Then P ( k , O ) is evident since
Fiirther, since
Hence if b(r, S , k ) = a(B(r,S , k ) ) then
and t ( 1 ,r + l ) = t ( l , r)+a(r+2)
TO f ~ r mthe sum ~ ( 14 -) 4 2 )+ ... + a ( n ) with a ( r ) and a ( r + s ) tra,ns?osed we introduce t * ( r , S , k , n ) and a*(r, S , p, q ) by the equations
I
therefore P ( k + 1 , r ) -t P ( k , r+ 1)
and P ( k , r ) follows b y the generalised induction schema 1,. Hence P ( 0 , r) follows, and this is equation 6.451, which completes the proof of 6.45. The proof of 6.46 proceeds along similar lines; mith q=O the
132
equation is eviclent. With q - 1 in place of q the equntiori is cqiiivalent to 6.461
t * ( r + 1, q + 2 . r - 1 2 , q ) = t ( r i - 2 . y )
siibstitnti~igr L 1 for r we derive and licnce, by the siibstitutioii theoreili. using the equation ( l A r = ( > )+ R ( r . q l 1 . r - 1 1 - q ~ p ) .
t * ( r + 1, q+E12, r 4 2, q ) = t ( r J - 2 , y).
Siibstitiiting s L l for y it follon-s tliat {(l's)=O}
\Ve denote 6.462 by R(E,q). ITe readily establisli
R(lc,0 ) and R ( k
1. q ) + R(l:, q
+ 1)
from which R ( k , q ) folloms by I,, comp1ctir.g tlir proof of G.46. Tlie proof of 6.47 is rather simpler. With p = O the equation is evident ; wit h p + l for p tlie equsttion js equivalent t o t * ( r + l , q + 1 , r + q + 3 , p ) = t ( ~ - t q + 3 p) ,
and this is proved by induction orer p. From 6.421, 6.431, 6.45, 6.46 and G.47 follows immediately
-t
( ( l A r ) = O + R ( , . . s. r f ( s ~ l ) - p ) : .
Finally siibstitiitiiig n L { r + ( s L 1)) for p we derive { ( r l ( s : l ) ) ~ n = O ')- [ ( l >s)=O -> { l ~ r = O -t R ( r . s. n))]
which is cquivalent to 6.482. Equation 6.482 prores that a sum of n terms is iinchanged by interclianging any two of them. Siiice a pcrn~utationmay be regariled as a result of repeated transpositions. i t reinains only to define reci11.sively a sequence of transpositions. Given three fiinctions r ( n ) . s ( n ) . and a ( n ) n e define B ( 0 , k )=a ( k ) C ( n-F 1, E ) = B(n, O(r(n).s ( n ) ,1:)) so that by 6.1, B is primitive recursire. For each n. B(n- l . E ) is a transposition of E(n. 8 ) such that
since b(r+l, q+1, r ; l ) = a ( r + q + 2 ) , b ( r + l , q f l y r f q f 2)=a(r+1)
and for any j, k , 1, m, n
Equation 6.48 is equivalent to
and, as we shall now prove, equivalent also to 6.482
1::s
1 - - ( r 1~) = 1 . - ~ (~1 r ) .
wliich follows from 6.462
KEDCCTIOXS TO PRIV~TIX-E ~LE('L-RSIOT
REDUCTIONS SO PRI3IITItLE EECURSIOX
{r>1 & s > 1 & n > ( r + s ) ' l }
4
B(n+ 1, k)= B ( n , k ) if k f r ( n ) ,E + r ( n )+ s ( n ) , = B(n, r ( n )+ s ( n ) ) if k = r ( n ) if k = r ( n ) +s(?E). = B(n, r ( n ) ) Define the sum function T ( n ,k, p) by tlie equations T ( n ,Ir, O ) = B(n, k ) T ( n ,k. p + l ) = T ( n , k , p ) - B(n, k + p - l ) , then taking B ( n . k) for a(]:) and B ( n ~ 11;) . for b(r, s. k) with r = ~ ( n s) = . s ( n ) in the proof of 6.482. we find
{ t * ( r ,S , 1, n ) = t ( l , n)).
1 ' ( n + 1 , 1. p ) = T ( z . 1. p ) T ( n ,1. p ) = T ( O , 1. p ) = t ( l , p )
Denote the equation t * ( r , S , 1, n )= t ( l , n ) by R ( r , S , n).
and so
Then, by 6.481, R ( r + 1, q+ 1, r+q+ p + 1 ) holds, and therefore
whicli shoms tliat the siim tr ( 1 )- a ( ? )- ... + a ( p+ 1 ) is u~ichaiige(l by sny rearrarigement of its terms.
R ( r + l , q + l , r + 1 + q + l + p ) holds;
ELIMINATION OF F 41: \JlETEItS
135
We start by showing how a single paraineter may be eiiminated. Consider tlie recursion 7.1
ELIMIX.\TION OF PXRAMETXRS 7. In tliis chapter me iindertake the elimination of the parameters in tlie sctieina of definitiori by recursion to wliich we referred iri
in which there may be parameters additional to tlie u and I'iiidicated. We proceed to derive this fiinctioi, f froni fiinctions defined by recursion in fewer parameters. Let the function F(w, x ) be defined b y tlie reciirsiori
$ 1.G.
This elirnination is efiected by establishirig an appropriate iiiimbering of the totality of pairs of natural numbers. A very simple nlinibering, sufficient for the first stages in the elimination is giren by the function J(u. = ( u+ v)?+7 ~ .To show that J ( u , u ) bssigns a unique number to each ordered pair (21. v) we introduce iri turn t.he functions
where d ( w )=a(O'w. Vu:), B(Pc,x. y) = h( C-11'. J * L ~ x. , y) so tlint F(w, O ) =f (Uzo. Vuj, 0 ) and
{F(w,x ) =f (Uuy, V w , x ) )
-
{P(rr. Sx)= f (l'lc.. 1 - 1 1 , . S.t)l
which proves that
Ex = X L - ( R t x ) ~ whrre R t x is the function defined in
S 1.6 as the greatest integer
whence, substituting J ( u . u ) for
io,
we find
u-hwe square does not exceed x ,
E.c is thr amoiint by which x exceeds the nearest square contained in if. sLl(:t? (u+v)"J(u, v ) < ( u + v + ~ ) ~
so that f is obtained by substitution froni F, aiid F is defined by the recursion R' with one fewer parameter than there is in the initial recursion R. By repeated application of this reduction process u-e may derive f by substitution from a function defined by recirrsion in a single parameter. We may suppose that tliis functioii is in fact F deñned by R' above, where the functions -4 and B contain no other variables than those indicated. 7.2 \Ve show nest that the variables ic aiid x niity be eliminated , by the recursion from B. Define ~ ( t ox)
Shcse equations prove that J ( u , u) assigns each ordered pair a iinique riiimber, for if J ( u , E ) =J ( u l ,u') then
R" mhere
cp(zc, 0 )= ~ ( P u, )~ ( wS"3) . = P(x, ~ ( z c x. ) )
m(w)=.J(w, Azu) , P ( x , y ) = J ( C y . B ( I 7 y ..r. V y ) ):
it readily follows that nnct
v = V J ( u . u)= VeJ(u.',u') = o".
u w b-
ELIMIS.\TION O F PAR4METERS
for the equation liolds for x = 0. and tlie equation iii x implies that mith S x in place of X . F(w, x ) is derivable from ~ ( z cx ?) by substitution only since
Since
where
v'(w9 S ) =J ( x , v(tu, 4 )
-
by observing that this equation holds for the value O of x, and that
'
J ( S U J ( x ,~
4 ), B(UJ(x,,d..
( 2 %
=J ( S x , j(x,
4)
I
V J @ >P ) ( ~ ~ .4 > )))
v f ( w ,x ) =J ( x , v(w, 2 ) )
implies
rpl(w,S S )=J ( S x , ~ ( wS x, ) ).
The function
v'
Rt S x = R t x whence and
is defined by iteration without parameters and
This completes the reduction of the schema of primitive recursion to the simple form
Z J S x = E R t S x = E R t x= Ux V S z = S x 2 ( R t S ~ ) ~ = ( l i x ) = x( R) t~ =
{ x1( ~x)"t - ( 1 {(Rtx ) ~ :2 ) )
= { x i ( R tx)?)+l=S V X siiice E S x > O entaiis (Rt Sz)*<Sx and therefore (Rt x)?< x so that x 0 (which by example ?.S3 liolds also when E S x = O ) . (Rt x ) ~ = = If we now define G ( x ) by the recursion without parameter 11
where so that
7.3
CJ(.u.C ) = I L , VJ(.zl.u)= v .
This second set of pairing f~inctionshas the additional property: if ESz>O then U S x = U x and V S x = S V x ; for if E Sx'O theri S x is not a square and so
rp(w, x ) ) )=J ( S x , ~ ( u i8x1) . ,
so that
GO=O , GSx=c(x, Gx)
c(x,y)=(l-1ESx)~USx+(1-(11-ESx))~b(y), c ( x , y ) = U S x if ESx=O =b(y) if E S x > O ,
A further simplification may be effected by taking
R*
F(w, O ) = w , F(zu, S x ) = b ( F ( w ,x ) )
so that
f (w, 2 )= F(a(zu),x).
To eliminate the parameter new pairing functions
1"
from the recursion R" we introduce
J ( u , .u) = ( ( u+ + U)' + v Ux=E Rt x , V z = E x .
.
E R.t J('IL.v) = I L .
so that
~ ' ( w=)J(0. a(uj)), @'y=J ( S U y , (8(Uy, V y ) ):
as before we prove
R t J(u,>a ) = (U + u)?+
E J ( x . c)= z: and
$(w, O ) =cc'(¿c) , rpf(.w,S x ) = ( 8 ' ~ ' 5( )~ ,
1
( ( u + ~ ) ~ + u ) ~ gLJ' )( < z ~( (. U + V ) ~ + U + ~ ) ~
therefore
F(20, x ) = V v ( w ,x ) . Next define ~ ' ( wx ), by the iteratioii
127
then we can show that
G(x)= F(Ux, V x ) ; the proof of the case x = O is obvious. If E S x = 0, so thnt V Sx = O then
F(U S r , V S x ) = U S X = c(x, Gx) =G S X ,
138
ELIXIXATIOS OF P . \ R A J m T E R S
E L I Y I S 4 T I O S OF i'1R iJIETERS
and if ESx>O then
139
aiisiliary functioiis al1 of wliich are definable by siibstitutioii in ternis of the functions
F ( U S x , V S r ) = F(C'.z. S V x ) =h (E'(C'x,V x ) ) =c(x. F(CIx, V x ) ) itnd so froni the Iiypothesis G(x)= F(U X . V x ) we derive
C ( 5 x )= F(C7Sr,V Sr) which completes the prc~of.F is ohtnined frorn (? by t,he substitution
whicli we shall cal1 the initinl fiinctions. We llave therefore proved tliut if t h r . foli ~ ~ r functionx nrc n cailnble fhen nll primitive recllrsire fztnctions rrrr ohtninnhl~hy s~rhstitiltion and clpplicntion of the ,;.ingZr schemn
F.r= -4 -0 fnr g~nerntin!]f~c?l('fion.rof »?ir vrrrinhlc.
-
L
.
The variable x may be eliminated froni c(x. y) exactly as from the functiori F! above. JITe define
H O=O, H S x = d H z
I where
cíy =J ( S l;yl C( a?/? vg)) , H x =J ( x , Gx). The proof of the case x= 0 is evident, and from the hypothesis HX=J ( x , Gx) follows
H Sx=d Hx=d J ( x , Gx) =J ( S x , c(x, Gx)) =J ( S x , G S x )
as required. Thus Gx is derived from H x by the substitution
G x = V HX and H x is defined by the schema
H O=O, H S x = d Hx, so that
H x =8 0 .
To carry oiit this reduction we have introduced a number of
7.1
It can be showii tliat in fact jii.jt tlie two initial fuiictions 11
-
11 ,
are sufficient but this rediictioii ir1 the initinl fiirictioiis tlioiigli of great interest iii sliowirig tlint a sirigle fiinctioii of twn variables enables us to reacli ail primitive recursive functions of aiiy number of variables, is iiot esseritial to oiir plirpose and xviil be omitted. 7 . 1 I n the coiistruction of primitii-e recursive functions of m e variable it is never necessary t o define by siibstitution a function of more than one variable. For instaiice if we can obtain a function f ( n ) from a function F ( x . y ) by taking. say.
f ( n )= F(g(n).N n ) ) with F ( x , y) defhied in ternis of given functions n(u. t . ) , b(z, y ) , c(x. y ) by the siibstitution F ( x , y) =a(b(x.y). c(x. y)). then since
f
( n )=a(b(ll(n).h(n)).c(g(n),
instead of definiiig the tnro variable functiori F ( x , y) we coilld define the one variable functions
S(n)==h(g(n),h ( n ) ). ;J(~L)
= c(g(n).h(n,))
aiid so define f ( n ) directly by
f ( n )= 4 P ( n ) .:,(n)).
140
ELIMINATIOX OP PAR.\XETERS
one variable prirnitive reciirsive functions niay be obtained usiiig only the schemata
It follows that.
Tlie sequence contains the initial functions 0. n, Sn. and R t n.
If it coiitnins the functions g(n) and It(n) then there are integers p. q such that
.
f (72) =.g(n) i1~(n) , f (n)= g(,n) h(n) j (n)= g(n).h(n) ,
--
f (n)= R t n , f (n)= g(h(n.)). f (n)= gnO .
, . a Tliis last result enables us to enumerate al1 oiie-rariable primitive recursive functioiis. Thc enumerating function is pm(n)defiiied as follo\vs: for the f i s t four fiiiictions iii tlie enumerntiori we take the fiiiictioiis O. n, Sn, R t n, so that
If \ve choose a value of n-e shali llave
Condition on m
I
for ~vhicli?,(m,1)- p, v ( m . 2 ) = q tlien
such a valuc of m is for instance
-. ..:
S e s t me assign five ranges of values of 7n to the operntions + , substitution and recursion. Denoting as in 5 4.61 the exponeiit of the greatest power of the kth odd prime in n by ~ ( nk), , we associate eacli operation with tlic cases ~ ( m0), = 0, 1, 2 , 3 and v(m. 0) > 3, for values of m > 3. Consider riQw tlie function qm(n) whose values are listed in the following table.
nl
where the factor 7 has been incliided to ensure tliat Y . 3P.5q.7 by m, S 0 tl1at
111 > 3.
Denote
v(nz,. O) = k . ?,(m,. 1)= p , v(n¿,, 2 ) = q and ni, > O
1
ancl therefore for tlie valiics l;= 0, 1, 2 . 3 the functions vmk(n)are respectirely g(n)-i- h(n) g(n) A 74n) g(n).h(n) , g(h(n)) ancl for 1: > 3 9
Value of vm(n)
7
Vntk(O) = O 3 ~rn,(n+1)=3(qrnk(n))
m > 3 v(nz, O) = O m > 3 &v(m,0 ) = 1 m > 3 &v(m,0 ) = 2 m > 3 &v(m.0 ) = 3 712 > 3 S v(m, O) > 3
-
1
-
(n)+ %(m.*](n) c 1 id t - L >, (m) A P)v<m,%>('2) ~ V l r n , ~(m). , ~ V 1 r n . 2 (7%) ) V V I ~ . 1)( ~ v i r n . ? ) (n)) {(l A (1A n ) } ~ , t ~(pm(nL m , l ~ 1)) (.vi..
whence it follows that tlie sequence cp,(n), m = 0, 1, 2. ... contains al1 one-variable primitive recursive functions. To represent ~ ( nconwniently ) in a single expression v e mrite 6,(.n) for 1 1 E, n / and ~ , ( n for ) 1 A (k: n), so that d,(n) = 1 if n = X: and is zero otherwise, and ~ , ( n= ) l if n > k and is zero otherwise. Then vm(n)may be summarised in the expression
1,
(PvIrn.i)
We shall sliow tliat the sequence %(n) VI@) VAn), - - is aii enumeration of al1 one variable priniitive recursive fuiictioiis, (with repetitions). 7
t
-4s a fiinctioii of the two variables m. n tlie fiiiictioii y,(n) is riot priiiiitive reciirsive: for if it weie tlieii y.,(n)l 1 woulcl be priinitire reciirsive. Biit g..,,(m)- 1 - r,,,(/lr) so thnt 7 ,,(H)-L 1 is not tlie fiiiictioii qm(n)fo; aiiy value of trl aiici so y.,(/¿) - 1 is iiot coiitainecl iri the eniimeratioii of oiie-v;iiiablr priiiiitirc. reciirsii-(1 fiiiictioris. 'rlie clefiiiition of' ~ , ( n )shows tli:~t tlie r:iliub of y.,,(n'l) is riveri in ternis of ~ , ~ ( nnrid ) (r,,,,( 1 1 ) . T,,,, ( n ) where n1,, ttr2 üre botli lcss thaii )ti: tiiiis ry,,*(n) is tfefiried by a course-ofdues cfouble recursioii wliicli rrii~'- be trniisforilied iritu i standard (!oiible iecursion by the rnetliocl of the previous cliapter. Kccordinglp ym(n)is cin rxczmple of n fitnction defined by dortblp rect~rsionwhich c(cnnol hc tlefined by pri~mitircrec~irsionr r l ~ ds7ibstitution alone.
CHXPTER Y111
The principal aim of this final chapter is to show thnt tliere is a primitive recursire functiorr f (n) sucli tlint eacli of tlie equations f (O) = O, f (1)=O, f ( 2 )=O, ... is prorable iri .2* biit tlie equiition f (n)=O. with variable n is riot pro~ablein W*. Tlie esistence of such an f shoms that B*is incomplete, tliat is to say that there is an eqt~ation S.
f(n)=@ tchich is verifinble but not provable in g*. This remarkable result. n-hicli n-as discovered by Kurt Godel in 1031, suggests that the natural niimbers may iiot be the only class of objects for mhich the provable formulae of W* are verifiable and that there may exist a class of entities wliich includes the natural numbers and other eritities besides for mhicli the prorable formulae of 9*are verifiable. Tliat such a class in fact esists was actually established three years after Godel's result, by Thoralf Skolem, the inventor of resiirsive arithmetic, m110 shomed (independently of Godel's methods and results) that not only systems like % . ! o r g * , but every formaliscition of arithmetic fails to characterise the number concept completely and admits as values of the number variables a class of entities of which the natural numbers form only the initial segment. 5.1
The construction of tlie unprovable equation f (?¿)=O
is effected by means o€ a code in which the terms of B* are expressed by numbers, aiid the syntas of 9* by primitive recursive relations. This code is called a Godel numbering of d*.
GODEL >TTIIBERIP~G AND THE ISCOMPLETESESS
We shall take the variables of .9* to be m , n, z,. %, 2, ... ; the functions to be yo(n),pl,(n), q,(n). .. . together mith m + n, m. n and m - n ; and the proof schemata to be
OF .\RITHXETIC
145
where p , is the kth odd prime. From the iiumber of the formula alone we can w i t e down the full forniula. simply by factorising the formula-number. For instance the number of the formula
is 2'5.39.5. y 3 . 1115 and the number of the formula
Sb, Sb, For axioms me shall take
since 'S' is standiiig for '9,'. Siiigle signs all have odd numbers, but the nuinber of a sequence is necessarily even. Of course 'nonsense' sequences of symbols like
also have numbers but this does not gire rise to any complications. the defining equations of the functions m.+n, nz2n and m . n , and the deñning equation of v,,(n) m l y . Thus the various primitive recursive functions of one variable are all deñned by the single asiom, the defining equation of plm(n). 8.2 The several elements of 9*are numbered as follows: to tlie functions pl, we assign the numbers 4k+ 17 , k= O, 1, 2, ... and t o the variables x, the numbers 41; + 10 , k = 1 , 2, ... . The numbers of the remaining signs are as indicated in tlie table
NOW any formula of g*is simply a sequence of signs of W * ;
are the numbers of the componeiit signs in a formula (in the correct order) then we assign to the formula the number
8.3 Kext we observe that since a proof iii .%'* is a sequence of formulae then proofs too may be numbered. Thus a proof which consists of a sequence of formulae x-itli numbers
is assigned the number
so tliat if N is the number of a proof and X: is the greatest value of r for which v ( N , r ) > 0 (and therefore iV is divisible by a non-zero power of p,, and p, is the largest prime for mhich this is true) then v ( N , k) is the number of the formula proved by proof number N. We shall see how t o single out the values of N which are in fact numbers of proofs. 8.31 Like all formulae the axioms of 9?* may be numbered, and we shall denote by A,, A-, A,, A,, A5 and A, the numbers of the 6 axioms. We shall have no occasion to require the actual values of the numbers denoted by these letters. We denote by Ax(n) the
-
CODEL S C M B E K I N < : A S D TTí!: : S C O M P L E T E N E S S O F ARITII~KETIC
dkjiinction n = A, v n = -4, v n 8, v n A4 v n = A5 v n = A6 which aays that 12 is the nurnber of ari asiom. ;=
The first step iii nritllnietisiiig tlie syiitax of .2*is to finci :he relstioii whicli expreses thc prvpcrty of bcing a rariixble. Sincc the riumbers of the \-ariables arc l G T 4X:wherc k ruiis tliroiigli the niinibers 0, 1, 2, ... tliereforc the property 'n is t!le iiumber of n vnriable' is espressed by tlie prirnitive rccursire rel. t'1011
147
instarice. thc iiiimber of 'q~,,(m)'is .'33.3".515.7' aiid tlie number of ' + n ' is 29.319 then the number of 'q,(m)+n' is
q.4
Thc fiitlctioii
rn
A
n is defined by the equa t 1011 '
CL
provided tliat neither ,rn rior n is the iiumber of :i sirigle terni. i.e. prorided that m aiid 71 are batli eveii. If n1 i!: ocict arld n eveii (thc~:e is ari m betweeti O aiid n siicli that' n = 4 m + 15). IYc denote this relatiori by V(n). w
The relation 'n is the number of a oiie-variable primitive :.ecur!:i\-e function' is similarly $.4i
w!iic!i we denote by t ( n ) .
and if »z is even but n is orld theii
aiicl if
aiitl n are brr:Ii ocld 111, A
SVe consider nest thc relntioii 'n is the number of a variable in forr:iula number f '. Tliis is exprcssed by the primitive recursive relntioii 8.42
8.51
The product f ( O ) A ... A f (n) is defined to be 17^ f (i) iSn
where I7^ f (i)= f (0) , D A /(i)= ( lTA f (i)) A f (.n~+ 1). i
\rhicIi we denote by Vi(n). 8.13
The number of terms in sequence number f is 1, $- 1 defined by
A',(n > m -t R(f, P,) > 0))) {R(f+ 1, 2)) where R(n, b) is the reniainder when a is divided by b. (Thus I , = O if f is odtl, and if f is even 1, is the least integer ín such that f is divisih!e by tlie mth odd prime but not by any greater prime). Tlie predicate F ( n ) which says that n is the number of a formula may now be expressed by R(n, 2) = 0 & Af.+'{R(v(n, r ) , 2) = 1). 1, = (L',{R(f,
P,)
=O
Bti.fore we consider the fundamental syntacticd operatioii of siibstitiitioii we introduce the primitive recursive function m A ri -,rI;íc!i determines the number of the espressioii formed by v~itin: es~ressionnumber n aftcr expression number m. For
n = 2'". 3"
i;nri
i<
tt
8.6 JVe are now r e d ? :o espress the relationship which holds between tliree numbers f, c, n and tlie number of the expression which is obtained by substituting object number n for the variable nriniber 1; in formula number. f. Xow f = 17 D:(~'~'and for each value of i for which ~ ( fi ), = v , i
the variable whose numknr is z, is to be replaced by object number n. Let us tlien deñne and
S.5
then Sub,.(c/n) is the desireci iiurnber of the espression obtained by substituting object riumber n for variable number t. in formula
the formulation of the predicate 'n is the iiumber of a proof'. To this end we must consider the relations between the numbers of premisses and conclusions ir1 the derivation schemata.
number f . For instante, the number of the formula
S.31 For Sb,. we take the primitive recursive relation S1(?n,n ) which espresses the relation: 12 is the number of an equation derived from eqiiation number m b y Sb,. &',(m, 7 1 ) takes the form
and the number of the formula
t,he resiilt of snbstituting x l - x 2 for m %("l
111 %(VI)
-
+ ~ 2 )
with number
333.3" 523.79.1127. 137
which is equal to
33 A 5 A 223. J 9 . 52'
ir
is
A
4
. .
.
Next me seek to characterise the concept 'recursive terms' or 'recursive function'. Nom t is tlie numbér of a recursive term if t is the number of a variable or the number of one of the functions m n, m - n, m .n,v,(n) or is the number of the espressiori obtained by substituting a recursive term for a variable in a recursive term. The numbers of the functions m + n, m n, m . n and q,(n) are 8.7
+
#,(m. n) says that there exist x, y. z between O tind m and rc between O aiid n , such that .E.y and zr9 are numbers of terms, 2nd z is the
number of a variable iii equation number m : m is the number of the equatioii between the terms numbered x and y. and n is tlie number of the equation which resiilts from it by substituting term number it7 for variable iiumber z . 8.82
For Sb2 we take S,(m, n) in the form :
-
and we denote these by o;, u2, u, and o ( k ) respectively. Thus the predicate T ( n ) ,which says that n is the number of a recursive term may be defined as
This is a course-of-values recursion so that T ( n ) is a prirnitive recursive relation.
It is now easy to formulate the condition E(n) that n is the number of an equation between recursive terms. E(n) may be expressed in the form
( x and y are term numbers less than m , n is a term number and u the number of a variable in term number u, m is the number of the equation between terrns numbered x and y, and n is the number of the equation between the terms obtained by substituting ñrst, term number x, then term number y. for variable nuinber v in term number u).
8.83
For schema T we take T ( n z ,n, p) in the form:
8.71
8.8 The most important part in the arithmetising of syntax is
(x. 9 and ;. are term numbers aiid m. n and p are the iiumbers of the equations between t,erms of riumbers x, y ; x, z and y. ;. respectively).
8.84 Fiiially for schci-iia, F: n-e tal;e E(ni. 71. 11) drfiiied as:
(.c is tlic iiiiriiber of a terni ai~il!/ t !ir iiiinibcr c,f :L \ 'rriable iri tliis trrni: n/ is tlle number of the r(l~~iti011 fos11i~:l eqi~i:tillg to zcro the residt of siibstitutiilg zero for the vnriitblc i r i tei.111 iiumber X, n is thc iiuniber of tlie espression fornicri !;y ta~;i:,>t ;ii: terin niiniber x to the result of p r e f i l i n ~',\' to t!,c \rict)lr: 111 I liis t e r ~ ntrnd JJ is tlie iiiiliiber of tlie espi.e~sioiicibtniiicid 11)- ccliiatiiig to zero terrn riiimber x). S.S.5
of
1
:L
\Ve are iio\\r readv to clcfinc the ~)redic.rrtc'71 is the number proof', dellotecl b ~ Pf - (n). We takc Pf(n) to he
n ( ~2¿) =.
a d~"{As(i~(,t,2))v E,"(S',(v(n,y),'zs(r~,x)) v S1(v(n, y). v(n, 2 ) ) )v E;E:(T(v(II,u ) , v(n. u), v(n, x)) v v E(v(n,ti), v(n, 2'). v(n, x))))
(n is the iiumber of a seqllerlce of equations eacli of which is either xn asiom or is derived from one or two prerious equations by one of t,lie derivation schemata). S.Sr5 As we llave already observed the rela,tion Pr(m, n) wllich says t,hat m is the number of a proof of formula number n, is esl.)r~ssedby
Since Pr(m, St,(lS/n)) is a primitive recursive relation it may be mitteii in terms of the SJ-nibolsof 3'*as an equation, and \ve may speak of ' N Pr(m. St,,(l!l,'n))' as an abbreviation for this equation in W*. Let the eqiiation in .9*for ~vhicli- P r ( n ~ . St,,(lg/n))is an abbrel-iation ha\-e iiiimber p. 111 forinula niiiiiber p sribstitute tlie niimeral niimber S, (Le. p,y, ... g,O witli g, occiii*i.ii:g p times) for tlie variable niimber 19 (i.e. the variable n ) : tlic :iiiniber of the resulting espresiioii is Stp(19/p) and tlie espresbion itsrdf maj- he abbreviated to 8.SS
ancl denoted by G(m).say. 8.S9 \Te shaU show the espresiori tliii* clcnotrcl by (?(m)is not provable ii-i ,B*. For if the espression \~-hicliG ( m ) deric,t<.s n-ere provable in d*. and if k mcrc the number of the proof tlicil k is tlie niimber of the proof of equation number St,( 1 9 ; ~ )anc i tlie relatioii
liolds; biit if G(m) is pro\-able then so i.; G\k)ancl tlierefore tlie relation
-
Pr(k. S t p ( l n l ~ ) )
liolds. As Pr(m, St,(lS/p)) is a primitive recursive relatioii. its represeiitiiig function f ( m ) , say. is priiiiitive recursive and the value of f (k) may be obtained from tlie dcfiiiing equations of f (m) by not more then k substitutions. This \-alue will be either O or 1: if the fornier then Pr(k. Stp(191p)) holds and if the latter then Pr(k, St,(l!l/p)) holds, but the tmo relations cannot hold simultaneousl~.Thus the espression denoted by G(m) is not provable. n-liich ib equiralent to saying tliat the equ atioii N
S.S7 The numerals O,SO,SSO, ... are representcd iii .%'* hy O. ylO. g2y20,~72%~)20, ... and so their numbers are l . 2". ::, 2". 3%.5. -2:. 325.525.7, ... ; if Fye denote these nixmbers by ,yo.S,.S,, S3, ... then N , is primitire recursive anci
xo=1, S 1 = 2 2 5 . 3 , NT+l=i?Tr.p?4.~I+1ir>1 . Write St,(i3,'n)as an abbreviation for Sub,(v/Arn)so that St,(u/n) is tlie niiiiiber of the expressioil whicli is obtairied mhen the iiiii!~:,rnl F~y2 ... yzO. mith y, occuring 9% times, is substituted for variable niimber v in espression number /.
-
is not prol-uble in B*. where !(m)is tlie teriii iii 9*b>- wliicli the
152
GODEL
GODEL NUMBERI'í(+ A S D THE ISCOMPLETENESS OF .lRITTIXETIC
P Í U J I B E R ~ - GA N D TUE LVCOJIPLETENESS OF ARITHJLETIC
representing function of the equation which G ( m ) denotes is expressed. However the assertion that this equation is not provable is equivalent to saying that noiie of the numbers O, 1, 2, 3, ... is the number of a proof of equation niimber St,(lS/p), i.e. that each of the equations
holds and is therefore derivable in g*. Thus there is a primitive recursive function the equations
1 such
with a free variable m is nof provable in g*. 8.9 The presence in B* of an unprovable equation is not a mere accident arising from the particular axioms of 9*; we may add as many more (verifiable) asioms as we please t o B* and still carry through the foregoing construction. I n particular me may add the equation f ( m ) = itself ~ as an axiom and still determine an unprovable equation in the extended system 9 2' say. This result suggests that the natural numbers O. 1, 2, ... do not exhaust the values of the variable. We can in fact readily show directly that the natural numbers do not constitute the largest class of objects which satisfy the axioms of recursive arithmetic. 8.91 To this end me prove that, given any sequence of functions fot, flt, f2t, f3t, and so on, (recursive or not) there is a monotonic increasirig function g(t) and a fiinction v(i, j) so that, for all i, j, one of the relations g(t)< f j g(t)
9
f
f i g(t)= j
g(t)
2
in a simple sequeiice, starting with the pair f,, fl followed in turn by f,, f 2 : f,, f 3 ; fl. f z : f,, f,: fl. f,; f,, f5: fl. f,; f2,f, alid so 011. thepair f i, f coming (e2 i + 1)"' or (e" e - i l)th in the sequence according as i + j is even or odd, where ?e is the greatest even number contained in i j. For any two fiinctions p(t). y(t) we can find a iiionotonic iiicreasing functioii v(t) siicli tliat one of the relations
+
that each of
ia provable in 9*: biit the equation
fi
153
f i g(t)> fjg(t)
holds for al1 t v(i, j). \Ve start by arranging all pairs of functions f,, f,, with r t s ,
holds for all values of t: to determine v me dix-ide the natural numbers t into 3 classes C,, C= or C ; according as the relation -, , = or > holds between pt and yt. ,4t least one of the three classes is infinite and we take the members of that class in increasing order of magnitude as the values of v(t) for the values 1 , 2, 3, ... of t, and we m i t e T- for the relation ( < , = or > ) which holds between pu(t) and yc(t) for all values of t. Next me define in tiirn vl(t), ~ ( t ..). with associated relations VI, V,. ... as follows; vl(t) and Vl are the function and relation determined as above so that
for all values of t ; then we take fozl and f,v, for q and y and determine v,, V, so that
for all t. Ir1 this way we determine step by step u,, V,; u,, V,, so that
77,.
Y,: ... ;
for all t , where n is the ordinal of the pair fi, f j in the orderiiig prescribed above. Write
g(n)= c1v2 ... ~',-~c,(n)
so that (taking t = n in equation E)
f is(n)vJig(n) .
Similarly substituting T,, -,u,,, ... I * , , _,(t) for t iii crliiation E ... i',, . ,,(O we fiiicl that fiy1r2... r , , - o (t) J-,, jl for ali t. and tliereforc. takirig t = ?r -' p.
for if thcrc \vere a fiinctiori Ii such that f i h
wliicli provcs that
holds for al1 f > n . Piiice t . , , .,(t) is monotoiiic ir1crci~>iilg( i i i i t l 1t.4 ~;!iil(>s are a!l . differc.iit wliole nurnbers) therefore t.,, ,(H -t 1 ) - n so that
xvhich proves tliat g(n) is moiiotoiiic iiicrcasirig.
I
S.!Y!!:y riieans of tliis theorem we niay introdiice a linear orderiiig betn c.en tlie functions fa. f,. f,. f,. ... . \T7e write
foi al1 sutticieiitly great, t. wliicli is of coiirse iinpossi!,lr. Tliiis tlie terrns of the seqiicnce f,, f,. f 2 , f,. ... share ~vit,litlie nntarsl iiiinibers tlie property of havirig a uiiiqiie siiccessor biit ttie seqiieiice has irienibers gieater tliaii evcry iiatiiral iiiiiribcr, silice for instarice the ideiitity fiiiictioii f (t) = f esccecls exrery const,arit. fiiiictioii. becaiisc j l ; ( f ) Ii11.i; tl!tl vnliie ! ~ ( t n-Iiicli ) grows n-itli f beyond aiiy assigiicti ü:iioiirit. 8.93 Let iis now take for fa. j,. f?. f,. ... tlic secjuencc of al1 orie variable primitive reciirsix-e fiiiictioiis ~rrit!i ,/, as tlie zeio fiiiictiori. We shall show that al1 tlie triic stntemcnts of rcciiisivc nrithinetic reniain valid if me tak? j,. f,. j2. f,. ... i i i p!ace of tlir iiatiiral nambers. Giveii ltiiy rcciirsive fiiiictiori
F(.c,. accortiiiig as T7,, has tlie value <,
=
l . tlien n.e
.-. ... . x,,)
of iiatural iiumbers, wc iiitroduce function variables g l ( t ) .y2(t),... xvhose 'valiies' are the fuiictions f1 . f 2, ... . For eacli assignment of raliies of tlie variables
or >, i.e. according as
for iili t , n. It is readily verified that f i = f i and fi=f, implies f i = f k
becoiiies a one variable fiinctioii
and f , íf , . f, f,:implies f i < f,, as our use of the notation anticipated. Amongst tlie functions f,, f,, f,, f,, ... xve may if we please include ail the constarit functions f(t)= O, f (t) = 1, and so on, which we may denote simpiy by O, 1, and so on, aiid tlie ideiitity function f (t)=t. Inequalities betweeii natural numbers become the same iriequalities htttn-een coiistant fur,ctions since for iristance if m t n then, re~arrlingm ttnd n as constant functions of t , m < n for al1 t. Each of tlie functions f ( t ) of course satisfies the inequality O -:j ( t ) . nncl each function f ( f ) has a mique successor f (t) 1,
+
say. whicli nve take to be the viilue of tlie function
... . jkn of tiie argumerits y,. q2. ... . T,,. Iii particular when me give for ralues of pl,v.. ... . v.,. n constant
for tlie values f fiiiictions
c!,.
rc?.
fq,
. .. . a,, say. F takec as valiie thc ~onctniitfiinctiori
+ + ir0
156
U ~ D E L1 í U U B E I ¿ I N U AND THE I N C O N P L E T E N E S S O F
AnITHUETIC
which is of course esactly the same as the value of F(xl, x,. ... . x,) for the values a,, a,, ... , a, of the arguments x,, x2, ... , 5,. Thus a functiori of natiirnl number variables may be interpreted as n function in the space of tlie furictions f,.f,. f2. ... q-ithoiit chaiigiiig its sigriificance. Under this interpretation a t,riic eqiiation (whetlier an asiom or derived from the asioms)
Solutions to Examples 1 1.
We define in tiirri
S.94
Peiit(x, O ) = 1 , Pcrit(x. Ry) =Tet(x, Pent(x. y)) Hes(x, O) = 1 . Hes(x. Ry) = Pcrit(x, Hes(x, y ) ) Hept(x, O ) = 1 , Hept(x. S y ) = Hes(x. Hept (.c. y)).
F(x,,s,, ... . ~ , ) = G ( s , . x ~... . . x,,) becomes a true equation in the function space; for if this-equatiori holds for all x,, x,. ... . .x,, and if i, j are the sufises for-which
*,
factors. for a given set k,, Ir,, k,. ... , k,,, f ,(t)= fj(t) for al1 t
then
which proves f i
=f
f ( x )= (1 (1l x ) ) g ( x )which is fornied by substituting the recursive functions 1 L ( l - x ) for u and g(x) for v in the reciirsive function uv.
1.31
I f k ( x ) = l ~ ( x - 2 )then f ( x )= k(x)g(x) (1 k(x))h(x).
1.4
Let Q(x,y), R ( x , y) denote quotient and remainder when x is divided by y and H f ( x , y), Lm(x, y) the highest common factor and least common multiple of x, y. Then R(0, 2)=O, R ( x + l , 2 ) = 1 ~ R ( xS,) , Q(O, 2)= O , Q ( x 1, 2)=Q(x,2) R ( x , 2), H f ( x , 2)= 2 R ( x , 2) , L m ( x , 2)= x { l + R ( x , 2 ) ) . Similarly R(0, 3)= 0 , R ( x ; 1, 3 )= (1 -I- ( R ( x ,3) l ) ] S R ( x ,3 ) , &(O, 3)=O, Q(x+ 1, 3)=Q(x,3)+ ( R ( x ,3 ) - 1 ) ) H f ( x t 3 )= 1 + 2(1 R ( x , 3 ) ), L m ( x t 3)=x{3 2(1- R ( x , 3 ) ) ).
and therefore
F(fic,, f k 2 >
' fkn)=G(fkl'fk,'
'.'
7
fk,,)
for all arguments fkr, r = 1, 2, ..., n . 5.95 Since we have defined equality and inequality independently it remains t o be verified that the equations x
+
-+
For the truth of f S f i we require fg(t) G fig(t) for t > v(i, j ) ; for the triith of f f = O, ~vritingf l f = f k , we require fd(t) = f,,q(t) for t > v ( k , O ) .
-
Since f&(t)= f;g(t)2 fjg(t) and f,,g(t)= 0, therefore the conditioiis fh54t) = fog(t) and f i ( t )G f s ( t ) are eqilivalent for any t, which completes the proof.
-
1.3
f d ( t )= f+j(t) for al1 t
and so
1.5
-
+
f ( x )= (1 A R ( x , 2))C/(x) +R ( x , 2)h(x).
153
S O L ; -YI<::;S
TO I..S.:SIPLI:S
If C(n,r ) is the coefficieiit of z' iii tli:. rspaiisioii of ( 1 + x ) theii ~ C(n, 0 )= 1 , C'ic). r -- 1)= U C ( n + 1, r $ l ) = c ( n .r)-;-(?(iz.r :- 1 ) : the function {.'(í?. ); is iiel-eriticicss píiniit Ii-e r.cciit.~i~.c.. Ir ! is pririiitiue recurciv~c:itl (,,'(;?. ,-) = n ! ir! ( n - r ! ! 1.6
This esample shorvs that a functiori G(p. n) defined by doubly recursive equations R, may be obtained by the substitiition G(p,n ) = y ( p $ 1 . n + 1) from the doubly recursive functioi: y(p, 11) n-liicli satisfies the simpler introductor' equations
It foliows that \ve iuay take tlie equations R: as tlie standard introductory equations of n function defined by double recursion.
f (n,-L 1 , n,J- l . n i - 1 ) = n(n,. n2. ?t3, ~ ( I Lnl, , n,). n,. ) 1 3 7 ) . f3(n1,n,, n3)) .wheie fl(ni. n,. n,) = f fi(n,. ~t,,12,) 1.9
C(O.n ) = y ( l , n + l)=?.(O, n , y(0, p(0, n,y(1, n ) ) ) y(1, , n))
and
=f
b(n,. n2.n,, f (nli- 1. n , 1.~n,)). c ( ~n,,~ n,, , f (nl T 1, 7 t l + 1, n3)))
( j ~ ,
+
(72, 1, n,, d(n,, n,,
72,.
f
l . nzL 1, %)))
SOLGTIONS TO EX.%JIPLES
161
- .-"32 a ( b ~ O ) = a b = a b - a . 0 ; a ( b - S c ) = a ( ( b ~ - c ) ~ i ) = a ( b ~ c ) - a 9
by 2.431. and a b . - a . S c = a b ~ ( a c + a ) = ( a b l n c ) - a . by 2.21.
Solutions to Examples 11
-
Let d x , y ) = ( f ( x , y ) , y / , then ~ ( 0y ), = 0 and q ( S x , y ) = ~ ( xy ), so that (1 q ( x , ?j))q(Sx,y) = 0 , whence q ( x , y ) = O and so f ( x . y) =y. Write f ( c ) , g(c) for a(b+c), nb+ac respectively: then f ( O ) = nb = g(0) aiid f ( S c )= a(b+ S c ) = aS(b+ c )Ff ( c )+ a , g(Sc)=ab+aSc=ab+ (ac+a)= y(c)+n. Thus f ( x ) and g ( x ) satisfy tlie sariie introductory equations, so that f (c)=g(c). (ab) O = O = a(b. O ) ; (ab)Sc= (ab)c+ (ab) and a(b.Sc) = a(hc + b) = a(bc)+ (nb), wlience the proof is completed as in 2.01. (ab). (cd)= a(b. (cd))= a((cd).b ) = (a(cc1))b
2.233
X I - x 2 = x ( lL X ) = O , by 2.232 and 2.1.
2.234
( 1 2 ~( )l ~ x ) = ( l ~ x ) ~ x ( ll ~ z x)bj-= , 2.232 and 2.1.
2.24
By theorem 2.63, (1 In. bl}(hI - n )= (1 I- la. bl >(b b) = O
2.241
Use 2.232 aiid 2.24.
2.242
If f ( u , b ) = ( b - a ) ( S a l b), theil f (a, 0 )= 0 . f (0, b) = b ( l b ) = 0 ancl f (Sn. S b ) = f ( a , b) so that f (a, b ) satisfies the same doubly recursire introdiictory equations as the zero function Z ( a , b) =O.
2.243
Proof as for 2.342.
2.244
as above.
(1I-O)+ (1-(1-0))=SZO, ( 1 - S x ) + ( l I - ( 1 I - S x ) ) = S Z S x so that (1-x)+{l-(1'-x))=SZx=l . a-(b+O)=a-b=(a-b):O ; a ~ ( b + ~ c ) = a - ~ ( b + c ) = { a ~ ( b +,c ) } ~ l and ( a - b ) - ~ S c = ( ( a - b ) ~ c ).~ l
Use esample 2.21. ( a + O ) ~ ( b + ~ ) = a - ;b
(~+Sx)~(b+Sx)=S(a+x)~S(b+x)=(a+x)~(b+x), whence the result folíows as in example 2. a ( O ~ l ) = O = a . o - ;~a ( S x - 1 ) = a ( S z = S O ) = ~ and a - S x 2 a = ( u x + a ) - a = a x , by 2.23.
.
2.2441 as above (using 2.201).
2.246
If f ( a , b ) = ( b - a ) + ( S u ~ - b ) and g ( a , b ) = l + ( a ~ b ) + ( b : S a ) then f ( a , O ) = S u =g(a, O ) , f ( O , S b ) =Sb = g(0. S b ) and f ( S u ,S b ) = f( a , b ) ,g(Sa, S b ) =g(a, b). Proof as in 2.245.
2.25
Use clouble recursion as in 2.245.
2.251
Substitute O and S r in turn for r.
2.26
a ( x ) - 1 = ( l - 1 ) - ( 1 - x ) , by 2.22, and so a ( x ) - 1 = O . 4 0 ) ( 1 -&(O)) = 1 , ~ ( S X {)l ~ ( S X= ) 1) so that u ( x ) (1 a ( x ) )= 1.. X ) O, proving a ( x )- x = O . &(O) O =~ ( S -Sx= x . a ( x ) = x - x ( 1 - x ) = x ; a(l-x)=l-(l1-(l-x))=l-ar(x). a ( a ( 0 )= ) a ( 0 )= O , u ( a ( S x ) )= ~ ( 1=)1 = ~ ( S X )so, that a ( a ( x ) ) = a ( x ) . Similarly 1 ~ a ( x ) 1=- x . a(x)-a(x)=a(x)~(l~x)a(x)=a(x)-{l~a(x))a(x)=a(x), by 2.1
2.245
= ((ac)d)b = (ac). (db) .
If f ( x ) = x ( l - x ) , f ( O ) = O and f ( S x ) = O .
-
+
+
-
+
Furthermore & ( o - y= ) 0 = a ( O ) - a ( y ) ,U ( X , O ) = O = a(x).a(O) and
2-29
K(SX-S~)=OL(S(X+~+X~))=~=~(SX)~K(S~). Finally from f .g = O follon-a a ( / - 9 )= 0 and so ~ ( f ..\(g) ) =O, n ( f ) . g - a ( q ) = O .* ( / ) . y = ( ) . c b ) , g(b. C) =c, theri ?..>ti1 If f (b, c ) = { b l b .&(c. b ) ) ~.'.(c. f ( 0 , c)=c=g(O. c ) : f(b. U)=O=y(D. U) : f (Sb. S C )= (b' 1 ) ( 1: & ( C . 6))7 ( c - l).u(c.b) =f
C) ,
so tliat f ( h . c ) =g(b, c).
1
.I
Substitute O nnd Sc in turn for C:
2.272
Use double reeiirsion as iri 2.245.
2.273
XTrite f ( a , b, c) for ( 1 ( 1-a)b) ( 1-ac)b and g(a, b, c ) for ( 1 ( 1 -a)h) ( 1 2 c ) b then f ( a , b, c ) g(a, b, c )
and
3.7=7.3=7.S2=7.2+7=(7-7)+7, aild 7+7=7+(3+4)=(7+3)-4=10+4, ( 1 0 + 4 ) + 7 = 1 0 + ( 7 + 4 ) = 1 0 - ( 7 + 3 ) + 1 = 1 0 + 10+1 = 2 . 1 0 + 1 so that 3 . i = 2 . 1 0 ~ 1 . Similarly 4.7 = 2.1 O -:-S. So el-aluate 34.27 me tnke (3.10+4) (2.107 7)=(3.10-4)2.10+(3.10+4)7, by 2.01, =6.102+S.10+(2.10+ 1 ) 1 0 ~ 2 . 1 0 + 8 = ( 6 + 2 ) 1 0 2 + ( S ~ 2 ) 1 0 ~ l . 1 0 + S = 9 . 1 0 2l.lO+S. + by 2.01 agaiii, and 2.03.
(6. c ) + { 1 = .\(c. b ) ) t n(c, h ) =Sf (h, C )
and g(Sb, S c ) =Sc=Sg(b.
1I
- -
1 l
l
ñ-hence it folloms that 1f , gj = O and therefore f =g. 2.074
Substitute O and S a in turn for a.
2.28
zm.fi=xmi.=xm+O
, xm.XSn = (xm- xn)x and p+sn=d(m+n)=xm+ , whence 2.281 (.")O = 1 = xm.O , (xm)Sn= ( x ~ xm )~. and ~ . S n = x m n + m = X m n . x m , by 2.281.
0.32
{ x i ( 1l x ) ) f = ( 1 :( . L . ' 1))f = f folio\rs by 2.31.
2.33
From ( 1 -L ( g 2 f )}h= O we derive (Sf - 9 ) ( 1 2 ( g f ))h = O and so (Sf -g)h = O, since, by 2.242, (Sf l g ) (g f )= O.
2.331
Use 2.246 and 2.31.
2.34
Use 2.2441.
2.341
O = ( l ~ f( )1 - a ( g ) ) = ( 1 2 f ) - a ( g ) ( l ~ f ) = ( l - f ) ~ a ( g ) ,
l
d a , b, c ) f ( a , b, c )
By repeatecl use of the formulae a + S b =SCL+ b. and l=SO, 2=S1. ... , I)=SS. 10=S9. we hare 3 . 2 = 3 - S 1 = 3 - 1 + 3 = 3 + 3 = 4 - 2 = 5 . 1=6. niid similar]? 4.2 = S. S e s t me proceed thus :
( x l l ) f whence the result
-
-
i,
since f.a(g)=O; but a(g)-1=Oandso ( l l - f ) a ( g ) l ( l = f ) = O whence m ( g ) ( 1 f ) = O. Accordingly In(g), 1 f 1 = 0 and so a ( g ) = l - f .
i
~O~~OWS.
(x2/)0= 1 = x0 yo , ( X Y ) ~="( x y y x y and 9". ySn= ( x n .z) .( y n .y ) = (xn.y"). xy , by 2.03.
I
2.351
To f"g=O add ( l = f ) f g = O aild use 2.32.
2.36
From fg
!
= O and ( 1 :g)h= O we derive fgh= O and ( 1-g)f h= 0 , and the result folloms by addition, using 2.32.
From the given equations foll0-a ( 1 l- f ) ( 9 + ( 1 - 9 ) ~ = ) 0, ( i - f ) p p = o , ( 1 - f ) ( l : g ) q = O and so ( i : f ) y ( g + ( i - g ) ) = O , n~hence( 1 - f ) p = O , sirice g ~ ( l ~ g ) = l + 1). ( g ~
2.451
= p ( ( ( b-ta ) 2 c ) '6 ) = p(n A 6). using 2.41. Siiiiilarlu p ( c - n ) = p ( b 2 a ) . Or from plb. cl = O me derire in timi 1 pb. pcl= o, pb =pc and hence from pa pb =;va pb follon-S p n ~ p b = p a ' p c , etc.
- -
-
write f ( x , z ) = ( x - - I / ) - z , g(x, z ) = ( x - z ) + { ~ - ( 4 : ~1 then f (O, z ) =I/ z = g(0, z):, f ( x . O ) = 1:+ y = g ( x , 0 ) alid f ( S x , S z ) = f (x.5 ) , ~ ( S X S-). =g ( x , 2 ) . Siibstitiite 1 for y snd x - a
-
-
From plb, cl = O \ve derive both p ( h 2 c ) = O and p(c- b ) = O ; then p(« c ) = p ( ( ( a1b ) b ) 2 c )
2.452
Tiiis follows from 2.451 aiid 2.1
2.453
This folloms from 2.451
1.461
( u ~ c ) (~ i( c r ~ b )( }{=( ab ~) - c ) i b ) { l ~ ( a - b ) } = ( ( a - ( c - b ) ) - b ) ( 1 - ( a - 6 ) : : sincc b - c = O . = ( ( a ~ b ) - ( c - b ) }( 1 - ( a - b ) ) = O .
2.462
p(n - c )
for z iii 2.41.
Double recursiori. using 2.42. The . common -value of a - ( a - b ) and 6 .- ( h L a ) is the lesser of a , b, denoted bj: min ( a , b). Or wc ma5r use equations 2.62. 2.61 1 a ~ i d exanlple 2.22 as f o l l o ~ ~ s : a-(a~ó)={(~+(6~a))~(b-a))~(a~b) ={ ( b + ( a ~ b ) ) + ( a - b ) ) ) ~ ( b : n ) =b.-(b-a). (a-b) (Sb-c)=(~(:b) { ( ( a + S b ) - ~ ) I - c ) =(a-b){((a+Sb)=~)~a}
= p { ( ( a$ b ) - c )
-- b )
=p{[(b~'c)~{a2(~-b))]~b) =p{(a-b)-(c-b)j=O. since p ( b - c ) = p ( a - b ) = O .
= (a:b){[(a-c)+
{Sb2(cLa)}lLa} =(U.-b){(Sb-a)-(c-a)}, since ( a ~ b( a) L c ) = O ,
since
From (1-a)c=O and (1-b)c=O we clerive c ~ b c = Oaiid bc-abc = 0 whence, by 2.462, c l n b c = 0 , t-hat. is (1 ab)c = 0.
By 2-41 and 2.32, ( a - b ) (c-b)=O follows fi.0111
3.4701 From ( 1 ~ a b ) c = O follows ( 1 -nb) ( 1 :a)c= O which added t o ab(1 :a)c = 0 yields ( 1- a ) c = O ; similarly ( 1 b)c = O.
= O , by 2.242.
-
It follows of course that ( a b ) ( b - e )
.
Sb-c=(b-c)+(l-(c-b))
2.47 = O:
( a - b ) ( S c - b ) = ~ ,a n d ( a 2 b ) ( c - b ) = ( a 2 b ) { ( ( c i a ) ~ a ) - b ) > SO
(by 2.233) a-b=O.
Double recursion, using 3.42.
-
2.471
From ( l = a ) b c = O and nbc(1- b ) = 0 %ve derive, by 2.462, bc(l~nb)=O.
2.472
( 1 - l a , Ll)c=(l - ( ( a - b ) (b-CL)))~ = [ ( l ( b a ) ]A ( C ! b ) ] c= O sirice ( 1 ( b a ) @= 0 follon-S from ( S u 2 b)c = O.
1.473
By 2 . 2 4 2 , ( a - b ) ( b - a ) = O a n d s o ( l - ( l ~ ( a 1 - b ) ) ) ( b ~ a ) = O whence b l a = O nnd t h e result foliows from 2.45.
-(a-b){((c-a)+a):b)=(a~b){a2b+((cLa)-(b2a)),
whence (a-b)z=O and
-
-
--
-
Siibstitute (1 '- ( 1 - x ) y ) y for c. x for (i and 2 for b in 2.441; tlien from { l ( 1 y ) y ] y ( l .r)z = O \Te tlrrire 2.4s.
{lila, b l k = [ { 1 ~ ( 6 ~ a ) ) . - ( ~ 7 - ~ b ) ] c ( b u ) ]1- ( r t \vliciice tlie rcasult follows. = [{l
1
2-
b ) ] ~ S¡TI(-C .. ($7 -.-h)r = 0,
We observe t l i n t i;ir?c.p ( 1 - t l ) t . = O , JVP Ii:i\ c {l-(1-a)b} { ( l ~ u ) ~ ( l - d ) ) c = { l ~ ( l - a ) b )( 1 - a ) c = ( l - ( z ) ( I ~ b ) c = O , and so 2.491 follows by 2.47.
2.5301 By theorem 2.6s ( 1 -15, p A ( p ~ x ) l )(x)=O. f since t ( p - (p-x)) = O . and so (1 ( x 2 p ) ) f ( x )= 0. Similarly ( 1 (Pp - x ) ) f (.r)= O , and the resnlt follows oii addition, iising 2.6.
-
e
,
,
Froiii (.\) we obtnin b - ( b - a ) _ { a + ( b - a ) ) - ( b l a ) = a and a - b = a - ( a + ( b - a ) ) = ( a - ~ - a ) - ( b l a ) = O ; from (p). ( b - c r ) + a = ( b - ~ a ) + { b - = ( b - a ) ) =b+{(b-a)-b)=b, and a ~ - b = { b - ( 4 - a ) ) - b = ( b - b ) - ( b - a ) = 0 ; from (y). a+(h.-a.)=b+(alb)=b and b~(b~a)=a-(a-b)=a. Double recursion.
-
2.7302 Por instance, consider the case p= 5 : me have f ( O ) = O, f ( 1 )= O , f ( 2 )= 0. f ( 2+Sr) = 0. Let ~ ( r=)f (& r ) ; from q ( 0 )= 0, q ( S r )= O follows g ( r )= O. Let y ( r )= f ( 1 + r ) : from y ( 0 ) = O, y ( S r )= ~ ( r=)0. folio\\-s y ( r )= O. Finally, from f ( O ) = O, f ( S r )=y ( r )= 0, follo\vs f ( r )= O. 2.7303 Let q ( r ) = f ( p l r ) then ~ ( 0=)0.
Z7t(x+SO)=Z7,(x)~f(Sx)=1T,(x)~~(z.0) and n t ( x + S S r ) = 1 7 ! ( x + S r ) .f (z-SS,.) & ( x ) y ( x , S r ) = 17,(x)- q ( x , r ) f ( s t S S r ) , which proves the first equation; it follon~sthat tlie second equation holds with r = O and with S r for 7 , and so for any r.
2.741
I f q ( x )= (1 f (x))lirt(x)then pl(0)= O and ~(SX = )( 1 2 f ( S X ) (Sx)17,(x) )~ = 0.
2.81
,41tS~.Altx=(l~~~ltx)~tx=O: Alt x+,MtSx=Alt x + ( l - A l t x)= 1 : (i=,lit2~)(1-Alt(5~+1))=(1~~Ut~x)(1---(1~Xlt.7~)) and similarly {l-(1-,ut ( 2 ~ + 1 ) ] ~ 4 l t ( 2 ~ + 2 ) = 0 whence by 2.36 ( 1~ B l 22) t Nt 5 . 8 =O ~
By theorem 2.65, since 1 f ( a , a + (bl a ) )= 1 ,
and the result follows by- 2.43. This follows from 2.701 by 5.33.
-
As in 2.701, (1-ISb+(a-Sb) , al)/ ( a , b)=0, and so ( 1 ( S b a ) ) f ( a , b )= O , whence rnultiplying by ( a b) the reslllt follows from 2.242. By 2.501 and 2.71
and tlie result follows by 5.32.
168
From (v) we obtain {I-I(SRX)~.Sxl) { ( R S X ) ~ ~ S X ) .=.O . . . . . . . . . . . . . . . . (C) Agaiii from the introductory equation we h d {(XRX)~LSX).IRX, RSxl = O . By theorem 2.68 (lLIRx, RSxl) ( ~ - { ( R x ) ~ - x )((RSx)?-x)=O ) and so bu esample 2.36 ( ( S R x ) b S x )(1 { ( R X ) ~x)) ((RSx)? x) = O whence by 2.332 ( ( S R x ) k S x )(1 ((Rx)?-x)) ((RSX)~-SX) = O . . . . . . . . (DI From (C), (D) (siiice 1 la, bl = (1 ( a l b)}2 (b -a) and prs = 0 foUows from qrs = 0 and (p.- q)r =O) (1 {Sx- (SRx)?))(1 { ( R X ) ~ ((RSX)~-T-SX) ~~)) =O and therefore by (vi) (12 ((Rx)~=x))((RSX)~-SX) =0 which together with the equation (RO)?I O = 0 proves (RX)~-X = 0.
which combined Mth at 2.0=0 proves d t -x=O; i t follows that ,41t (2x+ 1 ) = 1. Finally, a ( N t O) = x(0)= O =at O cr(Alt SX) = % ( l Alt - x) = 1A Alt x = Alt SX, proving a ( S t x) = Alt x. 2.82
We observe ñrst tliat H f S S x = H f x - ~ A l xt + A l t S x = H f x + l whence Hf(-?.Sx)=SHf (Zx), ~vhicli,mitli Hf (2.0)=0, proves Hf (2x)= x. Hence Hf ( 2 x + i ) = H f (2x)+illt ZX=X.
2.83
By example 2.243 aiid the introductory equation 6f R t x, and denoting Rt x by Rx for brevity, (1 (SSx- (SRX)~J)RSX = (1 {SSx- ( S R X ) ~ }and ) so by
'
-
-
2.35.
(1-{SSX-(SRx)?)).(Rx. RSxl = O ...:............... (i) By theorem 2.68 (1 1 Rx, RSx1) (1 (SSx- (SRX)~)) (SSx- (SRSX)~) = O . . (u) Multiplying (i) by SSx- (SRSX)~ and adding to (ii)we find (12(SSx-(SRx)2)) {SSX-(SRSX)~}=O . . . . . . . . . . . . . (A) By esample 2.241, multiplying the introductory equation 8x1 we obtain by 1 1 (SRX)~, (1 -I(SRX)~,Sxl}. RSx= 1-.-{I(SRX)~, SzI}-SRX and so (1 21 (SRX)~, 8x1). 1 RSx, SRxl= O . . . . . . . . . . . . . . . . . . . . (E) Again by theorem 2.68 8x1} - 1 (RSX)~, SxI = O .....fiv) (1 1 RSx, SRxI ) (1 1 (SRX)~, and so adding I(RSx)2,SxI times (iii) to (iv) we find (1 -I(SRX)~,8x1). ~ ( R S X8x1 ) ~ ,= 0 . . . . . . . . . . . . . . . . . . . . (v) It folloms that 8x1 j {SxL (RSX)~) =O (1 I(SRX)~, and therefore (1 1 (SRx)" 8x1) {SSX (SRSX)~} = O . . . . . . . . . . . . . . . . (B) From (A) and (B) by example 2.34 (1 (Sx (SRX)~}) (SSx (SRSX)~) =O ; 80- (ASR ' O)~ = O and therefore 8 x 2 ( S R X )=~O. . . . . .(vi)
-
-
-
169
S O L U T I O S S T O ES\JíPLES
SOLUTIONS TO EXAMPLES
-
*
2.9
-
-
R o m (1 ~ (~ 7 71, ! ) ) (1~ n !=)O { l ~ ( l - n ! ) }(1-Sn)=O follows, by esarnple 2.47 (1 (1-n!)} (1 (n!)Sn}= O i.e. (1 (1~ n ! ) (1 ) 2(Sn)!)= O which, Tvith 1 O! = 0, proves 1-n! = O. Further, (SO)! = 1=%(O) ; (SSn)!= ((Sn)!}SSn and IT,(Sn) = (n).SSn mhich proves that (Sn)!=1T,(n).
-- n,
2.91
- -
-
From (1-c)(l-(1-c)}=O and (1 (1-a) (1 b)} (1- a ) (1 b) = O follow the two equations ( 1 - ( l l n ) ( l ~ b )(1-c) } {(l.-(1.-c))-b}=O (1-(1-a) ( l = b ) } (1-a) ((12b)'-(1-c)}=O whence by esample 2.47 (1- (1A ~ (1) 2 b)} (1~ a c {l) 2 (b + (1l c ) ) } = O
Solutions to Examples 111 3.01
Use esamples 2.25 and 2.26.
3.02
Examples 1.25, 2.47 and 2.26
3.113
Examples 2.351. 2.1
::.O31
Esnmples 2.25, 2.1
3.032
Associative property of addition.
3.04
Esample 2.1
3.063
Esample 2.45
3.054
Esample 9.351
3.055
By esamples 3.081, 3.OS2, 3.083 a ~ i d3.064
3.09
Example 2.25
3.0911 Esaniples 2.25, ?.:Di aiid 2.232
t
3.041
Examples 1.25, 2.1
3.05
Commiitative property of multiplication
3.051
Example 2.25
3.052
Example 2.25
3.06
Commutative property of multiplication
l
Example 2.232
3.093
Examples 2.25 and 2.232
3.1
Exarnple '3.242
3.11
Esamples 2.331 and 2.244
3.12
Exarnple 2.244
3.121
The formula is equivalent to ( S b2 0 )( S c 2 b ) ( a c ) = ( S b ~ a()S c - b ) [ ( ( b - c ) + ( a - ( c - b ) ) } - b ]
1
3.0601 Example 2.31 3.061
3.092
Esample 2.36
= ( S b ~ a()S c ~ b ) . ( ( a ~ b ) ~ ( c ~ b ) ) = O . 3.13
Equivalent to 3.12
3.131
Theorem 3.43 and example 3.052
3.132
Theorem 2.63 and example 2.35
3.14
Example 2.1
3.1.5
Example 2.4701
3.16
If f ( x ) = 2 x 1 ( 1 + x 2 ) tlieri f(O)=O, f ( S x )= ( 2 2+ 2 ) ( 2 12x + x?)= O.
3.161
If / ( a , b)=Zab-(a2+b2) then f(a, O ) = / ( ( ) , b)=O and f (Sn,S b ) = f ( a . b ) so that f ( a . b) = O .
3.1
Esample 2.43
1
3.0611 Example 2.25 3.062
Example 2.26 (last part)
3.063
Example 2.471
3.064
Example 2.351
3.07
Example 2.36
3.071
Xs for 3.07
3.05
As for 3.07
3.081
As for 3.07
I
-
172
3.171
+-
SOLUTIOSS TO EXAYPLES
n < b -+ a ,-b 4 c we derive P ( n ) -+ { ( xa n ) -t ( f ( x )< S , ( S n ) ) .
By theorem 2.63 and example 2.25 (1-la. A l ) - I f ( a , b). f ( A ,b)l = O aiid (1-lb. BI).lf ( A ,b). f ( A . B ) ]= O ;
Hence by example 2.47, P ( n ) -> { ( x ~ nv )(x= S n ) -+ ( f ( x )á Z , ( S n ) ) )and tlien by esample 3.11 n-e derive P ( n ) -t P ( 8 n ) rrhich proves P ( n ) by incluct,ioii.
iiiultiplyiiig the first eqiiation by 1 1 b. BI and the seconcl by 1-t- la, Al and using esaniple 2.453 ( l L l a , 91)(1-lb. B l ) l f ( a , b). f(*4, B)I = O whence the resiilt follo~vsby esample 2.25. 3.1s
Since k; b = k + ( h i k ) and l g a - t a = l + - ( a - 1 ) , kn lb = k ( l + ( aA 1 ) ) 2 l(k -E ( b 1;)) = k ( a L l ) A l ( b - k ) , by esample 2.13. . .. = (k (6 k ) ) ( a 1) (1 -E ( a 1 ) ) (6 E ) , by t.lie same example =b(a~l)~a(b-k).
+
3.14 3.101
3.101
3.193
- -- - -
3.2
-
-
h > : c + b = c + ( b l c ) aiid b=c+(b'c) -+ ( n + b ) ~ c = { n i(IJ~c)+c):c
1 ~ 4n n - l + ( n ~ l, 7) n . g n -t n=rn+(u-m.) and so (by exainple 3.131) 1 < n & m 2 12 + (1 -+ (12 1) = nt + (n m ) ) -+ ( l + S ( n - l ) = S m + ( n - m ) ) ; b ~ i tn ~ l < n - r n + S ( n ~ l ) < n - r n -> { S m +- ( r ~ m ) ) - S ( n 2 1) =sm+ ( ( n - m ) - S ( n ~ 1 ) ) and thercfore l g n & m g n&n-l l>vt. P>Q
-
-
3.31
Substitute O and Sn in tcrn for n.
3.32
Since p íz p me llave to prove ~ ( n= Z,(n) ) .17,,,(n) = O (where ( 1 f ) ( n ) is taken to denote the function 12 f (n)). The result follows from tlie equations d o )= f( O ) - ( l - f ( O ) ) = O , ( 1 - t - ~ ( n ) ) ~ (= S n( 1) +n)) {Z;(n) + f(Sn))lT,-,(n)(1l f( S n ) ) = ( 1 - c p ( n ) ) ~ ( n () 1-t- f (Sn))= O.
3.311
If y ( n ) = (1 - n f ( n ) )(1 -t-Z;,,(n)) then y ( 0 ) = ( 1 f ( O ) ) ( 1 2 ( 12- f ( O ) ) )= O and ( 1 -7y(n)}y(Sn) = { l A ( 1 2 1 7 , ( n ) ) ( l L Z I A f ( n ){ )l }~ I 7 , ( n ) - f ( S n ) }
-
{p=q+(plq)) {r=s+(r-S)) + ( ( p + s ) ~ ( q ~ r ) } = ( ( q + s ~ ( p - q ) ) :' (s q +(rLs))} =(p-q)=(r-s)
I
& Y > S ->
Denote the implicatioii by P ( n ) .P ( o ) is siriiply (""){f(x)-f(o))=o. lii-0m ( z=Sn) -+ {f ( z )= f ( S n ) )and a = b -t c[ < b -- c follon-s ( x= S n ) { f ( x ),-Z ; ( S n ) ) and from this \ve derive P ( n ) { ( x= S n ) -+ ( f ( x )á Z , ( S n ) ) ) similarly fi-om +
Lct p(x, u ) denote the equatioii f ( x , a ) = O ; froni a proved cquatioii f (.r, a ) = O wr derive f (O, a ) = O aiid f (Sn,a ) O and tlierefore = { l L Z,(n)}{Z;(n) +f ( S n , 0 ) ) = O. (1 S,(n))Z;(Sn) For tlie second schema nre observe that ( 1 f (z))fl,(x) = 0 by esample 2.741
-
[ l L { Z l l f ( n ) + -( fl ( S n ) ) ) l
=O, by example 2.91.
e
e
3.322
This follows from the previous two esarnples.
3.33
( 1 -- Z,(O))f( O ) =O and { l - Z , ( S n ) } f (Sn) { l - f (Sn))f (Sn)-t- Z,(n)f (S??)
-
=o
3.31
Induction over n, iising esaniple 3.011
3.35
Lct 1 =L:p(x) then EZy(x) p(1) & 1 .L 71 l ~ nl = = n ~ ( f ~ - . - ? ) 1 = n (11: 1) -- ;))(!) p(ri aiid SO (iisiiig esaiiij~le:3.002) E(p(z)-+p(n = (,L 1)) + E:p(n 7 ) .
whence taking b L r for a, since (b~r)+{b~(b~r))=b+{(b~r)~b}=b, {b = b ) -+ ( p ( b )+ q(b r ) ) and so . p(b) -+q ( br). ~
-
-
( n 1))) 3.42
-
3.37
t
3.38
-
Write g;(u, 2 )=f(x, u ) : hy essniple 3.36 we liave botli -4: ' { f ( t , u+c)=O) - + / ( u . n -c)=O
'
a11tl -I:-'[p(t, a+c)=O) -> p(a. (1 +c)=O ;.e. ApAe(f(a+ c. t )= O } -+ f (a+ c, a ) O Tliiis if P ( x , a ) denotes 4; ( f ( t , a )= O ) & d;{f ( x , U ) = O )-+ f ( x ,a )= O we have proved both P(u+- c, a ) and P ( a , a + c) whence P ( x . a ) foliolvs by esample 2.272
Denote L r ( f ( x ) = a ) by b, so that E,"(f(x)=a)-+ f (b)=a. Since f (b)= a + {p(a)+ p(f ( b ) ) ) i.e. { f (b)= a ) a p(a) -+ p(f ( b ) ) , and b a m , t>herefore EEy{f(x)= a ) a p(a) -t EFp(f (x)),by example 3.42.
3.39
f (a)=O foUows from { I L l a , al)f(a)=O
3.391
1-l-f(n)=O foiiows from ( 1 - l - f ( a ) ) { l - l a , al}=O
3.392
From p(a+r) -+ q(a) we derive ~ ( a(bI a ) ) -+ q(a); by theorem 3.43 @=a+ ( b ~ a )+) {p(b)-+ p ( a + ( b - a ) ) ) and so {b=a+ ( b - ~ a -+ ) ) {p(b)-+ q ( a ) )
Let P ( b ) denote the formula. a g b p(n) + EEp(x) nrliere p(x) is tlie propositional fiiiiction f (x)= O. P(O) liolds by theorern 3.43 Since EEp(x) -t E'bp(x) (by esaniples 3.322, 3.31) therefore P ( b ) + {a g b a p(a) -t g b p ( x ) ;) but {(a=Sb) a p(a)) -+ p(Sb) , (theorem 3.43) , and so (a=Sb) a p(a) + E:bp(z). Siiice (a <Sb) -> (a6 b ) v ( a=Sb) . ( ~ s a m p l 3.1 e l ) , it follows, bx esample 3.081, that P ( b ) P(Sb), ~ h i c hcompletes a proof of P ( b ) by induction.
-
3.5
Let Q(r) denote A:+"q(x) tlieii Q(O) follows from A:q(x) and q(p+ 1 ) ; since &(Sr) is equivalent t o Q(r)a q(p+SSr) then q(p+SSr) + ( Q ( r )+ Q ( S r ) )and so Q ( r )+ Q(Sr)foliows from q(p+SSr) which prores &(r), that is AE+Srq(x).By examples 3.34 and 3.392 me liave A:"q(x) foliows from A;q(x), and so AEq(x) follows by esample 2.7301.
3.6
We have b < 1 -+ .up ( n L b) b? a characteristic property of the L-operator. Sinceg
-
+
3.7
-
Sincea=O+ab=Othereforeub>O+a~1&b~1.~oreorer a > 1 (y+l)a>(y+l)>y. From ( y+ l ) a > y follows En+'- {nb . ;x a na q y ) and there-
l \
sOLC;~IONSTO ES4UPLYS
SOLlJTIONS TO EXA>lPL,ES
have fore if N=L;+l { n b a x a naay), -{1Vbgx & N a á y ) ; since O á x therefore N > O a n d s o f l ~ l < But ~ . n;iY+{nb<x a n a ~ y and ) therefore N - 1 is the peatest vaiue 0f n for whicll n b ~ ax ntL
wliich is proved by observing it is true for n= O and that zr(n+l)=&(n)+p-q(nT1) and p.Zp(n+ l ) = p - Z q ( n ) + p q ( n T1).
N
3.81
From a proved equation ( 1= P ) ~ ( x=) 0 we derive both (1 p)q(O)= O and ( 1 p ) q ( x L 1 )= 0 and therefore ( l L ( l 1 - p ) Z q ( x ) () 1 - p ) Z q ( x - ; 1)= ( l L ( l - p ) Z q ( x ) )( 1 -p)S,(x)=O so that, by induction, ( 1 p ) Z u ( x )= O folloms.
-
-
3.82
I
3.831
From a proved equation ( l L q ( x ) ) p =0 follows ( 1 q ( 0 ) ) p= O aiid ( 1 - q ( x -L 1 ) ) p= O and hence, using example 2.272, { 1 - ! - ( l A f l q ( x ) ) p()l L D q ( x + 1 ) ) ~ = {1 L ( 1-17,(a))p) ( 1 ~ q ( x l+) ) p = O which completes a proof by induction.
-
3.834
Take
3.84
By 3.81, p -> A,"(q -+ r ( x ) ) foiloms from p -+ (q -+ r ( x ) ) , and by 3.834, AE(q + r ( x ) )-+ (q -+ d,"r(x)), whence the truth of the schema follon7s.
3.91
Bs in 3.833, if r ( x ) = p . q ( x ) , S,(n)=p.L',,(n), ( 1 - p ) Z r ( n ) = ( 1 -p)pZ,Jn)=O
3.93
Take r ( x )= p+ q ( x ) , then &(n)= ( n l ) p + Zq(n),the functions on either side of the equation satisfying the same introductory equations, and therefore { l A X r ( n ) ){PT Z g ( n ) ) ={ ( l ~ ( p ~ q ( n ) ) ) ~ n pZQ(n)) ](p= +O "nd f 1 ( P f Zq(n)))&(n) = ( 1 (P+ Z q ( n ) ) ) n p =( ( 1 - p ) ~ ~ ( n ) ) p = o .
3.93
The functions Z',+,(n)and Z p ( n )+ Zu(n)are readily shown to satisfy the same introductory equations and from the equality
-
From the true formula ( n = r ) > n + (0=0) and the previous example, folloms E:((n-r)>n) -+ (O=0) whence ( 0 = 0 ) -+ E:((n-r)>n) -+ EZ(x>n) , by example 3.35, 3 AZ(x
p for p in 3.833.
N
N
3.832
3.833
-
3.931
Negate both sides of 3.93, with
3.94
Let r ( x ) = ( l ~ p ( x ) ) q ( x )so , that we must prove y ( n )= ( l L Z ; ( n ) ) ( 1 - Z p ( n ) ) Z q ( n = ) O. Clearly y ( 0 ) = O, and y(n+ 1 )= ( 1-Zr(n+ 1 ) ) ( 1 -Zp(n+ l))Zq(n+ 1) = { ( 1 2 r ( n + 1))-Z,(n)) { ( l = p ( n + 1))-Z,(n)) {q(n-!-1)+
-
I n the form (Y
aild so
+
&+u(n)= I=,(n> + Zu(n) the example follows.
N
-
lii
p,
N
q in place of p, q.
-L Zq(n)> 1)) { ( l; Z p ( n ) ) > p ( n + l)}Zq(n), since ( 1~ r ( n + 1 ) ) ( 1- p ( n + l ) ) q ( n + 1)= 0,
={ ( l ; & ( n ) ) ; r ( n +
+ +
+
+
=y ( n ) = ((1 - Z r ( n ) ) p ( n 1 ) ( 1 Zu(n l ) ) r ( n l))L'&n) and therefore ( 1 - y ( n ) ) y ( n + 1)= 0, prol-ing y ( n )= O.
3.95
AZ(x < n + p ( x ) ) + (AZ(x9 n) -t A;p(x)), by 3.94,
17s
SOLUTIOXS TO EXAMPLES
and so AE(x g n) -- (d:(x < n -t ~ ( x -+ ) ) A,"p(x)) whence oiie part o f the esarnple inay be derived from example 3.531. For the secorid part. by 3.532, A:p(z) -+ ( y < % ]-'(!/)) and so 9Xp(x) -4;,y -n:-- p : g ) ) rlliz < ?L -+ ?(x)).
-
we derive in turn A:(@g n) p(x, y ) ) d:l?p(z. y) E,"(x< n -4yp(x,y ) ) -t AiE:p(r, y) E ! d i p ( x .y ) -+ AiE:p(x. y).
--
-\
--
p iii place of p.
3.96
Negate both sides of 3.95, with
3.97
Froni the equatioii ( 1 - ( 1 - p ) q ) ( l ~ ( 1 ~ (l=p)=U q ) ) (proved by taking O and S p in turn for y ) we &rive ( ~ (+4q ( x ) )-t (- q(5)-+ ~ ( 4 ) and from this proved equation we obtain '
-
3.90
E;E;(p(x)&p(y)tkx>g) t .E;E:(p(u) & p(v) (t 7: > L?) -t q E y ( p ( y )& p ( r ) a y r.r) 4q E Y ( p ( x )81 p(3) & y: - x ) . 11. 3.983 ; from the equation ( 1 - ( ~ + b c )()n i - b )( a i c ) = ( a - c ) ((1-a)-bc)b= = ( ( 1L bc) -a)bc = O me derive p(x) & p(y) (i ( ( x> y) v ( x< 3)) ( ~ (a 4 P(Y) a ( x > Y ) ) ( ~ (& 4 P(Y) ( x < y ) ) and so, by two applications of esarnple 3.931. a n d the equation ( 1 '- (1:b)a) ( 1 -ab)a = 0, ~ ~ ; ( p (a x?)(y) ) a x y ) ++ E:E;(P(x) p(y) a x > y).
+
Combine 3.97 mitli the formula obtained from it by interchanging p(x), q(x) and then apply example 3.93.
3.95
and
SO
(AZASp(i6.v) y ~ n )( ( x g n )+= p(x, y)) therefore (A{A:p(u, v ) & y ~ n -+ ) A;p(x, y), and so -4UAUp(u,v ) -t ( y
-+
AEA;p(u, 3.953
21) -t
A;A:p(x, y).
Segatc both sides o f 3.952 witli - p
in place of p.
{R(P.a ) = 0 b) = O pq = nb{Q(p,a ) .Q(q,b ) ) ) so that nb > O -+ { R ( p ,n )= O & R(q, b)= O -t R(pq. ab) = O]. ab=O -+ n=O v b=O; a=O & R ( p , a)=O + p=@ arid h=O R(q, b)=O--q=O so that ab = O ( R ( p ,a )= O a E(q. b)= O -+ R(pq, ub) = R(O.0)= O ) . -+
Solutions to Examples IV
-%
4.
R(0, 0 )= 0 and, b y theorem 4.03, n>O & n = n - l + O -+ R ( n , n)=O.
4.01
If ~ ( a=)R(a, O ) , ~ ( 0= O) and q(Sa)=Sp(a)-aSq(a)=Sq(a,).so that q(a)=a. Similarly Q(a, O ) = 0.
f .O12
R(a, b)= O -+ a = b &(a,b) R(b, c)=O -+ b=c.Q(b, C ) and so R(a, b)= O & R(b, c)= O + a 7c{Q(n,b) -Q(b,c ) ) whence c > O -+ {R(a,b)= O & R(b, c) = O + R(a, c) = O). But c=O & R(b,c)=O + b=O b=O & R(a, b)=O +a=O and so c = O -+ {R(a,b) = O & R(b, c) = O -+ R(a, c)=R(0, O ) = 0 ) which completes the proof.
4.013 x=O&R(a,x)=O+a=O so that x= O + {R(a,x ) = O + R(ab, x ) = R(0, O ) = 0 ) x>O R(a, x)=O -+ ab=x{b.Q(a, x ) ) -+ R(ab, x ) = 0. 4.014
It suffices to prove the equivalent formula R(ax, bx)=O & R(a, b)>O +x=O. Since a = bQ(a, b)+ R(a, b) therefore m = bxQ(a, b)+xR(a, b) ; if b > O, R(a, b)O + {x>O -+xR(a, b) O -+ { x> O -+ xR(a, b)=R ( m , b x ) ) whence b>O -+ {x>O -+R(a, b)=O v R(ax, bx)>O) so that b> O --+ (R(ax,bx)= O & R(a, b)> O -+ x= O). Purthermore b=O R(ax, bx)=O --+ ax=O; b=O & R(a, b)>O -+ a>O so that b = O -+ (R(ax,bx)= O & R(a, b)> O + x = 0).
R(tr, bc)= O -t a = b{c &(a,bc)J arid so Ir > O -+ { R ( a ,bc)= O + R(n, b) = O } ; biit b=O R(a, bc)=O +a=O so that b = O + { R ( n ,bc)= O -+ R(a, b) = R(0, O ) = 0 ) R(a+b, c)=O -+ a+b=c.Q(a+b, c) R(a,c)=O+ a=c.Q(n.c) arid so R ( a + b, c)= O & R(a>c.) = O -+ b = c{Q(a; b, c) &(a.c ) ) whicli completes the proof for the case c> O. If c= O me deduce as above that b = 0 and so R(b, c)= O. ab=b.atO so that b>O-tR(nb, b)=O; if b=0, R(ab, b) = R(0, O ) = 0. a + l = b - & ( a , b)+R(a, b)+l=b.Q(a, b)+1 if R(a, b)=O, so that b> 1 + R(a+ 1. b)= l .
This follows from 4.2 and 4.21.
By example 2.473, b > O -+ a= b +cd and so b>O - S a=d(c+Q(b, d))+R(b,d ) & R(b, d) O -+ R(n, d )= R(b, d).
+ + R(b, d ) whence R(a, d )=R(b, d).
a = {Q(b,d ) x)d
R(c, d )= R(b, d )= R(a, d).
Since ar = &(a,d)dr + R ( a , d)r br = Q(b,d)dr . ;R(b, d)r therefore, by 4.32, nr = R(a, d)r (mod d ) br = R(b, d)r (mod d ) mhence, since R(c, d) = R(b, d ) the result follows by 4.321.
If r, denotes tlie conirnon value of R(a,. d ) , R ( h t d ) a ~ i dr: that of R(a2,d). R(bl. d ) then al-~a2= { Q ( Q ~(1), 4 &(a2,d)).d+ T I crbi + b2= (Q(bl,d ) 7- ()(Ol, ( 1 ) ) ~ l ri 5 whcrice a, +a? = ( r , - r2) (niotl d ) =b, - O 2 .
-
then n> 2 & R(n. 2) = 0 -+ P ( ? ): but n> 2 P ( 2 ) -f E:{- P ( u ) ) A:(P(a)} and so n r 2 & X(n.2 ) = 0 -+ 1471) - - O . 3
+
If g ( n )= R(Z7,(r+ n). r 7 , ( r ) ) then ?(O) ~ ( 7 1= )
o
-t
=O
1.61
By 4.6, since p ( n ) = 0 + ,i > l . ail!i R(tr. 2 ) .: 1
4.7
If y ( m ) = ( 1 + i n x ) (~1 ~
aiicl, by 4.01::.
'p(I\'?1.)= ( ).
U,(U-- ( n a ) ) is divisible by Z7,(a), arid CI % n-* a + ( n : a ) = n .
-
. .
If I'(tr) denotes tlie giveii formula, theii P ( 0 ) holds t~iid since ( S n ) != & ( n )
-
( b y 1.42) x < n -> R ( ( S a ) ! S, x ) = O R ( ( S t ¿ ) !S. x ) = O & o = S n -+ R ( ( S n ) ! ,a )= 0 i.e. P ( S x ) holdu.
i.e. Sx<Sn SO tliat O
x tlieii ) (;(O) ~ = O anrl (~~x)9)(m)={l-(m+l)x-z2)~~l--~)'J'-' =q-(m+ l ) + [ x Z - [ ( l - - ~ ) , " - ' : ( l-(1>1;-l ) x ) ) ] so tliat, multiplying by 1 ~ p ; ( t 7 1 ) u.e fiiici (1 F(rn))F(m+ 1 )= O n-hich pro\-es p;(tn) 0
1.701
-
B y 4.7, a = l + x & x > l -tam>l-,nx& and so a > l -+nl">m.
l--nlr-.l-ni
4.702
For a > 1 & p = q + S r -+ rc"=a'~i":-Sr.rr"r~~.
4.8
If P ( k ) denotes tlie given foriiiiila, theii P ( 0 ) holds, and R ( g k f', p) = O + R ( g k ,p) = O v R(x. p) = O
q ( 2 ) L- 2 , q ( 2 )>: 1 so that q ( 2 )= 2 and so p ( 2 )= O. q ( 3 ) < 13 , q(2)> 1 ; since 3 = 2.1 + 1 therefore R ( 3 , 2 ) = 1 siid siiice R ( 3 , q ( 3 ) )= 0 it folloms by theorem 3.43 that ( ~ ( 3=) 2 ) holds, and therefore q ( 3 )= 3, mhicli proves tlint, 3 is prime. SO that p(1) > 2 ; from p(n t- 2 ) >p(nt- 1) and p(n T 2 ) > p ( n + 1) & p ( n + 1 ) > 2 -+ p ( n + 2 ) > 2 follows p(n :-1 )> 2 b y induction.
p(0) =: 2 , p(1) > p ( O )
P ( O ) = = :2! > , 0 ; and p(n ' l ) > S p ( n ) & p ( n ) > n+ p ( n ; - l ) > n + i .
Let P ( n ) denote the formula (n. 1
) v ( n = n ) vR(n.a)>O
which
and so, using the schema is proved by observing t,hat if ( 1-a)bc
= O then ( 1 - ( 1 1 - b ) c ) ( 1 -a)c=(l-cr)c- { ( 1 - ~ ) ~ 2 - ( l ~ a ) b c ~ } = ( 1 = a ) c ( l - c ) = O , it follows that P ( k ) -+ P ( S k ) , which completes the proof.
-1.801 If P ( k ) denotes R ( a , p) > O & R(ab, $ ) = O -+ R(b. pk)= O then P ( 0 ) holds ( b y 4.03) and since ( b y 1.01) R(ab. pk+') = O -+ R(ab. pk) = O
therefore P ( k ) -+ { R ( n . p) > O
&
R(ub, P';")
=0
-+ R ( b . p"
=
01;
SOLUTIONS TO ES4XPLES
but, if c = Q(b. P"), R(ú, pk) = O -t b = c$, and SO (by 4.014) P ( k ) a R ( a . p) > 0 a R(nb. p"') = O -+ R(ac, p) = O. However R ( n , p)>O a R(nc. p)=O -+R(c, p)=O and R(c, p) = O + c = pQ(c. p). Accordingly P ( k ) & R ( n . p) > O & R(ab. pk+') =O+ b = pk+' &(c, P ) 3 R(b, pk+') = O that is, P ( k ) -t P ( ~ l I) , which proves P ( k ) .
and AS,"R(f( x ) ,p) > 0 -+ AeR(f(x),p) > O a R ( / ( S k ) ,p) > O whence F ( k ) A + k R ( f ( x )p)>O+R( : 17 f ( x ) , p ) > O a R ( f ( S k ) , p ) > O z O, b y theorem z
zgk
R(m, a ) = O -+ m = @Q. ( m , a). and, by 4.801, R ( Q ( m ,a ) , pk)= O . Since p, is prime R(p,, x ) = O - t x = l vx=p,: and so, since p , > l .
i.e. P ( k ) -+ P ( k + l ) , so that P ( k ) is proved.
zSk
4.87
For R(pE,, p ,) -+ (k=Z).
= O -+ R(p,,
Denoting LF{R(b, p,) = O ) by 5 we have =(R(b, p,) = O } R(b, p,) = O a 5 á m . -+
(Pl=Pk)
From x g m -+ R ( a , p a ) = O and E { R ( b ,9,) = O } we deduce therefore
-+ ( k = 1).
4.82
We have R ( m , pZrn."')= O ; let P ( k ) denote the given formula (4.86). Then P ( 0 ) holds. Br 4.82 and 4.85 R( 17 pZrn."' , pk+J > O and so by 4.802, since R ( m , p E + l l ) = O , P ( k ) + R(m, { 17 p~m.z')p";k+'') = 0
R(@k, P1) =
p l ) =0
4.83
Since q ( m )á m, p(q(m))= O -+ E:(q(m) =pz) ; but R(,m,q ( m ) )= O and m > 1 -+ p(q(m))= O and so m > 1 -+ E:R(m, p,) = O , by example 3.38.
4.84
By theorem 3.91 we derive A:R(m, &)>O, i.e. ErR(m, p,) = O , from the condition x <m + R ( m , 9,) > O. From 4.83
R ( a , p:') = O and so, by 4.02, R(ab, p;"') =O which in conjunction with ( < m yields
p 5 + ' ) =0 ) .
e{R(ab,
I n particular,
N
-E,"R(rn,p,)=~-+ m ~ 1 . Since m = O + R ( m , p,) = O therefore m =0 ETR(,m,p,) = O , mhence the result follows. 4.55
185
Denote the given fornlula by F ( k ) ;clearly F ( 0 ) holds. Furthermore R( 17 f (x):p) > O R( 17 f ( x ) ,p) = O + R ( f( S k ) ,P ) = 0 zSk
xgSk
if c = { 17 pzoz)b and if c { R ( b ,P,) = O ) zgm
then E:{R(c, 4.9
=O }
.
We have v ( N , 1) > O and 1 < k -+ v ( N , k ) = O and, since N > 1 , v ( N , g ( N ) )> O and g ( N )< k -+ v ( N , k ) = 0. Hence g ( N )< 1 -t v ( N , 1) = 0 and so v ( N , 1) > O -+g ( N ) 1 and similarly, since 1 < g ( N ) -+ v ( N , g ( N ) )= O, therefore v ( N , g ( N ) )> O -+ g ( N )< 1, whence it follows that 1 = g ( N ) .
4.91
Denoting R(a, bc) by r \ve Iinve n = bc(J(cr. be) r. r= bQ(r.b ) + R(r, O ) , mhere R(r. h ) < b siiice b,0, and so a= b(Q(r.h ) +cQ(cc. b e ) )-- R(r. h ) lvhewe, since R(r, b)< b. R(r. b ) = R(u. h ) n hich coiiipletrs the proof. J-
Bibliographical Xotes
4.92
Ii divides h, c nnd so El c1ivitit.s
h nnd c. If f divides a, O, c tlien it diricles A , c aiicl so f tliritles 11, so that f
Recursive i~rithmetic\vas cliscovcred by the Soruegiuii rnatlicmaticiun Thornlf Slíolaiii antl his first accoiint ( i r i Uermaii) m a s piibiislicil t i i 1923 i r i Skolem [ l ] *). Sl;olem's work isdescribrd i i i Hilbert-Bernays [l], pp. 286-Rlíi. which contains also fiirthcr devclopin~ntsof rcciirsi\.c ;irithmctic-.
4.9:
JVe define * L(x. y) = L",{@ - - . O a E(,:. 1.1 = tl & n>('l. ?), - 0). Since .ry > O -t {R(xy..t.)= R(.cy. y ) = 0) therefore mj> O + [R(L(x,y ) , x) = O = R(L(x.y), y)) and a < L ( x . y) +u=O v R(n,x ) > O v R(a,!/)>O SO tlint L ( s , y) is the least co~nrnonmultiple of x and y, for xy > O .
The Bist account of the developnicnt o f lo~ic:from iirithmetic w,is @\-en by R. L. Goodstein in Goodstein [I]. Tlioiigh printed i r 1 194.7 this paper \vas presented to the London JIatlirmaticnl Society in J ~ m e1941. -1system somrwliat similar to the cqiintinn culculiis m a s dcvelopetl iiidc~cndently by H. B. Ciirry, in Ciirry [ l ]
,
(1.
The most complete account of the properties of recursive frincti«ns, the reduction to normal forms and the elimination of parametcrs is @ven by their discoverer Rúzsa Péter in Púter [l]. This book contains a coinpreliensive bibliogaphy of writings on reciirsive f~mctionsup to 1950, antl is the only book nn the subject. The proof of the schema of Gen@ralise
A brief siirvcy of recursive arithmetic
is
given in Skolem [4].
-. .
*) The iiiinibers in scluare brackets refer to tlit. bibliography ori the next pazr.
Pase
Bibliography ACKER.\XANS, \V. [ l ] Z u r lC'i&rwruclisfreil teit der Zahlentheorie. Math . .4malen. 117 (1940) pp . 162-194 . BERXAYS. P [ l ] cber das Indz~kstions-schemai n der rekursiven Zahlentkeorie. in h'onirolliertes Denken. München 1951. CURRY.H B . [ l ] A fonnalisation of recursive aritlbmetic. American Journal of Ikfathematics. 63 (1941) pp . 263-282 . GOODSTEIN. R . L . [ l ] Iit6n~ti0.nt h e in ~ u n axicwwfree equatioñ cakz~lua . Proceedings of the London Math Society (2) 48. pp . 401-434 [2] Constructive Formalism. University Collego Leicester 1951. GOODSTEIN. R .L . [3] Permutation in recursive arithmetic. Math . Scandinavica. Vol . 1 (1953). pp . 222-226 . I GOODSTEIN. R . L . [4] Logic-free f o r m a l i s a t i m of recurrive arithmetic. Math Scmdinavica. Vol . 2 (1954). pp . 247-261 . GODEL.K . [l] Über formal t~nenhcheidbareSatze der Principia Matl~ematica u n d vemandter Systeme I . Monatshefte fiir Math und Phys., Vol 38. pp . 173-198 . HILBERT. D . and BERNAYS.P . [l] Grundlagen der Mathematik . Vol . 1. Berlin 1934. Vol . 11. Berlin 1939 &ENE. S. C. [ l ] Introduction to Metamathematics. Amsterdam . 1952. Mos~owsm. A . [ l ] Sentences Undecidable in Fonnalised Arithmetic . Amsterdam . 1952 PETER.R [ l ] Rekursive Funktionen . Budapest 1951. [2] Rewiew of S k d e m [2]. Journal of Symbolic Logic. Vol . 5 (1940) pp 34-35 ROBINSON. R . M [l] Primitive recursive functiwns . Bulletin of the American Math. Soc Vol 53 (1947). pp . 925-942 SKOLEM.TH. [1] Begründung der elementaren Arithmetik durcl~die rekurrierende Denkweise ohne Anwendung scheinbarer Veranderlichen mit unendlichern Awdehnungsbere.ieh. Skrifter Norske Videnskaps.akademi. 1. No . 6b. Oslo SEOLEH.TH. [2] E i n e Bemerkungüber die Induktionaacl~ematnin der rekursiven Zahlentheorie . Monatshefte für Math . und Phys., 48 (1939) pp . 265-276 . [3] Uber die Xcht-charakterisierbarkeit der Zahlenreihe mittels endlich o ~ ~ a b z a l ~ l b a r u n e n dvieler l i c h A ~ s a g e ntit n awsc1~1~iesslichZahlenvariablen Fundamenta Mathematicae. Vol . 23. (1934) pp . 150-161 . [41 T h e develolmzent of recursive arithmetic . S t h Congress of Scandinavian Mathematicians . Copenhagen 1946.
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e
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. . . .
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.
.
Paga
Sddition . . . . . . . . . 12 associativity of . . . . . 29 coininutntivity of . . . . 35 asiom A . . . . . . . . . . 11.i . P . . . . . . . . . . . 1 1.7
.-.
Boole. C+. . . . . . . . . . bound variable . . . . . . .
8.7
.
25
Cüininutatirity of additioii . of miiltiplication . . . constants: 0 . . . . . . . . 1 . . . . . . . . . .
&
. . . ++ .
. . . . counting . v
N
. . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
- operator . . . . . . coiirse-of-raliies function - recursion . . . . . .
Deduction theoreni divisor. least . . . double recursion .
. . . . .
. . . .. . 3.. 1 . 7 :)1
112 S9 22
. . . . .
-.
. . . . . .
. . Factor. greatent common . . - prime . . . . . . . . .
.
Frege. Gformal system functions . . .
.
v
.
.
.
.
.
.
e(x)
. . . . .
.15. 18.
. . . . . . . . . .
3s 7 . . 35 . . 3s 134. 136 134. 136 . . 21 . . 146 . . 110 . . 21 . . 62
Sx . . . . . . . . . . .
140 27 27 27 44 16
-.-. -.
z, . . . 0.(x) . U(x) . . V(x) . . Z(x) . . A . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . . course.of.vaI~ies. initial . . . . . propositional . .
. . . . . . . . .
99 93 2
Gijdel. K . . . . . . . - number . . . . . greatest common factor
30
Inequalities . induction . . - scliema 1,
. . . . . . . 10 . . . . . . . 17 . . . . . . . .14. 18. 28
-. .+ .
17. 33 m(x) . . . . . . . . . . 37 Alt(z) . . . . . . . . . 20 E(i) . . . . . . . . . . 134
G
. . . . . . . . . .
-.-.
31
1. 1 . . . . . . . . .
57 57 57
73 119 119
. . . .
17.
11
. 9, . . . .
Knumeration of primitive recursive functions . . equation calculus . . . introductory . . . . provable . . . . . . verifiable . . . . . exponentiation . . . . .
. . . . . . . .
. 20 . G
. . . . . . . . . . . .
-i
. . .
189
. . . . . . . . .
. . . . . . . . .
. . . . . . . . . . . . . . . . . . . . .
143 144 97 46 66
123
Pge
. . . . 125 - . r,. . . . . . . . . . ( 2 ; initinl fiinc.tiolis . . . . . . . . 1 iiidiictioii sclieiiia 1.'
.l
iiitroductnry rquations . . . iteratiori . . . . . . . . . . ~fultipliciition . . . -, roiiiiniitnti\.ity of
2; ld
. . . . 15 . . . . :!O
Siimbcv . . . . . . . . . . 1 \-nrinlile . . . . . . . . fj i:otlcl . . . . . . . . . 14.1 , priine S3
..
.
.
.
.
.
V
.
.
.
riiimernl .
. . . . . . . . . Operatoi. countiiig . . . . 0.
3
:I
op rators: A; . . . . . . . 64 FE; . . . . . . . . . . . 64
. . . . . . . . . . .
A':
. . . . . . . . . . .
.
P a r a m ~ t r isrihatitution. . . rcciirsiun witb . . permutstion . . . . . pentatioti . . . . . . Péter. X . . . . . . . prcdicntca: E(17) . .
E ( ~ T pL) . ~ . I..
. . . . . . . 16. . 24.
. . .
. . . . . . Pf(.ri . . . . . . . . . . Pr(r~f.tl) . . . . . . . .
S,!??t.n). S,(III. 12) t
.
)
Scb,,(l./tt)
T(n)
. . . . .
. . . . . . . . . . . . . . . .
. . . . . . . . . .
T(nr.11.p ) . . . . prime ni:iilbers . . . ir~iinitiideof . - - scqiience of . - factors . . . . . primitir? recursion . proot' . . . . . . .
-..
....
. . . . . .
. . . . . . . . . . . . . . . . . .
64 73
120 129 17 121 14s 150 150 150 149 150 147 148 149 S9 90 90 93 99 , ,
27
Pxae
prot'f srlit?riin . . . . . . . . . . . . !~r'~l)~>%itic;n ! ) ~ ~ t i v n l ~ l c . c c ~ i i a t.i ~.! i i.
. . . < . i t ~ r ica:it . . . . . . . .
. . . .
34 .it j 27
i ..t.c.ii!.>;io n. uuiirs<.-ot-vnliics . i 19 ,.7 .. ! b e . . . . . . . . .-~~ririiitivo. . . . . 19, 1 19
.
.,>;iiipl c. . . . . u-it li pcirnlnetcr >;ob.utiti i t iori . . rt.ii~aindci.. . . .
. . . . .
. . l~ussc11,13.A.l\'. . . S.'c.icm:ti. 1wuUf . . . sclicmtit~i:. l . . . .
. . . . .
. . . .-. . . . .
Tctration . . transposition .
. . . .
. . . . . . . . . . . .
l!) 120 S6 .
3
. . . . . .
34 115 E . . . . . . . . . . . 116 E,. 13, . . . . . . . . . 105 T., . . . . . . . . . . . 10s 1, . . . . . . . . . . . 109 1. . T . . . . . . . . . . . 110 K . . . . . . . . . . . 105 S . . . . . . . . . . . 11s Sb.. Sb2 . . . . . . . . 104 T . . . . . . . . . . . 104 U . . . . . . . . . . . 104 U. . . . . . . . . . . . 105 single recursion . . . . . . . 19 Skoleni. Tli . . . . . . . . . 143 system. formal . . . . . . . 10 - S: 9 . . . . . . . . . . 112 W * . . . . . . . . . . 115 9?+ . . . . . . . . . . 152
. . Variable . . . . bound . . .
-.-.
. . . .
nurnber . . . vcrifiable equatioii
. . . .
. . . . .
. . . . .
16 129 13 65 G 47
