Recursive Number Theory: A Development of Recursive Arithmetic in a Logic-Free Equation Calculus

100 Pages·2000·15.96 MB·English

by  Reuben Louis Goodstein

UDIES IN LOGIC
1
i
1
AND THE
FOUNDATIONS OF MATHEMATICS
L. E. J. BROUWER / E. W. BETH / A. HEYTING
EDITORS
Recursive
Number
R. L. GOODSTEIN
NORTH-HOLLAND PUBLISHING COMPANY
AMSTERDAM
S T U D I E S I N L O G I C
R E C U R S I V E NUBIBER
AND T H E O R Y
THE FOUNDATIONS OF
:I DEVELOPhIENT OF RECURSIVE ARITHJIETIC
MATHEMATICS
IN A LOGIC-FREE EQC.4TION CALCULCS
R. L. GOODSTEIN
L. E. J. B R O U W E R
Professor of .tlcthema&ics
E. W. BETM Linioersi<y of Leicestcr
A. H E Y T I N G
NORTH-HOLLAND PUBLISHING COMPANY
AMSTERDAM
The nature of niimbers. .%rithinetic- antl tlie gRme of chess. I>t,fini-
tion of counting. Eiolution of the concept of 5 forriiul system.
CHAPTER.I . . . . . . . . . . . . . . . . . . . . . . . . .
The arithmet,ical operations. Deíinition b-i tertition und recursion.
Contracted notatiori. Single and niultiple rrc.ur.~ii>~is.
Dehition of proof. Commutative and associativil properties of ad-
dition. The equations
+
z j- (y-z) = y (2-y) and (1 ,r,yl)j(x)= (1 jz,yj)!(y).
Tlie f~mctipnZs ,, L', and p,. Inequalities. Verifiabdity and freedom
frorn contradiction.
C-R 111 . . . . . . . . . . . . . . . . . . . . . . . . .
The propositional calculus. Propositional functeion,s. The limited
universal, existential and minimal operators A:, and LE. Mathe-
metical induction. Tlie counting operator S:.
The fundamental theorems of arithmetic. Prime numbers. Unique-
ness of the resolution into prime factors. The greatest common
factor.
EXAVPLESI V. . . . . . . . . . . . . . . . . . . . . . . . .
Formalisations of recursive arithmetic. Tlie Deduction Theorem.
Reduction of the uniqueness schema.
CHAPTERV I . . . . . . . . . . . . . . . . . . . . . . . . .
Reductions to primitive recursion. Course-of-values recursion.
Recursion with parameter substitution. Simultaneous recursio~w.
Generalised induction schemata. Permutstion.
W COSTENTS
CHAPTER VI1 . . . . . . . . . . . . . . . . . . . . . . . 134
Elirnination of parameters. The initial functions alid tlie generat ion
of al1 prirnitiie recursive functions by the single scliema Fz = -4'0.
The enumerating function. A doubly recursive fuxiction which INTRODUCTION
cannot be defined by primitive recursion and substitution alone.
CHA~ERVI 11 . . . . . . . . . . . . . . . . . . . . . . . . 113 The Nature of Numbers
Godel numberirig and tlie incompleteness of aritlimetic. Tlie aritli- The questioii "lJTllat is t!ie natiire of a iiiathematical entitg?"
metisation of syntax. The constriiction of a verifiable unprovable 1s oiie which has interested thinkers for over two tliolisand yearc
eqiiation. Skolem's non-sta~idardm odrl for aiitliinetic. 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 at 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 to 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 to 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~p roperty 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?" iii the follo~~inwga y. We define the property of beirig ail empty
-4 good may of makiii? proreas with a question of this kind is to collectioii as the property of not being identical with oneself. and
nsk oiwselves "Hom do \\-e know tliat a collection has severi then the number zero is defined as the poperty of beiiig similar to
ilienibers?" because tlie aiiswer to thi.; cjiiestion shoiild certainly the einpcu collection. Sest me clefine the standard unit coUection
briiig to liglit sometliiiig nliich collcctions of seven objects sliare as the collection mhose only member is the niiniber zero, and tlie
iii coinmoii. An obrious iiiisxer is that \ve find out tlie number of riumber one is defiiied as the property of being similar to the unit
a collection by co?t~ztin!t/l ie collectioii biit tliis ansmer does iiot collectioii. Then the standard pnit is talcen to be tlie collection
seeni to lielp us becaiise, \rlieii \ve courit a collection, me appear to m-hose merribers are the numbers zero nnd uriity and the number
cio no more thaii '1:tl)í-1' c:rcli ;iieriibt>r of tlie collection mitli ;L trro is defined as the property of bniri~s imilar to tlie standard pair,
i:iini;>er. (Think of a li~il\o l Uoltiitrsr iui::Scririg off.) It Flcnrly dnes rintl so oii. This is. iii effect. tlie de!i~ii:ion of niirnber which mas
i:ot provide a definition of iiuiiiber to say tliat iiurnber is a property discovered by Frege iii 1SS-k aiid. indepeiiciently, by Russell in
of a collection which is found by assigning numbers to the members 1904. It cnnnot, homever, be acccnted as a coniplete aiismer to tlie
of tlie collection. problelii of the iiatnre of number.~- 4ccordiiiq to the definition,
iiumber is a similarity relatioii betwecn collc.ctions in mhich each
The Frege-Russell Definition element of orie collection is made to correspont7 to a certain elernent
To label each meniber of a collection with a number, as we seem of tlie other. and vice-versa. Tiie ~realíriessi n tlie definition lies in
to do in counting, is in effect to set up a correspondence between this notioii of correspondence. Hov; do \\-e know vhen tn-o elements
the members of two r~111ectioiis.t he objects to be counted and tlie correspond? Tlie cups and saucers in a collectio:i of cups standing
natural niimbers. Iii counting, for esample, a collection of seven iii their saucers have an obvious correspondence, but mhat is the
objects, we set up a correspondence between the objects counted correspondence betmeen. say, the planets and the JIuses? It is
and the numbers from one to seven. Each object is assigned a no use saying that eTen if there is no patent correspondence
unique number and eacli number (from one to seven) is assigned bet~veent he planets and the Iluses, \re can easily establish one,
to some object of the collection. If we say that two collections are for how do we h o w this, and, ~ h aits more important, what sort
simikr when each has a unique associate in the other, then counting of correspondence do me allow? In deñning number in terms of
a collectioii may be said to determine a collection of numbers similarity me have merely replaced the elusive concept of number
similar to the collection counted. Since similarity is a transitive by the equally elusive concept of correspondence.
property, that is to say, two collections are similar if each of them
Xumber and numeral
is similar to a third, it follows that in similarity we may have
found the property, common to all collections of the same number, Some matliematicians have attempted to escape the difficulty
for which we have been looking, and since similarity itself is in defining numbers, by identifying numbers nith numerals. The
deñned without referente to number it is certainly eligible to serve number one is identified with the numeral 1, the number two
in a definition of nirmber. To complete the definition we need only with the numeral 11, the number three with 111 , and so on. But
to specify certain standard collections of numbers one, two, three, this attempt fails as soon as one perceives that the properties of
and so on; a collection is then said to have a certain number ody numerals are not the properties of numbers. Xumerals may be
if it is similar to the standard collection of that number. The blue or red, printed or handwritten, lost and found, but it makes
numbers thernselves may be made to provide the requíred standards no sense to ascribe these properties to 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 Anthmetic and the Game of Chess
attempt to dehe numbers iii terms of numerals, makes niimbers, Tlie game of chess, as has often beeii ohserved, affords an escelleiit
not the same thing as, but the nnmes of the numerals ; tliis escapes paralle] with niathematics (or. for ~liatm atter. witli language
thc absurdities mhich arise in attempting to ide~ltifyn umber and itself). To the numerals correspond the cliess pieces, and to the
numeral but it leads to the eqiially absurd conclusion that some operations of arithmetic, the riiox-es of the game. But tlie paralle1
one notation is the quintessence of number. For if numbers are is even closer thari this, for to tlie problem of defiiung number
tlie names of numerals then we must decide whicli tiumerals theg corresponds tlie proble~no f defining the entities of tlie game. If
name; we cannot accept the number ten for iiistance as botli the \Te ask oiirselves the questiori "TVliat is the king of cliess?" me
name of the roman numeral and the arabic .numeral. ,hd if it is finct precisely tlie same difficiilties nrise in trying to h d a n ansxver
said tliat tlie number ten is the name of all the iiunierala ten then whicli we met in oiir consideration of tlie problem of clehing the
we reach the absurd concliision that tlie meaning of a number word concept of iiiimber. Certainly the king of chess, mhose moves tlie
changes with each notational innovation. rules of tlie game prescribe, is not the piece of characteristic shape
The antithesis of "number" and "numeral" is oiie which is n-hich we cal2 the kiiig, just as a numeral is not a number, since
bommon in language, and perhaps its most familiar iiistance is to aiiu other object. a matchsticlr or n piece of coal. mould serve as
be found in the pair of terms "proposition" and "sentence". The me11 to play the king in any gane. Instead of the question "Wliat
sentence is some pliysical representation of the proposition, but is tlie king of chess?" let us ask "TVhat makes a particular piece
cannot be identified ~vithth e proposition sirice di£ferent sentences in the game the king piece?" Clearly it is not the shape of the
(in different languages, for instance) may express the same proposi- piece or its size, since either of these can be changed at will. What
tion. If, however, \ve attempt to say just what it is that the sentences constitute a piece king are its mores. That piece is king which has
express we find that the concept of proposition is just as difficult the king's moves. And the king of chess itself? The king of chess
to characterise as the concept of number. It is sometimes held is simply one of the parts which the pieces play in a game of chess,
that the proposition is something in our minds, by contrast with just as King Lar is a part in a drama of Shakespeare's; tlie actor
the sentence, which belongs to the externa1 morld, but if this who plays the King is King in rirtue of the part which he takes,
means that a proposition is some sort of mental image then it is the sentences he speaks and the actioiis he makes, (and not simply
just anotlier instance of the confusion of a proposition with a because he is dressed as b g ) a nd the piece on the chess board
sentence, for whatever may be in our minds, whether it be a which plays the king-role in the game is the piece which makes
thought in words, or a picture, or even some more or less amorphous the king's moves.
sensation, is a representation of the proposition, differing from the Here at last we find the answer to the problem of the nature -
written or spoken word only because it is not a communication. of numbers. \Ve see, first, that for an understanding of the meaning
In the same way we see that the view that number is indefmable, of numbers xve inust look to the 'game' nrhich numbers play, that
being something which we know by our intuition, again confuses is to arithmetic. The numbers. one, two, tliree, and so on. are
number with numeral, that is confuses number with one of its characters in tlie game of arithmetic, tlie pieces which play these
representations.
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 ~vordsm ore suitable to the conteirt. what constitute
+ + ".
a sigil t!ie sign of a particular niimber are tlie trrr?asformation mtles "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
of tlie sisil. It fouo~vst,h erefure, that tiie OiJECT OF OUR STODY IS
the siibstitution of "xll" or '.Oe' for "x" iii effect allow the
SOT NOSIBEP. ITSELF BCT TIIE TI¿-tSSI:OCJIATIOX RrLES OF THE
substitution of nny nxmernl for x; the object of tlie formidation
rr-i\imn SIGXS, aiid in the cliapters \1 liicli foilo\i* me sliall have
n-e have adopted is that it scrres to define tlie concept of nny
iio fi,rtlicr occasion to refer to the iiiinihcr coi~c~pBt.u t jiist as the
numeral and the coricept of a nzo?1erul t.nri«ble siinidtaneously.
rii1c.- of clieus are cwreritly forniulated i~tie rrus of the entities of
CIICLL~S. SO tliat we say, for iristtziice, the kiiig of chess inoves only In the sequel, not oidy the lcttci. .c. hut otlier letters, too. will be
used as numeral variables.
one scluare at a time (escept iii castli~ig)i.r istr;xrl of tlie completely
tqi~ivalentf ormulatioii '%he piece p1;iyiri~t l:c part of king (or Tlie numeral formed by n~itiiigso me iiiinieral for -.z" iii ..x-L1 "
is called the svcces.sor of that iiuii~er;il.F or iiistance, n~iting
>iliiply tlie kiiig-piece) is inoved oiily orie arii:ti.re :~ta tinie (exce@
iii c,is¿liiig)'' so we shall conti~iiie,i ii piirely descriptil-e passages, "O+ 1 ;1" for "x" in "x; 1" n-e obtain ..O - 141 - l", tlie successor
to fjrniiilate the operations of aritlimetic in terms of aritlimetical of "O+ l L1 ". For this reason .'.zL1 " is called the (sign of the) .
successor function. The definite article is somewhat niisleading,
riititivs iristead of arithmetical signs. For instance, \ve niay speak
*of "ihe sii~iio f the iiumbers two and tlu-ee" ratlier than confine 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
'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 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
"::" are niimerals ~vhoser oles are thoh~o f tlie numbers two and
t!ree. To put it another way. in defiiiing the part played by a function, the uniqueness ir1 qiiestion heiiig tlie uniqueness of tlie
si1 -ii 1iPe g,in aritlimetic, we shali say that what we are defining sign which results when we mite some definite numeral for trhe
:.
is tlic suiu function, but the dehition itself will refer only to rariable, be it denoted by z. y or
For purposes of standardisatioii of notatioii we sliaii have
operations for transforming expressions which contain the sign f.
"+
occasion to introduce, insteaci of the 'alphabet' "O", "1" and "
Sumber Variables for writing numerals, the 'alphabet' "O". '.S" iii which the numerals
Tlie paiallel between chess and arithmetic breaks down when become "O", "SO". "SSO", "SSSO" ai~dso 011. 111 this notation the
sign of the successor function is ..SxYa' nd the transformation rules
wc contrnst the predetermined set of pieces in the game of chess
for a numeral variable x are (i) Sr may be mitten for x, (ii) O may
n-ith the licence granted to arithmetic to construct numerals at
be written for s.
:vill. In tliis respect arithmetic more closely resembles a languqe
Another notation in current use employs 'cx'" for the successor
liich places no h i t , in principie, upon the length of its words.
fiinctíon. so that the numerals are vrrítten "O", ..Of", "O"", "0'""
-1 familiar notation for numerals expresses them as ~vordss pelt
"+"; and so on.
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", '<O+ 1 + l", "O + 1 + 1+ 1". The forniation Definition of Counting
Il'o theory of tlie natural numbers is complete mhich cloes not
of nunierals may be fztlly characteriscd hy means of two operations,
also take into account the part which niimbers play outside
as f~llows.T T'e estend the alphabet by the introdiiction of a new
arithmetic. It is not only a propertj- of the number nine tliat it
sijn, '.x", aiid form 'words' by writing either "O" or "xf 1" for
'.x": for example me may form in turn, "x". "xf l", "x+ 1- k l", is a square but also that it is the number of tlie planets. and this
and one is two". "two and one is three", "tliree and one is fourW
latter property is not a consequence simply of the laws of arithmetic.
and so on. It is the recitation of these rules (in an abbreviated form
According to the Frege-Russell dehition of number, tlie number
in which each 'and one' is omitted, or replaced by pointing to,
of a collection is found by testing it for similarity with the standard
or touching, the object counted) mhich gives rise to the illnsion
unit, pair, trio, and so on, in turn, this testing being carried out
that in counting we are associating a number witli each of the
by the process of counting, biit as me llave proposed a definition
elements counted, mhereas me are in fact making a trarlslation
of number mhich does iiot rest upon tlie iindeñned concept of a
from the notation in which tlie number signs are "one", "one and
similarity correspondencc we cannot nccept counting, in the
one", "one and one aiid one", and so on, to the notation in mhich
Frege-Russell sense, as a nieans of fincling tlie nurnber of a class,
without readmitting this undefined concept. Tliere is, however, the signs are "one". "two". "three", and so on. The true nature
an entirely different interpretatioii of the pi-ocess of cóunting, of counting is perhaps most clearly brought out if me re-introduce
which makes counting arailable to us as a means of recoraing the.. the older process of making a tally. 3Iaking a tally of a collection
number of a collection. without transcending tlie limitatioii we consists in some formalised representation of the elements of the
imposed iipon ourselves of expressing tlie properties of numbers collection, say by means of dashes on a sheet of paper, so that in
in ternis of tlie transformation riiles of the iiiimber signs. We making a tally we are copging a number sign in some standard
Jtart by separating tivo distiiict stages in the process of eo~inting. riotatiori - finding the number of the collection, by treating it
The kst of tliese is m-liat we shall cal1 "using a collection as a as a number sigii and copying this sign. Thus a tally of the planets
numeral" mhich consists in overlookiilg the individual 'idio- consists in the row of dashes
syncrasies' 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 If we now proceed to traiisform this sign by means of the trans-
familiar in all acts of reading, writing or speaking; the letters "a" formation rdes 11=2, 11=3, 31=4, 41=5, 51=6, 61=7, 7 1 ~ 8 ,
on a printed page, for instante, have their several differences and, 81=9weobtaininturn111111111=21111111=3111111=411111=
subject to sufficiently close scrutiny, are as different as say the = 511 1 1 = 61 11 = 71 1= S1 = 9, which completes the transformation.
soldiers in a platoon, but for the purposes of reading me ignore In counting as we teach it today, the processes of tally making
these differences aild treat the vnrious a's as being the same sign. and sign tzansformation are carried out simdtaneously, thus
And SO too, in speaking, we treat as the same a variety of slightly avoiding the repeated copying of the 'tail' of the number sign in
differeiit sounds. In a differerit contest, signs which we would transforming to an arabic numeral. It is important to realise that
accept as the same for reading purposes, are carefully distingiiished, counting does not discover the number of a collection but transfo~ms
as, for esample, when we test the quality of printing. The process the numeral which the collecticun itself instances from one notation
of overlooking some differences, but not others, is fundamental in to another. To say that any collection has a number is just to
language; it is tlie process by mhich ive subsume objects with a say that any collection may be used as a number sign.
'family likeness' under a generic name and tlie process ~~1iich
makes possible the use in language of universal \trords. Without it, Formalisation of Counting
the concept of tlie number of a class could never have arisen. The Counting may be formalised in a system of signs by formulating
second stage ili the process of counting consists in a transformation the transformation rules of a counting operator "X". JVe represent
from one number notation to another by means of the rules "one t,he objects in the collections to be counted by letters a: b: c, .. ..
10 ISTRODvCTION INTRODUCTIOS
and collections by conju~lctionsl ike a S: b, n S: ZJ LP: C; a single considera1)le developmerit during tlie past centiiry. 3uclid7s
object being regarded also a collection. The letter 1 we use as a interition iri the "Ele~nents" was to decliice the 1vliole bod- of
variable for an object. that is. ii letter for which any object may geometrical lmowledge of his time from tt few self crideiit trutlls
be n~ittent;h e capital letter L berves as n variable for a collection (calied asioms) by purely logical reasoning. Euclid did not. Iiowerer.
2nd maj-. in any contest. be rcl~lncc(1b y a alletiiiite coilection or specify the naturc of 'logical rea?oiiiriq' nrid tlie tirst attenipt to
by "1; S: 1". The nunier~lso f tlie systcin are tlic signs (witliout X) 1 do so was made by Ckorge BooIe. iii Is17. iii Iiis JInfho)inficnl
obtairied from l. x ariti the succrssor furiction n:-- 1 hy sribstitution. i dnnlysis o/ Lqic. Boole coristriicted a ;:.-rnbolíc lurigiinge. in ~vliicli
Thcn Tre define l the 'laws of tliought'. formiilater! iis axioriis, niny be sti~cliedb u
Y(l)=l, S(L&l)=S(L)--l. mathematical tecliniqiies. Iii thc cnri~;lletct lcx-clo1)nicnt of the
notion a forinal systcni is ari a-icrilhl:icr. of slgnc qcbpa:.nted into
'Lliebe equations sufice to determiiic tlie iiiiiiiber of any c~llectionl
various categories. tlieir usage it~nilr! 1 ; ~\: irious con\-eiitioiis (tlie
For instance, substituting "a" for tlic variable-sign "l", in the
asionis and transformatioii rules) tiie objvct of the systeni being
kst. \ve obtain N(a)= 1, and tlieii, substitiiting "o" for "L"
to arraiige sequences of formiiIne (rhicii are tliemselres svqiiences
and "h" for "l" in tlie secoiid. \ve obtaiii
of signs mith certuin >pecified forniation r.iilcs) iii certniri relatiori-
+
! N(a S: b)= N(n) 1 ships to one anothcr to forin a particular pattern callec1 proof.
and so. X(n & b) = 1 -L 1. I A formal systeni ninu contaiii l,otli !ii,tthc.rnatical nrici logical
YPS~s, iibstituting "a & h" for "L" and "c" for "1" me find 1 sips (the distinctiori is an arbitrar'- ori:?). aiitl nintheiiintical and
N(a &O &c)=X(a & b)+1=1-~-1+1, logical asionis; its essential featurc, q~1f,o irnal systeni. is that its
and so on. operation does not presuppose any !íiiu:r letige of the significance
' ITc observe that the definition of X(L) is by recursion, that is of the signs of the system thali is givcii hy the nsiomq aiicl trans-
to say. X(L) is not simply an abbreviation for some other expres- formation rules. Tiie mathematical a~ioilisa re no loiiger "self
sion, as, fbr instante, when we define 2= 1+ 1, the sign "2" may evident truths" biit arbitrary initinl ;io.;itions in a ,rran;c. and tlie
+ logical aasioms espress. not the "la-\\-=o f tlioiig!it" but arbitrary
be replaced by "1 1" for which it is merely an abbreviation, but
conrentions for the use of the logical .igns.
X(L)i s determined only step by step, by introducing the members
In the formal s~stenmi itii ~rhichn -e :iiall first be coiicerned in
of tlie class to be counted one at a time (or shedding them one at
this book, the equation calculus. the or:!y :igils are sigris for functions
a time). R'e may express this by saying tl~atfo r the variable L, l
S(L)i tself is undefind, only the result of substituting a definite aiid numeral variables. and tlie equality sigii. Tiirie are iio asioms tri
escept tlie introductory equations for function si;ns. and there
class (lilíe a & b & c) for L being defined by the reciirsive definition.
v
is no appeal to 'logic7. the operation of tlie sj-steni being specified
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 l siniply by the transformation rules for the mathec?,ztical signs. u
by siibstitution for any particular class a & b & ... & k. It is sho~r-nth at a certain branch of losic is defi?iable in !he equation u
calcl~lusa nd logical signs. and theor?:i?s, are introduced as conre-
I
Evolution of the Concept of a Formal System 1 nient nbbr~z.icttionsf or certain f1inctio:is ancl formulae. This brancli w
i
In the following chapters we shall set up arithnietic as a formal of logic is characterised by the fact that it caii assert the ~xistence
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SY.S~P-BI.T lie idea of a formal system is one which derives from of a number mith a giren propcrty uiily miien tlie niimber iir m-
Euclid's presentation of geometry, but the notion has ~indergone question can be found by a specifiahle riomber of trials.
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CHAPTER 1
DEFINITIOX BY RECURSIOK
1. Variables
We have already had occasion, in the definition of a numeral.
to refer to the use of a letter as a sign for a variable. In terms of
two operations
(1) replacing x by x i1 ,
(2) replacing x by O,
\ve defined numerals to be tlie signs constructed from x by the
repeated application of tlie operation (1)f olioíved 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.
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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 introducto-ry 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