1. 2012/2013
History of Mathematical
Induction & Recursion [B].
[Anaccount of the historyof mathematical inductionandmathematicalrecursionwithan
expansiontotransfinite inductionandtransfiniterecursion]
Damien MacFarland
Alexandre Borovik
Math40000

2. History of Mathematical Induction & Recursion [B].
1

3. History of Mathematical Induction & Recursion [B].
2
Introduction:
On the 3rd
March 1845, GeorgFerdinandLudwigPhilippCantorwasborninSt Petersburg,
Russia.Cantorwouldsoonmove toGermany where he wouldspendmostof the remainderof his
life.Havingstudiedtobecome anengineerat hisfathers’ request,Cantorwouldturn to
mathematics.Cantorwassaidto have shownexceptional skill inmathematicswhilstatschool,
particularlyintrigonometry.However,whilstatuniversityinBerlin,Cantorcame underthe influence
of three greatmathematicians –Karl Weierstrass,LeopardKroneckerandErnstKummer – and it was
undertheirinfluence thatCantorshowedaninterestinnumbertheory.
Havingfinisheduniversity,Cantorwouldeventuallyfindapositionatthe Universityof Halle
where he workedalongside HeinrichHalle,whochallengedCantortoprove the problem of the
uniquenessof representationof afunctionasa trigonometricseries.WiththisCantorturnedhis
attentionfromnumbertheorytoanalysis.Cantorsolvedthe problemwithinayearof takingup the
challenge.Trigonometricseriesrepresentationwoulddraw Cantor’sinterestawayfromanalysisand
back towardsnumbertheory.
In 1873, Cantor producedthree ground-breakingresults.The firsttwoshowedthatthe
rational numbersandthe algebraicnumberswere bothcountable,i.e.couldbe putintoaone-to-
one correspondence withthe natural numbers.The thirdresultwastoshow that the real numbers
couldnot be put intoa one-to-one correspondencewiththe natural numbers,thusmakingthem
uncountable.The resultwas verycontroversial atthe time andwouldleadtoa lot of toughyears for
Cantor as he triedto justifyhisresulttothose whodoubtedit.
One reasonfor the ridicule whichCantor’sworkreceivedwasbecause of itsintroductionof
newconcepts suchas his transfinite numbers intomathematics.Anotherreasonwasitslevel of
abstractionand use of philosophical arguments.WithCantorintroductionof new conceptshe often
arguedextensivelyfromaphilosophical pointof view,distancinghisreasoningfrommathematics.
Howeverthe maincause of confrontationtowardsCantor’sworkcame fromhisuse of the infinite.
We knowthatboth the natural numbersandthe real numbershave nogreatestnumber.
Both are infinite totalities.Cantor’sresultthatthe real numberswere notdenumerableessentially
saidthat the real numbershad a differentinfinite entitythanthatof the natural numbers. What
Cantor impliedisthattheyare more thanone type of infinity. Inordertojustifythis,Cantor
embarkedona long,enduringjourneytodevelophistheoryof transfinite numbersandsolve his
ContinuumHypothesis.

4. History of Mathematical Induction & Recursion [B].
3
The concept of infinityhasalwaysbeenadifficultone.Fromthe time of the earlyGreek
mathematiciansupuntil the time 18th
Century,infinitywasalwaysseenasthe potential infinite;
everywhere finite butstillwithoutend.Thiswasdue toAristotle whohadbannedthe actual infinite
inthe 4th
CenturyB.C.due to itscontradictive characteristics.However,inthe 19th
Century,
philosopherssuchasBolzanobeganto workon the infinite andinvestigatedthe paradoxeswhichit
ledto withinmathematics.The maindigressfromthe use of infinityinmathematicscame fromits
paradoxes.Dedekindturnedthisnegativeintoapositive andwasone of the firstmathematiciansto
challenge the Aristotle stance onthe infinite.
The paradoxesthatwere producedas a resultof infinityoftenstemmedfromtheir
contradictionsof lawsanddefinitionsthathadbeenconstructedonlytoincorporate the finite.
Dedekindtookone suchcontradiction andusedittodefine aninfiniteset.Fromthishe thendefined
the natural numbersas one such infiniteset. However,before the time of CantorandDedekind,
mathematicianshadconstructedtheirownprocedure of dealingwithinfinitequantities.
Withinmathematicsthere are avaried collectionof mathematical proceduresandconcepts
whichwe frequentlyuse.Some we use tocreate anddevelopnew results,otherswe use toverify
and prove onesinwhichwe alreadyhave.Asmathematicshasdevelopedwithtime,ithasseenthe
introductionof newconceptswhichhave servedinhelpingwithsuchdevelopment. These new
conceptsare notalwaysperfectintheirbeginningbuthave beenmoulded intomore desirable
forms. Withregardsto handlinginfinite quantities,the principlesof inductionandrecursionmade it
possible tomanage the infinite withrespecttofinite terms.
In thispiece of workI wishtoinvestigate the originsof the principlesof inductionand
recursionandhowthe twoconfine the natural numberstoa couple of finite steps.Ishall give
account forhow mathematical recursionwasnaturallyintroducedtomathematicsandhow itwas
usedto produce one of the most importantresultsof the philosophyof mathematicsbyGödel.
I shall thenworkmy way back, fromPascal whogave structure to the conceptof induction,
throughthose whowere the firstto realize the benefitsandusesof sucha principle andshowed
traces of the conceptthroughouttheirwork,usingtheirownsimilarversionstoobtainresults,and
mostlyproof to results,inshort,finite,structuredmanners.ThenIshall addressthe name of
mathematical inductionandhowthe principle becamepopularinthe 18th
and 19th
Century’s,paying
particularattentiontohowbeneficial the conceptwastoPeanoand Dedekind.
The principlesof inductionandrecursionare of amajor benefitintacklingthe totalityof the
natural numbers,however,theycannotbe appliedtoCantor’stransfinite numbers.Thus,Ishall give
an account of the originsCantor’stheoryof transfinitenumbers,the reasonshe felttheywere

5. History of Mathematical Induction & Recursion [B].
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needed,the struggle he hadfortheiracceptance,the differentpathshe tookintheirdevelopment
and the resultingtheoryhe leftbehind.AfterCantor’slastmajorcontributiontothe theory inwhich
he produced,itwas takenupby otherswhotriedto solve anyproblemswhichCantorleftbehind.
The subjectwouldcome underthe influence of ErnstZermelowhowouldaddaxiomstoCantor’s
naïveset theory.Throughoutthe developmentof the theoryof transfinite numberswe shall
produce the conceptsof transfinite inductionandrecursionandgive anaccountof themtowards
the end.
I expectthe readertobe familiarwithabasicunderstandingof settheoryandhave a
universitylevel of mathematical understanding.

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Contents:
Introduction:............................................................................................................................ 2
[1] Induction & Recursion:............................................................................................................. 6
[1.1] The Natural Numbers:.......................................................................................................6
[1.2] Mathematical recursion:....................................................................................................8
[1.3] Gödel’s Theorems:.......................................................................................................... 12
[1.4] Mathematical Induction: ................................................................................................. 16
[1.5] The Origin of Mathematical Induction:............................................................................. 20
[1.6] Maurolycus & Gersonides:............................................................................................... 25
[1.7] The Name – Mathematical Induction:............................................................................... 30
[1.8] Bibliography:................................................................................................................... 33
[2] Set Theory and Transfinite Numbers:...................................................................................... 34
[2.1] Giuseppe Peano:............................................................................................................. 35
[2.2] Richard Dedekind:........................................................................................................... 38
[2.3] Arithmetices principia and Was Sollen:............................................................................. 43
[2.4] Dedekind’s Interest in Infinity:......................................................................................... 46
[2.5] Georg Cantor:................................................................................................................. 49
[2.6] Cantor’s Quest for the Transfinite Numbers:..................................................................... 54
[2.7] Cantor’s Struggle:............................................................................................................ 61
[2.8] Post-Grundlagen:............................................................................................................ 65
[2.9] Contributions to the Founding Theory of Transfinite Numbers:.......................................... 72
[2.2.1] Part I:....................................................................................................................... 72
[2.2.2] Part II:...................................................................................................................... 82
[2.10] Aleph-One and Transfinite Induction:............................................................................. 88
[2.11] Post-Beiträge:............................................................................................................... 92
[2.12]: Axiomatic Set Theory and Transfinite Recursion:............................................................ 94
[2.13] Summary:..................................................................................................................... 99
[2.14] Bibliography:................................................................................................................103

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[1] Induction& Recursion:
[1.1] The Natural Numbers:
In mathematics,there are numerousthingsthatwe take forgranted.Asmathematicshas
progressedthroughthe ages,ithasgrown andbecome more abstract. Whenwe say ‘5’ whatdo we
actuallymeanby‘5’; 5 apples,5 minutes,5metres,5 grainsof sand? Whenwe say that ‘ten equals
twofives’we knowthatwhenwe have 2 sets/groups/clusters,eachof size/quantity/length5, and
thenour total is 10, regardlessof any 10 possible concrete examples. Backinthe agesof the ancient
Greeks,where mathematicshad essentiallygrownfrom, philosopherslike Plato,Socrates,the
Pythagoreansetal, beganto ignore concrete numbersof ‘things’andtooktocounting‘things’
philosophically,withnoconsiderationof what actuallycountinganything,andthatliveson today.
Buildinguponthis,there ismuchmore we take forgranted withregardsour number
systems e.g.the definitionsof odd,even,prime,complex,irrational,integer.Anditdoesn’tstopat
definitions,we cancontinue toinclude basic,essential propositions,(1 + 1 = 2; 4 isaneven
number;Pythagoras’Theorem).There are formulasandsummationsthatwe expectanyotherpier,
withsimilarlevelsof mathematical knowledge asourselves,to know andto believetobe true.
One thingwe alwaysuse,withoutanyconsiderationastowhyit istrue,is the notionthat,as
an example, 2 < 3.Of course thisdoesseemobvious;if one has2 applesbut3 orangesthenof
course one has more orangesthan apples,whichistrue syntactically.Butwhydowe considerthe
number2 to be strictlylessthan3? Andwhyin turnis 3 strictlylessthan4, 4 lessthan5? We
overlookthe factthat the natural numbers are in a naturalorderingi
.Startingfrom1, 1 is succeeded
by 2, whichissucceededby3, and that by4 and that by5, then6, 7, 8…
We obtainanynatural number 𝑛 from1. The Greeksthoughtof 1 not as a number,butas
‘unity’oras the ‘One’.Thusany numberisa collectionof units.We obtainone numberbytakingits
predecessorandadding 1 tothat. But how do we obtainsaidpredecessor?Well,fromits
predecessor;andthatpredecessorfromitsownpredecessor;thisinturnhasits ownpredecessor.
Eventuallywe arrive at 1,which,thinkingasthe Greeksdo,iswhere we can go no furtherwiththe
natural numbers.
The natural numbersare a simple concept,beginningfrom 1,we add 1 to it 𝑛 − 1 timesto
obtainthe natural number 𝑛. Thenadding1 to 𝑛 we obtain 𝑛’s successor, 𝑛 + 1.Thus everynatural
i In fact they are ‘a well ordering’ and we shall cometo that later.

8. History of Mathematical Induction & Recursion [B].
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numberhasa successor;eachdependsonthe one definedbeforeit. Thislatternotionof defining
somethingintermsof howwe define anotherbringsustomathematical recursion.
Accordingto the Oxforddictionary,recursionisthe processof ‘repeatedapplicationof a
rule,definition,orprocedure tosuccessive results’.We see resemblancesof recursionineveryday
life,e.g.the Russian MatryoshkaDolls,The Droste Effect;howeverwe mainlyuse recursionin
mathematics inorderto define suchthingsas sequences,series,relationsandfunctions.We canuse
recursiontodefine sets,ortodefine unionsandintersectionsof sets. Andthere are manymore
usesof recursionindifferentareasof mathematics - logic,statistics,graphtheory…

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[1.2] Mathematical recursion:
Mathematical recursion isthe processof definingamathematical process byrepetition;a
functionorprocedure definedintermsof itself. We define the natural numbersbyrecursive
definition:startingfrom 1,add1 to obtainthe nextnatural number.Fromthisnew number,namely
2, add 1 to it to obtainthe nextnatural number, 3.From3, we
add 1 to obtain 4. Etc.
Before we startto look furtherintomathematical
recursion Iwouldlike tolookat a well-knownandcommon
example,the Fibonacci Numbers, attributed toLeonardoPisano
Bigollo(LeonardoFibonacci) [ca.1170- ca.1250] of Pisa. The son of
a wealthybusinessman,Fibonacci hadanextendedinterestinthe
mathematicsof the East and the Arabs.In 1202, after returningto
Italyhavingvisited Egypt,Sicily,Greece andSyria,Fibonacci
published Liberabaci,a textmainlyfocusingonthe base 10
arithmeticof al-KhwārizmīandAbūKāmil.In Liber abaci appearsthe Fibonacci Numbers.
The Fibonacci Numbers are definedas:
 Define 𝐹1 = 𝐹2 = 1;
 For the natural number 𝑛 > 2,define 𝐹𝑛 = 𝐹𝑛−1 + 𝐹( 𝑛−2).
Thus the 1st
fewtermsof thissequence are 1,1, 2,3, 5,8,13, 21,34,55, … .
Thisprocess for definingastructure throughan arbitrary numberiswhatisknownas
Recursive Definition inmathematics. Notice thatwe have essentially2parts to the definitionof the
Fibonacci Numbers:
(1) A startingpoint,indexedbythe natural numbers 1 &2;
(2) A rule forthe formulationof greaterFibonacci Numbers,indexed bythe naturalsgreater
than 2. The rule correspondsto the 2 previouslydefinedFibonacci numbers.
Generally,todefineamathematical procedurerecursively,we:
(1) assigna Base Case;
(2) setup a Recursive Step.
The Base Caseservesas ourstartingpoint. 1 beingthe base case of the natural umbers; 𝐹1
and 𝐹2 serve as the base case of the Fibonacci Numbers.The base case isour reference pointfrom
where we can continue, toformulate the remainderof mathematical procedure.Andwe dothis
Figure 1: Leonardo Fibonacci.

10. History of Mathematical Induction & Recursion [B].
9
fromour Recursive Step.The recursive stepallowsustocontinue to
formulate more examplesof a procedure;itextendsourdefinition
towardsa possible infinite numberof termsforour procedure.The
recursive stepallowsustodescribe aninfinite numberof instances
ina finite quantityandthe natural number 𝑛 holdsthe infinite
factor.
We can thinkof a recursive definitionasa sequence,
𝑢1, 𝑢2, 𝑢3, 𝑢4,… , 𝑢 𝑛,… indexedbythe natural numbers.Thisallows
us to visualizethe sequence inanorderparallel tothe naturals and
see the sequence asbeing acountable setof terms.We can draw up
results,formulas,ratiosetc.betweenthe natural number 𝑛 andthe
term 𝑢 𝑛 of our sequence orevendefine the our 𝑢 𝑛 intermsof the
natural number 𝑛. Anexample of the latteristhe factorial function,
whichwe know to be 𝑛! = ∏ 𝑘𝑛
1 .
Howeverwe candefine the factorial functionrecursivelyasfollows:
 Base Case:let1! = 1;
 Recursive Step:define,for> 1, 𝑛! = 𝑛 ∙ ( 𝑛 − 1)! .
Nowif we were to rename ourfactorial functioninthe structure of a sequence,thenwe
couldlookat the previousdefinitionas:
 Base Case:let 𝑢1 = 1,
 Recursive Step:define,for 𝑛 > 1, 𝑢 𝑛 = 𝑛 ∙ 𝑢 𝑛−1
Thus the 1st
fewtermsof the sequence are: 𝑢1 = 1, 𝑢2 = 2, 𝑢3 = 6, 𝑢4 = 24, 𝑢5 = 120...
The Degree of recursion issaid tobe the numberof predecessorsthatare usedindefining
any termby the recursive stepi.e.itisthe numberof termsdefinedinthe base case.
Lookingback at examplespreviouslydefined,the Fibonacci numbersare of degree 2
whereasthe factorial functionisof degree 1. Beginningwiththe 1st
3 Fibonacci numbersasa new
base case,we can define the Tribonacci Numbersbytakingthe sumof the previous3 defined
numbersas opposed to2. Thus the Tribonacci Numbersare of degree 3.Similarlywe candefine the
Fibonacci 𝑛-stepNumberSequence whichwillbe of degree 𝑛.
Nowthat we have lookedatthe degree of recursionIgive a standard procedure for
definitionof recursivedefinition:
Figure 2: The Sierpinski Triangle –
showing iterations of the Recursive
Step.

11. History of Mathematical Induction & Recursion [B].
10
1. Base Case:let 𝑢1 = 𝑛1, 𝑢2 = 𝑛2,…, 𝑢 𝑚 = 𝑛 𝑚 where 𝑚 isthe degree of recursionfor this
definition and 𝑛1, 𝑛2,…, 𝑛 𝑚 are 𝑚 pre-definedterms.
2. Recursive Step:let 𝑢 𝑚+1 = 𝑓( 𝑢1, 𝑢2,… 𝑢 𝑚)where f is an 𝑚-dimensionalfunctionorrelation.
Setsare anotherimportantentityinmathematicsthatcan be definedbyrecursion;we can
define ℕ,the setof natural numbers,recursively inthe followingmanner:
 Base case:1 ∈ ℕ;
 Recursive Step:if 𝑛 ∈ ℕ, then 𝑛 + 1 ∈ ℕ;
 ExtremalClause:ℕ is the smallestsetsatisfyingtheseconditions.
There are otherpropertiesthataccompanyourset ℕ of the natural numbers,namelythe
Peano-DedekindAxioms;howeverif we looksolelyat ℕ asa setof elements,withoutanyregardsto
the ordering,thenwe can define ℕrecursivelyasabove. Otherexamplesof setsthatwe can define
by recursionare the positive evenintegers,the integers, the setof triangularnumbers, andthe set
of squaredintegers. Insettheoryall of these exampleswouldbe accompaniedwithextracriteria
withregardstheirordering,butfornow I choose to overlookthis aswe are still justlookingatsets
withspecificelements.
Notice thatI have addedanotherpiece of criteriatothe definition of ℕ,the ExtremalClause.
We dothisto distinguishbetween 2differentsetswhichmay satisfy boththe Base Case andthe
Recursive Step butyetmay still be 2 differentsets.Forexample,the set {1,1.5, 2,2.5,3, 3.5,4, …}
satisfiesthe Base Case andRecursive Stepafore mentionedyetitisnotthe setof natural numbers
as it includessome rational numberswhich are not‘whole’numbers.
Takinga (possibly infinite) collectionof sets,we candefine recursivelytheirunions,
intersectionsandCartesiancrossproducts. Thisispossible becauseof the associativityof these
actions. For example,suppose we have the collectionof sets 𝑆1, 𝑆2,𝑆3,… 𝑆 𝑛,… where the indices
are randomlyassignedtothe setsof the collection,then
a) The union can be definedas:
 ⋃ 𝑆𝑖 = 𝑆𝑖
1
𝑖=1 ;
 ⋃ 𝑆𝑖 = (⋃ 𝑆𝑖
𝑛
𝑖=1 )⋃ 𝑆 𝑛+1
𝑛+1
𝑖=1 .
b) The intersectioncanbe definedas:
 ⋂ 𝑆𝑖 = 𝑆1
1
𝑖=1 ;
 ⋂ 𝑆𝑖 = (⋂ 𝑆𝑖
𝑛
𝑖=1 )⋂ 𝑆 𝑛+1
𝑛+1
𝑖=1 .
c) The Cartesiancross productcan be definedas:
 ∏ 𝑆𝑖 = 𝑆1
1
𝑖=1 ;

12. History of Mathematical Induction & Recursion [B].
11
 ∏ 𝑆𝑖 = (∏ 𝑆𝑖)𝑛
𝑖=1
𝑛+1
𝑖=1 × 𝑆 𝑛+1.

13. History of Mathematical Induction & Recursion [B].
12
[1.3] Gödel’s Theorems:
Mathematical recursionisa tool usedthroughoutmathematics;afew exampleswe have
seen.Itcan be usedineveryareaof mathematics, fromlogicto geometry,fromstatisticstograph
theory.Recursionseemstobe anatural processof mathematics,one withoutanyorigin. Itappears
to be a tool forbuildingstructuressuchthatwe can produce theorems,propositions,oralgorithms
baseduponthese structures.
Recursive definitionseemstofollow fromthe natural numbers since we use the naturalsto
structure thisform of definition.Aswithanydefinitioninmathematics,we canjustmake any
assumptionorassigna certain criteriaor a specificprocedure; the truthof whichwe justassume to
hold. We can do thisbecause we are more interestedinthe resultsthatwe canproduce from these
assumptionsthanthe assumptionsthemselves.We use the definition asourpointof reference from
whichwe aimto buildupon, toexplore andinvestigate. There isnorequirement toverify thatour
definitionistrue,the truthvalue followsfromthe natural numbersandthe axiomsexhibitedthere.
The originof recursiontherefore cannotbe pinned
downto one specificpointintime.The Fibonacci numbers
were developedbyFibonacci inthe 12th
centurybutthey were
saidto be knownto the Indians before that. Fibonacci wasalso
knownto have readand studied alotof IndianandArabictext
inhis time,tracesof whichcan be foundin Liber abaci.Each
row of Pascal’sTriangle, usedtodefine binomial coefficients
(whichcan themselvesbe recursivelydefined),canbe
recursivelydefinedbythe row above it.However,althoughit
isnamedafterBlaise Pascal wholivedduringthe 17th
century,
‘his’triangle wasinvestigatedbythe Geeks,Chinese,Hindu
and Arabicmathematiciansbefore him.
The originof mathematical recursioncanbe hard to
trace but itspopularityandimportance iswell known.One
such importantresulttocome fromrecursion isthat of Kurt
Gödel.
Born 1906 inBrno, nowof the CzechRepublic, Gödel wasaphilosopheraswell asa
mathematicianwhofocusedmainlyonthe logical aspectsof mathematics. Withthe Czechoslovak
Republicdeclaringitsindependencefromthe Austro-HungarianEmpire in1918, Gödel movedto
Figure 3 - A young Kurt Gödel ca. 1922.

14. History of Mathematical Induction & Recursion [B].
13
Viennain1924 to studyat university,havingalwaysconsideredhimselftobe an Austrianlivingina
Czechoslovakmajority.
In 1931, Gödel had publishedthe paper‘ÜberformalunentscheidbareSätzederPrincipia
Mathematica und verwandterSystemeI’whichcanbe translatedas ‘On Formally Undecidable
Propositionsof Principa Mathematica and Related SystemsI’.The paper,originally appearingin
‘MonatsheftederMathematikund Physik’ Vol.38,wasof significantimportance tomathematical
logicand philosophyasitcontainedGödel’s highlyimportantIncompletenessTheorems.
The theorems were usedtoanswerthe 2nd
problemof DavidHilbert’s “bootstrapping”
Program whichasked“to provethatthey (the axiomsof arithmetic) arenotcontradictory,thatis,
thata finite numberof logical stepsbased upon themcan never lead to contradictory results.”
The axiomsof arithmeticthatHilbertisreferringtoare the axiomsof the PeanoArithmetic
for the setof natural numbers, ℕ,whichappearedinPeano’s 1889 book, The Principles of Arithmetic
by a New Method:i
1. 1 ∈ ℕ;
2. 𝑎 ∈ ℕ,∃𝑎′ ∈ ℕ 𝑠. 𝑡. 𝑎′ = 𝑎 + 1;
3. ∄𝑎 ∈ ℕ 𝑠. 𝑡. 𝑎 + 1 = 1;
4. 𝑎, 𝑏 ∈ ℕ, 𝑎 = 𝑏 ⇔ 𝑎 + 1 = 𝑏 + 1;
5. 𝐹𝑜𝑟 𝑀 ⊂ ℕ, if
 1 ∈ ℕ;
 𝑎 ∈ ℕ ⇒ 𝑎 + 1 ∈ ℕ;
then 𝑀 = ℕ.
In effect,the problemisasking –isthere anyproof that thisarithmeticis consistenti.e.yield
no contradictions?Gödel usedhistheoremstoprove thatthe answertothisquestionwasinfact,
no. Hilbert’sProgramwanted tosecure the foundationsof mathematicsandthendevelopits
foundationsfurther.Hilbertwantedtofindoutwhatlayin store forthe future generationsof
mathematicians,whatpossiblenewtechniquestheywoulduse, andmostimportantly,whatresults
wouldtheyyield. Butsome thought Gödel’sanswersoughttodestroysucha possibility,whereas
Gödel himself sawitas a route to developfurtherHilbert’swork.
i I only give 5 of the original 9 axioms as they are sufficientto describethe Peano Arithmetic. The remaining4
axioms deal with the transitive,reflexiveand symmetric properties of equality in the natural numbers.

15. History of Mathematical Induction & Recursion [B].
14
The IncompletenessTheorems:
1. Anyeffectivelygeneratedtheorycapableof expressingelementaryarithmeticcannotbe
bothconsistentandcomplete.Inparticular,foranyconsistent,effectivelygenerated
formal theory thatprovescertainbasicarithmetictruths,there isanarithmetical statement
that istrue,but not provable inthe theory.
2. Anyformal effectivelygeneratedtheorythatincludesbasicarithmetical truthsandalso,
certaintruthsabout formal provability,if the theoryincludesastatementof itsown
consistency,then the theoryisinconsistent.
Gödel'spaperconsistedof 11 propositions.PropositionsVIandXIare now what are called
the IncompletenessTheoremsanditispropositionXI,the 2nd
of the Incompletenesstheorems,that
answersHilbert’sproblem.Once Gödel hasestablishedthe 1st
of the 2 IncompletenessTheoremshe
proceedstobuilduponitto produce the 2nd
.
FirstI state PropositionV:
“Every recursiverelation is definablein the systemP (interpreted asto content),
regardlessof whatinterpretation is given to the formulaeof P.”
AfterprovingPropositionV,Gödel statesPropositionVI.He doesso inthisorderas he
requires the recursive elementof PropositionV toprove VI. The 1962 Englishtranslation,byB.
Meltzer,of Gödel’spaperisintroducedbyRichardB Braithwaite,whohighlightsthe importance of
recursioninmathematicsandthe importance ithad for Gödel inhispiece of work:
“Recursive definition enablesevery numberin a recursively defined infinitesequenceto
be constructed according to a rule, so thata remarkabouttheinfinite sequencecan be
constructed asa remarkabouttherule of construction and notasa remarkabouta
given infinite totality.”
“For the proof of Gödel’s‘Unprovability’theoremtheimportanceof recursivenesslies in
the fact(Proposition V) thatevery statementof a recursive relationship holding between
given numbers 𝑥1, 𝑥2,…, 𝑥 𝑛 is expressibleby a formula 𝑓 of theformalsystemP which is
‘provable’within Pif the statementis true and ‘disprovable’within P...if the statement
is false.”
Recursionremainsanessentialandvaluable assetthroughoutmathematics.The abilityto
representthe infiniteinthe termsof the finite allowsfor anefficientand more abstractpractice of
mathematics.Its quickandeffectivemethodhasbeenknownthroughoutthe historyof
mathematics,fromthe time of the ancientGreeks,throughtothe Renaissance andonwardsto

16. History of Mathematical Induction & Recursion [B].
15
today,where itremainsincommonuse,frommathematicsusedinprimaryschool,tothe highest
and mostintellectual of levelsof mathematics.The mainpurpose of mathematical recursionisfor
statingdefinitionsordefiningfunctions,sequences,series,etc.However,asmathematicsbecame
more abstract and everclosertologic,the needformathematical proof became greater.Through
this,a close relative of mathematical recursiondeveloped;thatof mathematical induction.

17. History of Mathematical Induction & Recursion [B].
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[1.4] Mathematical Induction:
“Even in mathematicalsciences,ourprincipal instrumentsto discoverthe truth are
induction and analogy.”
– Pierre-SimonLaplace, EssaiPhilosophiquesurlesProbabilités.
Mathematical induction,oftenreferredtoas The Principle of Induction, hasclose tiesto
mathematical recursion; if recursionisthe processof buildinginmathematics,theninductionisthe
processof checkingthat build.Essentiallyinductionisaformof proof formathematical procedures
that are definedrecursivelyinmathematics.Alternatively,once we have definedaprocedure
recursively,we cancheckits validitybyinduction. Mathematical inductionisnota methodof
discovery,butamethodof provingthat whichhasalreadybeendiscovered. Inarithmeticinduction
provesthatsome propertyholdsforall positive integers.Inlogicitprovesthatthata propertyholds
for a language baseduponthe lengthof a sentence.We candefine setsrecursively,andprove
propertiesaboutthese recursivelydefinedsetsbyinduction.
Mathematical inductionbearsaclose resemblance tothe inductionusedinthe other
sciences.The inductivemethodsof otherscienceslookatgeneralitiesfromspecificexamplesin
orderto formulate ageneral andcommonconjecture thatcan be putforwardand usedinother
casesto testits strength. Similarlyinmathematicswe canusuallysee that if propertyholdsordoes
not hold forarbitrary cases. Butthis isnot proof that that propertydoesinfactholdfor everycase
but itis an indicatorthatit mightjustdo so.
Where mathematical inductiondiffersisin the factthat one case dependsuponanother due
to the fact that we have recursivelydefinedourprocedure,andthatiswhere we formthe ‘proof’
aspect. If we findacertainpropertyassociatedtoa certaincase thenwe lookto the nextcase that
followstosee if the same propertyholdsinthatcase as well. Howeverwe usuallyneedastarting
point,sowe prove that a propertyholdsforthe most trivial of casesfirst.Fromthat we lookat the
nextcase afterit. If the property holdsforthat case as well,thenwe lookatthe succeedingcase and
see if the propertystill holds.Andsoon. Butcan we testthat a propertyholdsfora structure of a
vast size,ora set of infinite cardinality?Itwouldbe verytime consumingtocheckif it istrue that
1 + 2 + 3 + 4 + ⋯+ 𝑛 =
𝑛( 𝑛+1)
2
for 𝑛 = 12 or 𝑛 = 12222.
In mostcaseswe use the truth whichwe install inthe axiomsforthe natural numbers. This
allowsusto prove thata propertyholdsforall the natural numbers 𝑛 withouthavingtoinspect each
case one byone. But in othercases,as withsetsor logic,we use our recursive definitions.These

18. History of Mathematical Induction & Recursion [B].
17
recursive definitionsdohave a startingpoint,namelytheirbase cases,whichwe have previously
seen.Andwe knowthe natural numbers have the base case 1i
. Mathematical inductionhasthe
same structure;a base case and the stepthat passes fromone case to another,althoughwe referto
thisas the InductiveStep as it isslightlydifferentfromourrecursive step. ItisthisInductive Stepthat
provesthata propertyholdsforeverycase inthe structure.
We can state The Principle of Induction forthe natural numbers inthe followingaxiom:
Supposethat 𝑃( 𝑛) is a statementinvolving a generalnaturalnumber 𝑛.Then 𝑃(𝑛) is
true forall forthe naturalnumbersif;
1. 𝑃(1)is true, and
2. 𝑃( 𝑘) ⇒ 𝑃( 𝑘 + 1) forall naturalnumbers 𝑘.
Notice the comparisonof thisaxiomandthe 5th
Peanoaxiomstatedinsection[1.3].They
are,albeitslightlywordeddifferently,the same. The Principle of Inductionisanextensionof the
algebraicandorderii
axiomsthat we have forthe natural numbers.Inductionismerelyanextension
of somethingwe oftenoverlookaboutthe natural numbers; startingfrom1,we can reach anyother
natural numberbysimplyadding 1.
The statement 𝑃(1) isthe Base Caseof The principle of induction,whereasoursecond
statementisthe InductiveStep.Notice thatthere isan implicationinthe inductivestep.So,if 𝑃( 𝑘) is
true,then 𝑃( 𝑘 + 1) isalso true.We may thinkthat thismeans thatwe have to prove that 𝑃( 𝑘) is
true and thus itfollowsthat 𝑃( 𝑘 + 1) istrue also,but whatwe have to do isshow that – if it wasthe
case that 𝑃( 𝑘) was true,thenfromthis 𝑃( 𝑘 + 1) is alsotrue. Thisis where we getourInductive
Hypothesis;we assume,asa hypothesis,that 𝑃( 𝑘) istrue andunderthat assumptionwe prove that
𝑃( 𝑘 + 1) is true.
The concept of mathematical inductionis – fora general property 𝑃( 𝑛) of anatural number
𝑛, whateveritmaybe,the base case checksthat 𝑃(1) is true and then,bythe inductive hypothesis,
𝑃(1) ⇒ 𝑃(2), and againby the inductive hypothesis 𝑃(2) ⇒ 𝑃(3) andthus 𝑃(3) ⇒ 𝑃(4) ⇒
𝑃(5) ⇒ ⋯ ⇒ 𝑃( 𝑛) ⇒ 𝑃( 𝑛 + 1) ⇒ ⋯
NowI wouldlike togive anexample of proof byinduction toillustratehow we use the
inductive hypothesistoprove the inductivestep.I shall prove thatthe followingstatementholds:
For all natural numbers 𝑛, the number 𝑛2 + 𝑛 is even.
i In some cases when usingmathematical induction,we can assumethat 0 ∈ 𝑵 as a property may hold for the
casewhen 𝑛 = 0.
ii I have not mentioned the orderingaxioms atthis pointbut they will appear in a later section.

19. History of Mathematical Induction & Recursion [B].
18
So 𝑃( 𝑛) isthe statement‘𝑛2 + 𝑛 is even’.Thuswe are lookingfora natural number 𝑚 such
that 2𝑚 = 𝑛2 + 𝑛. We proceednowwiththe structure laidout inthe inductionaxiombefore.
Base case:When 𝑛 = 1, 𝑛2 + 𝑛 = 1 + 1 = 2 thus we have verifiedthat 𝑃(1)istrue.
InductiveStep:Assume nowforan InductiveHypothesis that,foran arbitrary 𝑘, ‘𝑘2 + 𝑘
iseven’, thenwe seektoshowthat ( 𝑘 + 1)2 + ( 𝑘 + 1) = 2𝑝 for some natural number
𝑝. From our inductive hypothesiswe canassume furtherthatthere exists 𝑞 belongingto
the natural numberssuchthat, 2𝑞 = 𝑘2 + 𝑘. However( 𝑘 + 1)2 + ( 𝑘 + 1) = 𝑘2 +
2𝑘 + 1 + 𝑘 + 1 = ( 𝑘2 + 𝑘) + 2𝑘 + 2 = 2𝑞 + 2𝑘 + 2 = 2(𝑞 + 𝑘 + 1). So if we take
𝑝 = 𝑞 + 𝑘 + 1 thenwe have a natural number 𝑝 suchthat 2𝑝 = ( 𝑘 + 1)2 + ( 𝑘 + 1) as
required. ∎
Notice that,fromthe assumptionthat 𝑃(𝑘) is true for an arbitrary 𝑘, we were able to prove
that 𝑃(𝑘 + 1) was alsotrue by manipulatingwhatwe knew about 𝑃( 𝑘 + 1) toreacha trivial
conclusionbasedon 𝑃( 𝑘)’sassumption andgeneral arithmetic.
We donot necessarilyhave tostartwith 𝑛 = 1 for the base case,there are certainfunctions
or theoremsthatonlysatisfy‘𝑓𝑜𝑟 𝑛 > 𝑛0’forsome certainnatural number 𝑛0 i.e. 𝑃( 𝑛) istrue for all
natural numbers 𝑛 ≥ 𝑛0.If thisis the case we justneedto make 2 slightadjustments toourusual
procedure – the base case ischangedto 𝑃( 𝑛0) and the inductive step nolongerappliestoall natural
numbers 𝑘,but onlyfor 𝑘 ≥ 𝑛0. Suchan example of thiscanbe seenwhencomparing 𝑛2 and2 𝑛.
When 𝑛 = 1,2,3 we have 𝑛2 > 2 𝑛. However,for 𝑛 ≥ 4,we have 𝑛2 ≤ 2 𝑛. So inprovingthat the
latterholds,we use inductionwiththe base case for 𝑛 = 4. Thenforhe inductive stepwe prove
𝑘2 ≤ 2 𝑘 ⇒ ( 𝑘 + 1)2 ≤ 2 𝑘+1 for all 𝑘 ≥ 4.
Anothervariationof mathematical inductionisthe methodof Strong induction. We require
thismethodwhenwe have the case that 𝑃( 𝑘) alone isnot enoughtoimplythat 𝑃( 𝑘 + 1) is holds
but we actuallyrequire some, ormaybe evenall,of 𝑃(1), 𝑃(2),… , 𝑃( 𝑘 − 1) also.
The axiomof Strong Induction isgivenas:
Supposethat 𝑃( 𝑛) is a statementforsomenaturalnumber 𝑛.Then 𝑃(𝑛)is true for all
naturalnumbers 𝑛 if:
1. 𝑃(1) is true,and
2. "𝑃(𝑛) ℎ𝑜𝑙𝑑𝑠 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑛 ≤ 𝑘" ⇒ 𝑃(𝑘 + 1) holds for all such 𝑘.
If anythingthisisjusta strongerversionof the axiomof induction,asall statementsproven
by our standardinductioncanalsobe provenby stronginduction.We maywantto use strong
inductiontoprove that 𝑛! = 𝑛 ∙ ( 𝑛 − 1)! .

20. History of Mathematical Induction & Recursion [B].
19
In PeterEccles An Introduction to MathematicalReasoning,he givesthe followinganalogyof
The Principle of Induction whichreallysumsupitsstrength:
Supposethatwethinkof theintegers lined up like dominos.Theinductivestep tells us
thatthey areclose enough foreach domino to knockoverthe nextone,the basecase
tells usthatthe first domino fallsover,the conclusion is thatthey all fall over.
Once it isknownthat the dominosare so close togetherinordertoknockanotherover,then
there isno needtopusheach one over, andwe justneedtoknockover the first.Inductionwith the
natural numbersisthe same,once we have seenthat 𝑃( 𝑛)shows that 𝑃( 𝑛 + 1) holds,thenall that
isrequiredistoshowthat the base case is true.
Mathematical inductiondevelopedaroundthe natural numbersandtheirordering.Then,
inductionwasappliedtootherareasof mathematicswhere objectscanbe put intoan ordering,to
prove propertiesandtheirtheorems,andtotackle the problemof representingthe infinite inthese
areas – we have seenthat we can define aninfinite collectionof sets,sowe mayuse inductionto
showthe propertyof inclusionamongstthese sets,basedontheircardinality.Inlogicwe canprove
a theoremfora language byproof byinductiononthe lengththe sentencesbelongingtothis
language.
Mathematical inductiontodayisanessential tool withineveryareaof mathematics.
Induction allowsustorepresentthe infinitewiththe finite.Itremovesall the complexityand
longevity associatedwith proofsconcerningthe infinite.Itisa mathematical tool thathasgrown in
importance until today.Inprovingtheorems,propositions,lemmas, whateveritmaybe,we have a
certaingroupof techniquesinmathematicsthatwe can use, andinduction isone of these;its
importance is unrivalled.

21. History of Mathematical Induction & Recursion [B].
20
[1.5] The Origin of Mathematical Induction:
As I have said,mathematical inductionisof great importance withinmathematics,but
where didthe principle originate from?Whowasthe 1st
persontodevelopthe principle?Whowas
the firstto realize itsimportance?Whoeven,wasthe firsttoname the principle mathematical
induction?What we alreadyknowatthispointis that inductionwasdevelopedfromthe natural
numbersasthat is the basisof its structure;the orderinginwhichthe natural numbersexhibits. The
exactoriginof the principle of mathematical inductionisunclearanditcouldbe debatedtobe
attributedtomanydifferentgreatmathematiciansortovariousdifferentperiodswithinthe history
of mathematics.
We knowthatthe methodbearsa close resemblance tothe inductive processesof other
sciences whichtendtobe basedon observationsandtoa degree,thatisone wayinwhich
mathematical inductionwasdeveloped.Inductioninmathematicsisgenerallyamethodof proof;we
do notjust developanyoldprocedure andsetouttoprove its validity.Insteadwe lookatpatterns
fromexamplesandtryto developanatural property.Once we have thisproperty,we use induction
to see if itis true.Thiscan be seenin Mathematicsand PlausibleReasoning,Vol.1– Induction and
Analogy in Mathematics (1954) by George Pólyai
:
Wantingto developaformulaforthe sumof the squaresof the 1st
𝑛 natural
numbers,Pólyacomparesthe ratioof ∑ 𝑖𝑛
1 and ∑ 𝑖2𝑛
1 . Here I wishto define ∑ 𝑖𝑛
1 = 𝑆 𝑛
and ∑ 𝑖𝑛
1
2
= 𝑆 𝑛
2.
In doingthis,Pólyastates,withoutproof,that 𝑆 𝑛 = 1 + 2 + 3 + 4 + ⋯+ 𝑛 =
𝑛( 𝑛+1)
2
,
whichis true andcan be proven,byinductionof all things.
Lookingat the 1st
6 cases forpossible 𝑛,Pólyaobservesthatthe ratiosare givenby:
𝑛 1 2 3 4 5 6
𝑆 𝑛
𝑆 𝑛
2
3
3
5
3
7
3
9
3
11
3
13
3
From thisPólyaclaimsthat 𝑆 𝑛/(𝑆 𝑛
2)= (2𝑛 + 1)/3 whichiscertainlytrue for 𝑛 =
1,2, …,6. Takingthisintoconsiderationandmultiplyingbothsidesthruby 𝑆 𝑛 =
𝑛( 𝑛+1)
2
,
it isobtainedthat
𝑆 𝑛
2 =
1
6
𝑛( 𝑛 + 1)( 𝑛 + 2).
Thisremainstrue for 𝑛 = 1, 2,… ,6.
i George Pólya was an Hungarian mathematician [1887-1985].

22. History of Mathematical Induction & Recursion [B].
21
What Pólyahasdone is establisha‘conjecture’ashe callsit,all fromtakingat a few
examplesandlookingforapatternin orderto presenthisequationfor 𝑆 𝑛
2.Now that he has seen
that itworks fromthe 1st
six cases,he goeson to prove thatit holdsfor 𝑛 = 7 by simplypluggingin
the numbersof his equation.Notwantingtoprove thatthat hisequationistrue for 𝑛 = 8, 9,10, …
Pólyaproceedstothe inductive step;assuminghisequationholdsforanarbitrarynatural number 𝑛,
he proceedsinshowingthatitdoesalsoholdfor 𝑛 + 1. Pólyaconcludes the proof of hisequationfor
the sum of the squaresof the first 𝑛 natural numbersbysaying:
“If ourconjectureis true fora certain integer 𝑛, it remainsnecessarily truefor the next
integer 𝑛 + 1. Yet we knowthatthe conjectureis true for 𝑛 = 1,2, 3,4,5, 6,7. Being
true for7, it mustbe true also forthe nextinteger 8; being truefor 8, it mustbe true for
9; since true for9, also true for10, and so also for11, and so on.The conjectureis true
forall integers;we succeeded in proving it in full generality.”
The procedure inwhichhe developedhisequationissimilartothat of inductioninother
sciences,bymakingobservations.Howeverinprovingthathisequationdoesinfactholdforall the
natural numbersPólyarequirestoshowthe inductivestep.Makingobservationsandshowingthat
theyare true for certaincases,doesnotprovide afull proof;thatis whyPólyarequiresthe inductive
stepand that iswhere mathematical inductiondiffersfromthe inductionof othersciences. This
inductionobviouslyinfluencedmathematicianstolookforrecurringpatternsinorder to formulate
‘conjectures’like Pólyaandthentryto prove theirvalidity. Like Pólyasays:
“Has mathematicalinduction anything to do with induction (inscience)?Yes,ithas.”
Howeverthiswouldonlybe the inspirationforthe base case;the inductive stepwouldnot
have followedfrom the observationof examples.
There are tracesof recurrentandinductive processesinthe worksof some GreekandHindu
mathematicians. However,ratherthantryingto generalize aparticularmathematical method,they
soughtto findone particularsolutionfromanother.Here theywere showingsigns of aninductive
step,althoughfromone specificcase tothe next,asopposedtoan abstract case to the following
abstract case. Examplesof suchare foundinthe work of both Theonof Smyra (fl.100 A.D.) and
Proclus(412-485 A.D.) ontheirworkon findingnumbersrepresentingsidesanddiagonalsof
squares. Bhāskara’s(1114-1185) cyclicmethodforsolvingindeterminate equationsof the form
𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 𝑦 alsoshows traces. Furthertracescan be foundinEuclid’sElementsIXin
PropositionXXwhere Euclidshowsthatthe numberof primesisinfinite.Howeveranytrace of
mathematical inductionthatdidappearinthe work of the Greeksand Hinduswouldnothave been
inthe same modernforminwhichwe see ittoday.

23. History of Mathematical Induction & Recursion [B].
22
Duringthe 17th
century,the FrenchmathematicianPierre
de Fermatwas knownto have focusedalot of hiswork on
infinitesimal calculusanditwasin a lettertoChristiaanHuygens
inwhichFermatclaimed touse a methodcalled la descente
infinite ou indefinite.The letterwastitled‘Relation des
découvertesen la science desnombres’ andwasnot discovered
until 1879, amongsta groupof papersthat had belongedto
Huygens.Inthe paper,Fermatclaims he had foundthat hisnew
methodwasapplicable in provingthe impossibilityof certain
mathematical statements,before realizing thatitwas also
applicable inprovingthe assertionof statementstoo. Fermat’s
methodof infinitedescentdidcontainrecurrentmodesof
inference,however,itwasnotmathematical inductioninits
purityas Fermatwouldoftenjumpfrom acertaincase 𝑛 to 𝑛 − 𝑖,
for some 𝑖 < 𝑛,skippingoverseveral cases ata time.Insteadof lookingtoprove aspecificbase case
and an inductive step,Fermatchoosestoprove astatementforone certaincase andthenmake a
connectiontoanotherinorderto prove thissecondcase.Here Fermat doesshow a trace of an
inductive step.
Fermatwas knownforpublishingonlyhisresultsandexcludinghismethods;itwasnot until
1995 that Fermat'sfamousLast Theoremwasproveni
;Fermathavingdiedin1665. Thiscouldbe the
cause for, bypure chance, notseeing Fermat’smethodof infinite descentuntil 1879.The only
occasionson whichFermatmade hismethodsknownwere inresponse tocritiquesof his work,who
didnot believe hisresultstobe true or whosought a clearerexplanationof hisresults.One of these
reasonscouldbe whyHuygenspossessedacopyof Fermat'smethodof infinite descent,andthe
reasonthat we nowknowthat a recurrentmode of proof wasknownto Fermat.There isfurther
evidence forthisinthe factthat Adrien-Marie Legendre,GustavLejeuneDirichletandLeonhard
Eulerall usedsimilarmethodsof Fermat'smethodof infinitedescenttoprove alot of Fermat’s
propositionsinthe space of time afterhisdeathandbefore the discoveryof Fermat’sletterto
Huygens.
There are alsotracesof mathematical inductionthatcan be foundinthe workof Blaise
Pascal. Alsoa Frenchmathematicianof the 17th
century,Pascal was workedcloselywith Fermaton
i Andrews Wiles proof of Fermat’s Last Theorem was published in 1995.The famous theorem had been written
in 1637 in the margin of Fermat's copy of Diophantus’ Arithmetica to which Fermat said themargin was not big
enough to contain the proof as well.
Figure 4 - Pierre de Fermat.

24. History of Mathematical Induction & Recursion [B].
23
the foundationsof modernprobabilitytheory. Pascal isfamoustodayforPascal’sTrianglewhichwas
mentionedearlier,howeveritwasknowntomanyin the daysof Pascal and Fermatas the
arithmetical triangle.In1665, Pascal's Traité du trianglearithmétiquei
waspublished.Pascal usedthe
treatise toshowthe applicationsof the arithmetical triangle inthe theoryof combinations,the
theoryof probabilitiesandtothe calculationsof the powersof the binomial coefficients.Pascal
showedthatthe binomial coefficient 𝐶 𝑘
𝑛
canbe foundbythe equation
𝑛 ∙ ( 𝑛 − 1) ∙ ( 𝑛 − 2) ∙ …∙ ( 𝑛 − 𝑘 + 1)
𝑘!
which,if we multiplythruby
( 𝑛−𝑘)!
( 𝑛−𝑘)!
we obtain the formthat we use today
𝑛!
( 𝑘!)( 𝑛−𝑘)!
.
Pascal usedthe arithmetical triangle tocalculate the share
of a total stake in a game of dice between2players whichhasbeen
stopped prematurely.Letthe firstof the 2 playersbe person A and
the secondperson B. Withperson A needing 𝑛 pointstowinand B
needing 𝑚 points,Pascal usedthe arithmeticaltriangletocalculate
that the ratio of A to B shouldbe givenas the sum of the 1st
𝑛
numbersof the 𝑗 𝑡ℎ row of the arithmetical triangletothe sumof
the remaining 𝑚 numbersof the same row,where 𝑗 = 𝑚 + 𝑛.
Alternatively,letnumbersof the 𝑗 𝑡ℎ rowof the arithmetical
triangle be givenbythe sequence 𝑎1, 𝑎2,… , 𝑎 𝑛,𝑏1, 𝑏2,… , 𝑏 𝑚.Then
the ratio of A to B is givenby ∑𝑎𝑖 ∶ ∑𝑏𝑗 .
Here 𝑗 = 𝑚 + 𝑛 is the numberof throwsof the dice remaininginthe game whenitis
stoppedearly.Pascal proved inhistreatise,throughalemma,thathisratioiscorrect for 𝑗 = 1 and
thenproved inthe nextlemmathat,if itis alsocorrect for some natural number,thenitisalso
correct for the nextnatural numbergreaterthanit.ThenPascal concluded thathisratiois correct
for everynatural numberthat 𝑗 can be.This ispreciselymathematical inductionof today.Pascal did
not refertoit as mathematicalinduction oruse a 2 stepprocedure;insteadhe hadprovena base
case inone lemmaandthenproven an inductive stepinthe nextlemmabefore he made aseparate
conclusion –that hisratio istrue foreverynatural number.
Due to the popularityof Pascal'streatise onthe arithmetical triangle,he couldbe deemedas
one of the reasonsas to howthe principle of mathematical inductionbecame knownat the time of
and inthe time afterthe 17th
Century. Pascal couldhave alsobroughtthe principle of mathematical
inductiontothe attentionof Fermatthroughtheirworktogetheronprobabilitytheory. Pascal was
i Translated into English as - A treatiseon the Arithmetical Triangle;it appeared firstin 1654.
Figure 5 - Blaise Pascal.

25. History of Mathematical Induction & Recursion [B].
24
certainlyone of the firstmathematicianstouse the principleof inductionsimilartoitscurrentform,
ina systematicway andto realise the implicationof the inductivestep. However,toPascal was
knownthe workof an ItalianmathematiciancalledMaurolycus. InaletterfromPascal to Pierre de
Carcavii
,Pascal refersto Maurolycusfor the proof that twice the 𝑛 𝑡ℎ triangularnumberminus 𝑛
equals 𝑛2 . Although Pascal doesnotmentionMaurolycusinhis Traité du triangle arithmétique, he
was well aware of Maurolycuswhose workcouldhave beenthe inspirationof Pascal'smethodof
induction.
i In the letter Lettre de Dettonville á Carcavi.

26. History of Mathematical Induction & Recursion [B].
25
[1.6] Maurolycus & Gersonides:
FranciscusMaurolycus(1494-1575) was an Italian
mathematicianwhoworkedontranslationsfromsome of the most
famousGreekmathematicians –Euclid,Archimedes, Theodosius.His
workwas of great importance forthe transitof Greekworkto Europe.
In 1575, Maurolycus publishedatreatise onarithmetictitled
Arithmeticorumlibri duo found inhis book D. Francisci Maurolyci
Opuscula Mathematica.Here,Maurolycususesamode of inference in
a systematicway,buildingupfromthe firstcase to the next,to
demonstrate simple propositionsbefore movingontoprove harder,
more complicatedonesinasimilarfashion.Proposition XIof this
treatise isthe propositiontowhichPascal credited Maurolycus inthe
lettertoCarcavi.Maurolycus provedthis propositionthrough2previouslystatedpropositionsand
variousdefinitions. Therefore Pascal didnotgethisideaforhisinductive proof fromthe Maurolycus
propositionthathe hadreferencedtoCarcavi.Howeverinthe same treatise,2otherpropositions
may have caughtthe eye of Pascal.
PropositionsXIIIandXV of the treatise asfollows:
(13)Every squarenumberplusthefollowing odd numberequalsthefollowing squarenumber.
(15)The sumof thefirst 𝑛 odd integers is equal to the 𝑛 𝑡ℎ squarenumber.
In modern notationthiswouldprecede asfollows:
(13) ( 𝑛 + 1)2 = 𝑛2 + 𝑂 𝑛+1 for 𝑂 𝑛 = 2𝑛 − 1.
(15) 𝑂1 + 𝑂2 + ⋯+ 𝑂 𝑛 = 𝑛2.
In the modernnotationitisclearto see that there isa connectionbetween the 2propositions.
Maurolycus stateshisproof of PropositionXV asfollows
“By a previousa previousproposition (namelyPropositionXIII) thefirstsquare
number(unity) added to thefollowing odd number(3) makesthefollowing square
number(4);and this second squarenumber(4) added to the 3rd
odd number(5) makes
the 3rd
squarenumber(9);and likewise the3rd
squarenumber(9) added to the 4th
odd
number(7) makesthe 4th
squarenumber(16);and so successively to infinity the
proposition isdemonstrated by therepeated application of Proposition XIII.”i
i This is the translation given by W.H. Bussey in American Mathematical Monthly, No.5, Vol.14, May 1917.
Figure 6 - Franciscus Maurolycus.

27. History of Mathematical Induction & Recursion [B].
26
Proposition XV isachievedby repeateduse of proposition XIIIwhichactsas an inductive step.
ClearlyMaurolycusproof isan example of mathematical induction.Againitisnotinthe current
structure as we use today butit isin a systematic,step-by-stepprogression fromthe firstcase tothe
next,andthento the next,andto the next,andso on towardsinfinity.One difference between
Pascal'smode of inferenceandthatof Maurolycus,isMaurolycusdoesnotmake an inductive
assumption/hypothesiswhereasPascal does,makinghismethodof proof more abstract.
Now I come to Levi BenGershon,(Gersonidesin
Latin).A Rabbi bornin 1288 inLanguedocinwhat would
now be the southerncoastof moderndayFrance,
Gersonideshadabroaderbackgroundto his
mathematical careerthanMaurolycus.Although
Gersonidesdid tooworkonEuclid’sElements, he also
worked ona lot of the oldArabicand Hindutexts,suchas
Bhāskara,who I mentionedbefore,who isknowntohave
usedhiscyclicmethod to solve indeterminateequations
of the form 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 𝑦.
Gersonides’1321 work – MaaseiHoshevi
, couldbe
calleda piece of work that wasaheadof itstime;
Gersonides usesletterstorepresentarbitrarynumbers,
onlyJordanusNemorariuswasknowntohave alsohave
done this at that time.Anotherreasonthat MaaseiHoshev
couldbe consideredasaheadof its time is because of Gersonidesuse of the methodof whathe
calledrising step-by-step whichhassimilaritiestoMaurolycus’workonrepresentingthe infinite.
Thisstep-by-stepmethodcanbe foundin Propositions9to 12 inclusive,whichInow give (inmodern
notation):
(9) 𝑎( 𝑏𝑐) = 𝑏( 𝑎𝑐) = 𝑐(𝑎𝑏).
(10) 𝑎( 𝑏𝑐𝑑) = 𝑏( 𝑎𝑐𝑑) = 𝑐( 𝑎𝑏𝑑) = 𝑑( 𝑎𝑏𝑐).
Gersonidesused Proposition9to prove Proposition10before makingthe followingstatement:
In thismannerof rising step-by-step,itis proved to infinity.Thus,… the result of
multiplying onenumberby a productof other numberscontainsany oneof these
numbersasmany timesas the productof all the others.”
i Title is taken from Exodus 26:1 to roughly mean The Work of the Calculator.
Figure 7 - A stamp of Isreal showing Gersonides'
invention, Jacob's Staff, which was used for
measuring nautical and astronomical
measurements.

28. History of Mathematical Induction & Recursion [B].
27
WhichI interpretas 𝑎𝑗( 𝑎1 𝑎2… 𝑎 𝑖 … 𝑎 𝑛) = 𝑎 𝑖( 𝑎1 𝑎2… 𝑎𝑗 … 𝑎 𝑛).
(11) 𝑎( 𝑏𝑐𝑑) = ( 𝑎𝑏)( 𝑐𝑑) = ( 𝑎𝑐)( 𝑏𝑑) = ⋯
AgainGersonideswrote towardsanextensiontowardsinfinity:
Similarly, it is shown to infinityby thesame kind of demonstration.Therefore,any
numbercontainstheproductof any two of its factorasmany times asthe productof the
remaining factors.
(12) In modernlanguage,Proposition12statesthat multiplicationisbothassociative and
commutative.Gersonidesprovesthisusingthe previous3propositionstoshow how factors
can be grouped intodifferentstringsof differentlengths,whichisnotreallyafull proof.
Althoughnone of these propositionsare provedby induction,theydoshow whatGersonides
meantby rising step-by-step anditdoesbare some resemblance toaninductive step. Gersonides
continued tostate and prove a longseriesof propositions;IskiptoPropositions 63– 65 on
permutations:
(63) 𝑃 𝑛+1 = ( 𝑛 + 1) 𝑃𝑛.
(64)𝑃2
𝑛
= 𝑛( 𝑛 − 1).
(65)𝑃𝑗 +1
𝑛
= ( 𝑛 − 𝑗) 𝑃𝑗
𝑛
.
Gersonidesprovedeachone of these propositionsinturn,usingProposition63to prove
Proposition65.Once thisis done,Gersonided concludesbysaying:
Thusit hasbeen proven thatthe permutationsof ordera given numberfroma second
given numberof elements equalsthe numberswhosefactorsareasmany asthefirst
given numberand they are the integersin their naturalorder,thelast being thesecond
given number.
Thus Gersonideshad proventhat 𝑃𝑗
𝑛
= ∏ 𝑖𝑛
𝑛−𝑗+1 .One can see that, whenwe take 𝑗 = 2 we
getProposition64,whichhad beenprovenpreviouslybyGersonides.Gersonidesclaimedthatthese
3 propositionsare enoughtoprove thisgeneral result,andtheyare.FromProposition64,we use
Proposition65to prove the case for 𝑗 = 3 and fromthat,we applyitagainto prove that the result
holdsfor 𝑗 = 4 andapplyit againfor 𝑗 = 5, andso on. Proposition 65acts as an inductive step,from
any arbitrary 𝑗 to 𝑗 + 1.
Thus Gersonideshad usedinductiontoprove hisresultonthe numberof permutationsof
order 𝑗 of 𝑛 elements. Againwe cannotsaythatthisishow we use inductiontoday,butGersonides
methoddoescontainthe essence of modern induction havingprovenaparticularcase as well asa

29. History of Mathematical Induction & Recursion [B].
28
recursive step. The proof of proposition42inthiswork wasalsoprovenina similarfashion.
However,there wasalack of an assumptioninorderto make the inductive step,similartothatof
Maurolycusand again,Gersonidesworkwasconstructive,buildinguptothe result,whereaswith
inductiontoday,we seektoprove whatwe have alreadystated.
The source as to Gersonidesinfluencetoinvestigate anduse arecursive mode of inference
inorder to prove mathematical procedures liesinthe Hebrew communityof Gersonidestime where
the subjectwasinvestigatedatan earlystage. Aswell asBhāskara,Gersonidescouldhave been
influencedbythe worksof SeferYetsirahandhis Bookof Creation,whichinvolvesarecursive mode
of lookingatthe permutationsof the twenty-twolettersof the Hebrew alphabetandisbelievedto
be from the secondcentury.Anotherpossible influence is Rabbi ShabbetaiBenAbrahamDonnolo,
(913-970), whoprovedthat 𝑛 letterscanbe arranged 𝑛! waysin a similarfashiontoarecursive
method.
In MasseiHoshev,Gersonidesmentioned thatthe readerof histextshouldbe aware,and
capable of understanding, the 7th
,8th
and 9th
booksof Euclid’s Elementswhere we have already seen
a small trace of a recursive mode of inference. However,itwouldbe possibletoassume that
Gersonidesfoundmodesof recursive definitionandinduction,throughouthismathematical career,
inthe worksof othersand we couldevenassume furtherthat GersonideshadsetoutinMassei
Hoshevto investigate the applicationof recursivelydefinedstructuresandthe applicationof
recursioninmathematical proof inastructured,systematicway.
Andit seemstome that Gersonideswasthe firsttodo this.I feel thatGersonideswasthe
firstto realize the significanceof the orderingof the natural numbersandhow he couldapplyit to
proofsinorder torepresentthe a propertyof the infiniteintermsof the finite.
There isonlyone furthermathematicianthatIfeel we couldconsiderforthe invention of
mathematical inductionand thatisCampanus. AnItalianmathematicianof the 13th
century,
Campanusof Novara workedontranslating
Euclid’sElementsintoLatin.Indoingthishe
includedhisownversionof the proof thatthe
goldenratioi
isirrational. The methodusedby
Campanusinhisproof was similartothat of
Fermat.Campanus,like Fermat,useda
descendingmethodof progression,jumping
i
(1+√5)
2
is the Golden Ratio, which is said to be found to occur naturally throughoutall of life.
Figure 8 - The Golden Ratio.

30. History of Mathematical Induction & Recursion [B].
29
sporadicallyovercertaincases,toprove othercertaincases.
However,Idonot feel thatthis methodisfullyrepresentativeof the mathematical induction
of today.Modern inductionrepresents everypossiblecase whereasthe methodsof Campanusand
Fermatappeardisjoint andfull of gaps.Althoughtheirmethodsdoshow tracesof induction, they
lack thatcontinuous,connected,one afteranothersequence.Ifeel thatthe methodsof Gersonides,
Maurolycusand Pascal are strongerand closertothe methodin whichwe now use today,not
because theyprecede fromacertain,finite,base case towardsinfinity,butforthe reasonthatthese
methodswouldnotskipoveranyparticularcase;theyrepresentthe continuousprogressionfrom
one discrete case to another.
In addition,Ibelieve thatthe same three men,Gersonides,MaurolycusandPascal were
more aware of the significance intheirmethod;Fermatcouldalsobe includedinthisregard but
since he neverreleasedhisarguments,we will neverknow if he wasaware of such significance.As
for Campanus,Iwouldhave toregard himas one of the menwhoinfluence the formerfour.Iregard
himin the same groupas Bhāskara, Theon,Proclus, Euclidandothers,whoshowedtracesof the
mode of inference andrecursion, whousedamethodsimilartoinductionasa one-off proof,
significantonlyatthat one particulartime.MoreoverI would considerGersonidestobe the first
mathematiciantounderstandthe meaningandsignificance of arecursive mode of inference andto
give ita stepby stepstructure. Furthermore IregardMaurolycusas the firstmanto see the
significance of applyingthisrecursive mode of inference toproofs whereasPascal broughtittothe
attentionof manyothers.

31. History of Mathematical Induction & Recursion [B].
30
[1.7] The Name – Mathematical Induction:
For the originof the term MathematicalInduction,agroupof othermathematicians who
were aware of the methodare to be credited. Fermatreferredtohisnew methodas la descente
infinite ou indefinite,Gersonidesreferredtohismethodof rising step-by-step.HoweverMaurolycus
and Pascal didnot assignanyparticularname to theirmode of inference. Itisevidentthatthe term
mathematicalinduction isderivedinitiallyfrom the observational induction of sciences asmanyseen
it as an adaptationof that concept.
In JohnWallis’1656 work, Arithmetica Infinitorum,
Wallisi
usedthe methodof inductionusedinscience and
simplyreferstothismethodas induction.InProposition16of
thiswork,Wallislooked tofindthe ratioof the 1st
𝑛 squared
numberstothe product ( 𝑛 + 1) 𝑛2. Wallisproceedsto
observe that,inthe 1st
6 cases,the ratioturns outto be
1
3
+
𝑥ii
where 𝑥 < 1 decreasesasthe size of 𝑛 increases.Wallis
thenconcludedthat lim
𝑛→∞
𝑥 = 0. Thismethod wasreferredto
by Wallisas per moduminductionis.iii
AsWallisproceeded
throughthe remainderof thispiece of work,he relied freely
on thismethodof inductionsimilartothatof natural science.
Wallisdidfeel stronglyaboutthe scientificinductionandthatitcouldeasilybe appliedto
mathematics.Inhis1685 treatise onAlgebra,Wallisstated:
“ThosePropositions...demonstratedby way of Induction:which isplain,obvious,and
easy;and wherethingsproceed in a clear regularorder (ashere they do),very
satisfactory.”
“I look upon Induction asa very good method of investigation;asthatwhich doth very
lead usto the easy discovery of a General Rule.”
However,in1686, JacobBernoulliiv
recommendedinhis Acta Eruditorum thatWalliscould
improve hismethodof inductionbyintroducingthe argumentfromanarbitrary 𝑛 to 𝑛 + 1. This
appearsto be the 1st
appearance of the InductiveHypothesis andthe beginningof the modernform
i John Walliswas an English mathematician,(1616 –1703).
ii Just likePolya noted in section [1.5].
iiiLatin for - by way of induction.
iv Jacob Bernoulli (1655 – 1705) was a Swiss mathematician and partof a largemathematical family.
Figure 9 - John Wallis.

32. History of Mathematical Induction & Recursion [B].
31
of Mathematical Induction. Bernoulli usedthisnew 𝑛 to 𝑛 + 1 argumentto prove the binomial
theoreminhis ArsConjectandii
.
FlorianCajoriii
,ahistorianinmathematics, referstothe methodusedbyWallisasincomplete
and referstois as IncompleteInduction,whichgivesrise tothe CompleteInduction thatCajori
defined asthe methodbyBernoulli. Formore thana century afterBernoulli’srecommendationto
Wallis, Induction wasbeingusedasthe name forboththe methodsof WallisandBernoulli. The two
methods were seeminglyunpopularinthistime anditwasin fact Bernoulli'smethodthatwasless
knownat the time Most whousedeithermethodactuallyusedthe methodwithoutassigninga
specificname.
Howeverinthe 1830s, thischanged.George
Peacock(1791 – 1858) wasan Englishmathematicianwho
publishedhisTreatise onAlgebrain1830. In thisTreatise,
Peacocktalked of a “law of formation extended by
induction to any number”. Inexplainingthe argument
from 𝑛 to 𝑛 + 1, Peacockreferred tohismethodas
DemonstrativeInduction.
AugustusDe Morgan (1806 – 1871) wasa British
mathematicianwhose name ismainlyassociatedtothe
lawsof negationonthe conjunctionanddisjunctionof
sets.In1838, De Morgan publishedinthe Penny
Cyclopaediahis Induction(Mathematics) inwhichhe described clearlymathematicalinductionand
itssimilarities/differencestoinductioninphysics.De Morganshowed how mathematical induction
shouldbe appliedthroughtwo clear,well-describedexampleswhereapropositionisstatedand
thenprovenviaan inductive step(usinganinductive hypothesisonlyinthe 1st
example),before
referringback to a base case.De Morgan referred toinductionas successiveinduction at the
beginningof thispiece of work, howeverhe laterreferstothe method asMathematical Induction;
the firstpublishedoccasiononwhichthe termhadbeenused.
Both the terms DemonstrativeInduction andMathematicalInductionbecame popularinthe
time afterbutthe formertermfell intodisuse asmostmathematiciansbeganto adoptthe latter.
The term VollständigeInduktion wasusedbyGermanmathematiciansinthe 19th
century,most
notablybyRichard Dedekindinhis1887 Was Sind und Was Sollen die Zahlen.It wasthisusage by
i Published in 1713,after Jacob Bernoulli’s death.
ii Origin of the Name Mathematical Induction,The American Mathematical Monthly, vol.25, number 5, 1918.
Figure 10 - Augustus De Morgan.

33. History of Mathematical Induction & Recursion [B].
32
Dedekindthatpopularizedthe methodinGermany,althoughthe methodwasslightlydifferentto
that of Peacockand De Morgan. In 1863, Isaac Todhunter(1820 – 1884), an Englishmathematician,
popularizedDe Morgan’s mathematicalinduction inhisAlgebra forBeginners usingthe methodto
prove variousexamples.One suchexample thatTodhunterspoke of was:
“The sumof (the first) 𝑛 termsof the series 1, 3,5,7, …is 𝑛2. This assertion wecan see
to be truein somecases...wewish to howeverto provethistheoremuniversally”.
Usinginduction,inthe same manneras De Morgan, Todhunterproved the above universally,
for all possible casesof 𝑛.Realisingthe full benefitof mathematicalinduction, Todhunterthen
stated:
“The method of mathematicalinduction may bethusdescribed:we provethatif a
theoremis true in one case,whateverthatcasemay be, it is true in anothercasewhich
may be the nextcase;hence it is true in the nextcase,and hence in the nextto that, and
so on;henceit mustbe true in every case afterthat which it began..... Themethod of
mathematicalinduction is asrigid as any otherprocessin mathematics.”
Todhunterreferred tothe method directlyas
mathematicalinduction –the title thatDe Morgan assignedtoit
and the title whichwe use today. Inthe centurythat followed
Todhunter’s Algebra,mathematical inductionhasbecome even
more abstract and has been acceptedacrossthe mathematical
worldas an essential tool formathematical proof. Todayits
structure has become more like thatof a procedure thatis
followed inastepby step manneras we have seenearlier.
Howeverthe name andthe basic concepthave remainedintact;
fromGersonides,toPascal,throughtoDe Morgan and onwards
until itsmodernformtoday.Figure 11 - Isaac Todhunter

34. History of Mathematical Induction & Recursion [B].
33
[1.8] Bibliography:
1. N L Briggs, Discrete Mathematics,Revised Edition,1989, OxfordUniversityPress,P8-10.
2. J L Hein, Discrete Mathematics,2nd
Edition,2003, JonesandBartlettPublishers,P145-146.
3. H Eves, An Introduction to theHistory of Mathematics,4th
Edition, 1976, Holt,Rinehart&
Winston,P209-212.
4. R C Penner, Discrete Mathematics:Proof Techniquesand MathematicalStructures,1999,
WorldScientificPub.Co.Inc.,P141.
5. K Gödel,on Formally UndecidablePropositionsof Principa Mathematica and Related
Systems, EnglishTranslationbyB.Meltzer,1962, Oliver&Boyd LTD. IntroductionbyRB
Braithwaite FBA.
6. S C Kleene,MathematicalLogic,1967, JohnWiley&Sons,Inc.,P250.
7. J W DawsonJr., Logical Dilemmas,the Life and Work of Kurt Gödel, 1997, A.K. Peters,P3-21,
P53-79.
8. DavidHilbert,MathematicalProblems, Bulletin of theMathematicalSociety, 1902, Vol.8,
Number10, P437-479, translatedbyM WinstonNewson.
9. I Grattan-Guinness, Search forMathematicalRoots1870-1940, 2000, PrincetonUniversity
Press,P227.
10. I Grattan-Guinness, Search forMathematicalRoots1870-1940, 2000, PrincetonUniversity
Press,P227.
11. P Eccles, An Introduction to MathematicalReasoning, 2007,Cambridge UniversityPress,
P39-51.
12. G Polya, Mathematicsand PlausableReasoning,Vol.1:Induction and Analogy in
Mathematics,1954, OxfordUniversityPress,P108-111
13. F Cajori, ÜberdasWesen der Mathematik, Bulletin of American MathematicalSociety,1909,
Vol.15, Number8, P407.
14. F Cajori, History of Mathematics,5th
Edition, 1991, Vol.2, ChelseaPublishingCompany,
P142. Also,whole textusedforbirth/deathdatesof Mathematiciansandtheirwork.
15. G Vacca,Maurolycus, TheFirst Discoverer of the Principle of MathematicalInduction,Bulletin
of the American MathematicalSociety,1909, Vol.16, Number2,P70-73.
16. N L Rabinovitch, RabbiLeviBen Gershon and the Originsof MathematicalInduction,Archive
forHistory of Exact Sciences, 1970, Vol.6, Issue 3, communicatedbyCTruesdell.
17. F Cajori, Origion of the Name‘MathematicalInduction’,TheAmerican Mathematical
Monthly,1918, Vol.25, Number5, P197-201.
18. A De Morgan, Induction (Mathematics),Penny Encyclopaedia,1838,Vol.12, London.
19. I Todhunter, Algebra forBeginners,4th
Edition,1866, Macmillan& Co.,P281-284.

35. History of Mathematical Induction & Recursion [B].
34
[2] Set Theory and Transfinite Numbers:
The historyof howmathematical inductionandrecursioncame tobe is an importantone.
The journeythatboth have takento progressthroughthe agesto theirmodernformshasbeena
labouringone.Bothmathematical applicationshave showntracesof theirimportance asfarback as
the time of Pythagorasand Plato,partlydue to the resultsthattheywere able toproduce,partly
due to the struggle withthe infinite atthe time. Mostof thishas beenaddressedinthe firstpartof
thispiece of work.One thingI didnot investigate fullywasthe significancethatinductionhadon
mathematicsinthe late 19th
and early20th
Century’sandinparticular,itseffectonnumbertheory.
Duringthistime,numbertheorybecame animportantinteresttoa lotof mathematicians,
the most importantbeingGiuseppe Peanoof Italy,RichardDedekindandGeorgCantor,bothof
Germany.The importance numbertheory hadat the time wasnot just froma mathematical pointof
viewbutalsofroma philosophical one.The mainaimwas to establish foundationsforthe theoryof
numbersanditsarithmeticsuchthat it shouldbe soundandfree of contradiction. The workof
PeanoandDedekindhadthe mostsignificance inthisarea.
Mathematical inductionwasanessential tool forGiuseppe Peanoinestablishinghisnow
famous,anduniversallyaccepted,axioms whichIhave previouslymentioned.The use of induction
by Peanohelpedhimconvert mathematicsintoasymbolic,logical form;infactthe simplicity of
induction shinesthroughinPeano’ssymbolicnotations. ThereforeIwouldlike tostartbylookingat
the work of Peano,mostnotablyhis Arithmetices principia,nova method exposita,asitcontinueson
fromthe workdone inthe previoussection.Thiswill thenleadme toRichardDedekind.
In hisday,Dedekindpublishedtwofamouspiecesof work – Stetigkeitund irrationalle
Zahlen and Was Sind und Was Sollen die Zahlen? The latterof whichhasa real significance onthis
piece of work,althoughIshall mentionthe formerbrieflyinvariousareasbeyondthispoint.
Although Dedekind’s WasSollen waspublishedbefore Peano’s Arithmetices principia,Iwishtolook
at both inthisorderas the workof Dedekind,baringhuge similarityandimportance tothe workof
Peano,alsohasan overlappingconnectiontothe workof Cantor.
The work of Georg Cantor isthe maininterestof thissection.The developmentof the
transfinite ordinalsandcardinalsbyCantorinthe late 18th
century as a new subjectinmathematics
was notan easyone but it doesremaintothisday and itbecame one of the mostimportantareasof
mathematicsof the 20th
century, bringingmathematicsinto alogical formandbringingmathematics
closerto itsphilosophical roots.Towardsthe endof thiswork,havingintroducedCantor’s naïveset
theory, we shall have come across transfinite inductionandrecursion, whichare slightlydifferent
formsof the inductionandrecursionalreadyinvestigated.

36. History of Mathematical Induction & Recursion [B].
35
[2.1] Giuseppe Peano:
Giuseppe PeanowasanItalianmathematicianwhowasborninCuneoinnorth-westItaly,
nearthe borderof Austria,on27th
August1858. Havingmovedto Turinin1871 withhisuncle to
furtherhisstudies,Peanoenrolledin agraduate programin mathematicsat the Universityof Turin
in1876. Upon graduatingin1880, Peanowasofferedaposition of workat the universitywhichhe
acceptedandwhere he remaineduntil hisdeathin1932. Peanosoonbecame a professorof the
infinitesimal calculusatthe university,althoughhe didhave otherinterestsoutside of thisarea,
mostnotably,the foundationsof mathematicsand,tofill the time,Peanohadakeeninterestin
linguisticstudies.
Peanopublishedinexcessof 200 papersduringhislife andthe firstof these came in1884,
entitled Calcolo differenzialee principii di calcolo integrale. Peanodedicatedthiswork toaformer
teacher,AngeloGenocchi,bypublishingitunder Genocchi’s name andassigninghisownname asa
subtitle of the work.Itwas inthispiece of work,and hisworkas a lecturer,thatthe needfora
higherstandardof rigour in mathematicsbecame cleartoPeano. Whenwritingtowardsa
publication,Peanolikedtokeeptohisownhighstandardof rigourwhile atthe same time making
hisworksimple andeasyto understandandfollow,soitwasusual forPeano,justlike inancient
Greekscripts,to showa demonstrative formof writing.
All of thisis evidentinPeano’s1889 Arithmetices principia,nova method exposita,whichcan
be translatedas The Principles of Arithmetic,Presented by a New Method.The needforsuch work
was apparenttoPeano,to worktowardsthat more rigorousand simple foundationof mathematics
that he thoughtwas necessary,andtoachieve this,Peano introduced symbolstorepresent full
Figure 12 - Giuseppe Peano (1858-1932)

37. History of Mathematical Induction & Recursion [B].
36
mathematical structures andalsoused letterstorepresentwhole propositionsandpropositional
functions;he wasthe firstto do so,there wasno needforPeanoto write full labouringsentences
explainingwhathe wantedtoachieve. Whenpublished,hissymbolicworkprobablyappearedasa
code to its readers,butthe simplicityandeasyflow of readingwouldhave beeneasytopickup.
Othersymbolsthatwere alsointroducedwhichwe stilluse todaywere ∈ forthe inclusionof an
elementinaset, ⊂ for the containmentof asubsetina largerparentset, ∪ and ∩ for the unionand
intersection,respectively,of two sets. Peanopaid specialattentiontothe distinctionbetween ∈and
⊂ as to leadto no ambiguity.
For Peanoto establishhisnew foundationshe hadtostart at the most essential andbasic
area of mathematics,the natural numbers.Peanowaslookingtoestablishanaxiomaticsystemfor
lookingatthe natural numbersandtheirproperties, presented throughhissymbols.Withthe
natural numbersaxiomatized,anypropositionortheoreminmathematicscouldbe verifiedby the
truth heldinthe axioms. Toquote Peano:
“with thesenotations,every proposition assumestheformand theprecision
thatequationshavein algebra”.
Arithmeticesprincipia was publishedasa 36 page pamphlet.The openingintroductionwas
16 pagesdevotedtodefiningandexplaining,rigorouslyof course,the new symbolsthatPeano
wishedtointroduce. The remainderbeganwithfour definitionsandthese were followed bythe
axiomsof whichtheywere nine intotal.Thus, Arithmeticesprincipia beganas follows:
“The sign 𝑁 meansnumber(positiveinteger).
The sign 1 meansunity.
The sign 𝑎 + 1 meansthesuccessorof 𝑎, or 𝑎 plus1.
The sign = meansis equalto.
1. 1 ∈ 𝑁.
2. 𝑎 ∈ 𝑁, 𝑎 = 𝑎.
3. 𝑎, 𝑏 ∈ 𝑁, 𝑖𝑓 𝑎 = 𝑏 𝑡ℎ𝑒𝑛 𝑏 = 𝑎.
4. 𝑎, 𝑏, 𝑐 ∈ 𝑁, 𝑖𝑓 𝑎 = 𝑏 𝑎𝑛𝑑 𝑏 = 𝑐, 𝑡ℎ𝑒𝑛 𝑎 = 𝑐.
5. 𝑖𝑓 𝑎 = 𝑏, 𝑎𝑛𝑑 𝑏 ∈ 𝑁, 𝑡ℎ𝑒𝑛 𝑎 ∈ 𝑁.
6. 𝑎 ∈ 𝑁, 𝑎 + 1 ∈ 𝑁.
7. 𝑎, 𝑏 ∈ 𝑁, 𝑖𝑓 𝑎 = 𝑏, 𝑡ℎ𝑒𝑛 𝑎 + 1 = 𝑏 + 1.
8. 𝑎 ∈ 𝑁, 𝑎 + 1 ≠ 1.
9. 𝐾 ⊂ 𝑁, 𝑖𝑓 ( 𝑎) 1 ∈ 𝐾 𝑎𝑛𝑑 ( 𝑏) 𝑖𝑓, 𝑓𝑜𝑟 𝑎 ∈ 𝑁, 𝑡ℎ𝑒𝑛 𝑎 + 1 ∈ 𝑁; 𝑡ℎ𝑒𝑛 𝐾 = 𝑁.”

38. History of Mathematical Induction & Recursion [B].
37
Axioms2,3, 4 & 5, are the axiomsthatwe have not seenbefore.Todaythese canbe considered
as trivial oras part of the underlyinglogicof equality. Notice thataxiom9,the axiomof induction,is
statedslightlydifferent fromthatof before.i
Withthe inductionaxiombeingstatedlast,Peano
followshisaxiomswithdefiningthe natural numbers,byinduction:
2 = 1 + 1; 3 = 2 + 1; 4 = 3 + 1; and so forth.
Withthis,Peanointroducesaddition,subtraction,maximumandminimumnumbers,
multiplication,powers,division andthenproceedstomove onto theoremsonnumbertheory,
rational andirrational numbers,theoremsonopenandclosedintervals.Mostof the work in
Arithmeticesprincipia is provenbyinductionorconstructedfrom 𝑁 and the successorfunctionii
undera mode of recursion.The detailsof these howeverIdonot wishtoinvestigate,aswe shall see
all of these froma set-theoreticpointof view whenlookingatthe work of Dedekind.
From the axiomof induction, Peanodoesnothave todeduce the validityof the propertiesof his
natural numbers.ThisiscontrastingwithDedekindaswe shall see.We know the axiomcarries
propertiesoverthe whole of the natural numberswhenthe propertyholdsforthe basecaseand,
underthe inductivehypothesis,itholdsforthe inductivestep.This axiom, alongwiththe,whatwe
may call logical, precedingaxioms,isall Peanorequirestodefinestructuresof,andpropertieson,
the natural numbers.
Today,Peano’saxiomsare consideredareference pointforthe natural numbers.However,they
are notalwaysreferredtoas exclusivelybelongingtoPeano;some refertothe axiomsasthe Peano-
DedekindAxioms.
i This is due to wanting to keep to the original translation of Peano’s own statement of the axiom, found in
Jean van Heijenhoort’s Frege to Gödel.
ii 𝑓( 𝑎) = 𝑎 + 1. This is simply the third of the four terms defined by Peano at the beginningand how I shall
refer to it from here.

39. History of Mathematical Induction & Recursion [B].
38
[2.2] Richard Dedekind:
Figure 13 - Richard Dedekind (1831-1916)
“One of the wholly great in the history of mathematics, now and in the past”
– EdmundLandauon Dedekind.
JuliusWilhelm RichardDedekind wasbornthe 6th
of October1831 in Brunswicki
,Lower
Saxony,Germany. Whenhe wasyounger,Dedekindfoundhimself more interestedinphysicsand
chemistrythanmathematics,however,atthe age of 17 and discontentedbythe lackof reasoninghe
foundinphysics,Dedekindturnedhisattentionstowardsmathematicsinsearchformore logically
soundreasoning.Itwasin1850 thatDedekind begantoattend the Universityof Göttingen.Here,
havingbeenborninthe same town andattendedthe same college,Dedekindwouldstudyunderthe
influenceof Carl FriedrichGauss.
Aftertwoyears,Dedekindgraduatedwithhisdoctorate before takingupa lecturingrole at
the universityin1854. It is believedthatDedekindwasthe first,in1857, to give a course on Galois
Theoryat a universitylevel;he gave the course onlytotwo students.Inthe same year,Dedekind
movedonto take up a positionata polytechnicinZurich.1862 saw Dedekindreturnto Brunswick to
take up a positionasa professorat a technical highschool.Thisseemslike twostepsbackwardsfor
Dedekind,butitappearsthatDedekindlivedanobscure,secludedlife,awayfromthe demandand
attentionthatother,lesscapable,mathematicianswere receiving. In1904, it appearedinTeubner’s
i This is an Anglicization of Braunschweig.

40. History of Mathematical Induction & Recursion [B].
39
CalendarforMathematicians thatDedekindhaddiedonthe 4th
of September,1899, although
Dedekinddidinformthe editorthathe wasin fact “in perfecthealth” onthat day havingspentit
withhis“honoured friend Georg Cantorof Halle”.
In 1888, a yearbefore hisfictional death,Dedekindpublishedhisfamous WasSind und Was
Sollen Die Zahlen? The secondeditionof thiswaspublishedin1893 and thiswas translatedin1901
by WoosterW. BemanintoEnglishandgivingthe title theNatureand Meaning of Numbers,
althougha directtranslationwouldbe WhatAreNumbersand WhatShould They Be?From this
work,like thatof Peano,Dedekindwaslookingtoestablishahigherrigourof mathematics,butin
doingso,he tacklesthe age oldelephantinthe room – infinity.Infinityplaysamajorrole inwhatis
to come in laterparts of this workand we have touchedon the infinitebeforewithmathematical
inductionandrecursion.Ishall sayno more on the complexitiesof the infiniteatpresentasI would
like togive a brief summaryof WasSollen, highlightingthe keypointsof interest.
Was Sollen differsinapproachfromPeano’s Arithmeticesprincipia as Dedekinddoesnottake
the axiomaticapproach of Peano,insteadinsistingonaset-theoreticapproachtodefiningthe
natural numbers.Indoingso,Dedekindbeginswiththe infinite anddefinesafiniteseti
asa set
contrastingfromthat of an infinite one;thispartof Dedekind’sworkiswhathe shareswithCantor.
Dedekindusesthismethodof workingbackfromthe infinitetodefinethe natural numbers.
Was Sollen consistsof 172 paragraphs;each assignedanumberinunisonbyDedekind.These
paragraphsare thengroupedtogetherin14 differentsections.The secondeditionhasa new preface
attachedwhere Dedekindwantedtoaddressessomeof hiscritiques;howeverhe couldfindno
justificationforanycriticismthatthe firsteditionmetandthe new preface ismerelyan
acknowledgementthatDedekindhadtriedtoaddressanyissues.
In the preface to the firsteditionof WasSollen,Dedekind addresseshisdesire foramore
rigorousfoundationtomathematics:
“In science nothing capableof proof oughtto beaccepted withoutproof.Though
this demand seemsso reasonableyetI cannotregard it as having been meteven in the
mostrecent methodsof laying the foundationsof thesimplestscienceii
… numbersare
free creationsof the human mind…It is only the purely logical processof building up the
science of numbersand by thusacquiring the continuousnumber-domain thatweare
prepared accurately to investigateour notionsof spaceand time by bringing theminto
relation withthis number-domain created in ourmind.”
i Dedekind referred to a set as a system.
ii Here Dedekind singles out,in a footnote, the work at the time of Kronecker, Schröder, and von Helmholtz.

41. History of Mathematical Induction & Recursion [B].
40
In WasSollen Dedekindreferred tosetsas systems consistingof things(Dinge) thatcouldbe
“considered froma common pointof view,associated in the mind”.Although,inlatersectionsof
Was Sollen,Dedekindreferredtothese thingssimplyaselementsand tookthe same conceptof
Peanobyusinglowercase letterstorepresenttheseelements.Dedekindalsoreferred tofunctions
as transformations.
As alreadynoted,DedekindandPeanotackle theirworkfromdifferentangles.Peano’s
axiomaticapproachallowsPeanotogive hisaxioms,alongwithhis4new definitions,andexpand
fromthere to produce desiredresults.ThusPeanoestablishesthe arithmeticof the natural numbers
fromhis axioms.Incontrast,Dedekind’sconstructionof the natural numbershasto be more
progressive.The foundationsof the workbyDedekindconsistof aseriesof definitionsand
correspondingtheorems.Thusthe earlysectionsof Dedekind’s WasSollen addressthese
requirements.
In the thirdsection, Similarityof Transformations.SimilarSystems,Dedekindsaysthattwo
setsare similar if theycan be putin to a one-to-one correspondenceundera similar function,one
that isbijective.Thisisfollowedbythe definitionof a class, by whichall setsthatare similartoone
anothercan be groupedtogetherandany one of the setsin the class can be the representativeof
the class as a meansof identifyingthe classfromanyotherclassof similarsets.The conceptof
similarity betweensetsishowDedekinddefinesinfinitesetsandfromthis,finite sets.
¶64. “Definition. A system 𝑆 is said to be infinite when it is similar to a properpartof
itself; in the contrary case, 𝑆 is said to be finite.”
Thus,a set 𝑆 is an infinitesetif ithasa one-to-one correspondence toaproperi
subsetof
itself.Onthe otherhand,if thisisnot the case,then 𝑆 issaidto be a finite set.Dedekindhasnow
definedwhatitmeansfora setto be infinite andhighlighteditsdifference fromafinite set.
¶66. “Theorem. There existsinfinite systems.”
Althoughin1872’s Stetigkeitund Irrationalle Zahlen,Dedekind hadalreadyusedinfinite
collectionswhencreatinghisidealsii
,the needtojustifythe existence of infinite sets inWasSollen
was because hisdefinitionof the natural numbersdependedupontheirexistence.Thus,according
to Dedekind,if infinitesets exist,thenthe natural numbersexist;asomewhatpressure toaccept
theirexistence.The proof of thistheoremiscompletelyphilosophical andnon-mathematical and
shall be returnedtoindue course.Dedekindcontinuedbydistinguishingbetweeninfinite andfinite
i 𝑈 is a proper subsetof 𝑆 if itis not equal to 𝑆, i.e. 𝑈 is strictly contained in 𝑆; 𝑈 ⊂ 𝑆.
ii Dedekind defined an ideal as an infinitecollection of algebraic real numbers;numbers that are the root of a
non-zero polynomial in 1 variable,with rational coefficients.

42. History of Mathematical Induction & Recursion [B].
41
sets,howeverbeforeDedekindcould define the natural numbersthere wasthe needtodefinea
simply infinite set.
A set 𝑆 is simply infinite whenthere isa similarfunction 𝜙: 𝑆 → 𝑆, suchthat 𝑆 can be
obtainedfromthe unionof repeatedapplicationof 𝜙 onanelementof 𝑆 thatisnot in 𝜙( 𝑆)i
, i.e. 𝑆 is
generatedbyanelementthatisnotin the image of 𝜙.
Thisdefinitionis thenreferredtoa particularset, 𝑁, andan elementthatgeneratesthisset;
the Base-element“which we shall denoteby the symbol1”. The definitionof asimplyinfinite set, 𝑁,
was simplifiedbyDedekindtothe existence of asimilarfunction 𝜙of 𝑁 and an element1 suchthat
the followingconditionsare satisfied:
𝛼: Define 𝑁′ = 𝜙( 𝑁),then 𝑁′ ⊂ 𝑁.
𝛽: 𝑁 = 𝜙(1), that is, 1 isthe base-elementof 𝑁.
𝛾:1 ∉ 𝑁′.
𝛿: 𝜙 issimilar,i.e.aone-to-onefunction.
It wouldbe possible totake the fourrequirements 𝛼, 𝛽, 𝛾, 𝛿, andassume theirexistence as
axiomsfora set 𝑁, howeverDedekind didnotfeel therewasaneedforaxiomsanddefined the
natural numbersas follows:
¶73. “Definition. If in theconsideration of a simply infinite system 𝑁 set in order by a
transformation 𝜙weentirely neglect thespecial character of the elements;simply
retaining their distinguishabilityand taken into accountonly the relationsto one
anotherin which they are placed by the order-setting transformation 𝜙,then are these
elementscalled naturalnumbers orordinalnumbers orsimply numbers,and thebase-
element 1 is called the base-numberof thenumber-series 𝑁.With referenceto this
freeing the elements fromevery othercontent(abstraction) wearejustified in calling
numbersa free creation of thehuman mind.The relationsor lawswhich are derived
entirely fromthe conditions 𝛼, 𝛽, 𝛾, 𝛿, in (71) and thereforearealwaysthe samein all
ordered simply infinite systems,whatevernamesmay happen to begiven to the
individualelements,fromthe first objectof the science of numbersorarithmeticii
.”
The remainderof thisdefinitionfocused onhow the elements andsubsetsof 𝑁 are closed
under 𝜙. It wasalso noted byDedekind that“thetransform 𝑛′of a number 𝑛 is also called the
numberfollowing 𝑛”.
i Thus 𝑆 =∪ 𝜙 𝑛( 𝑠), 𝑠 ∈ 𝑆, where 𝜙 𝑛
is the 𝑛th iterate of 𝜙.
ii From here the elements of the set 𝑁 arereferred to as numbers.

43. History of Mathematical Induction & Recursion [B].
42
Thiswas howDedekinddefined the natural numbers;it wassufficienttohave the simply
infinite set 𝑁,awhole,singleentity,orderedbythe repeateduse of the function 𝜙 suchthat,for
𝑛 ∈ 𝑁, 𝜙( 𝑛) ∈ 𝑁.
The definitionof the natural numberwasfollowedbyaseriesof theoremsthatcoincide with
thisnewdefinition of the natural numbers.Thesetheoremscan be seenasPeano’saxioms.For
example,Peano’s8th
axiom, 𝑎 ∈ 𝑁, 𝑎 + 1 ≠ 1,can be comparedto paragraph 79 of WasSollen
whichsays “every numberotherthan thebase-number1 isan element of 𝑁′.”
The followingparagraph,paragraph80, iswhenDedekinddefinedinductionuponthe
natural numbers.Dedekindhadpreviouslydefinedinductiononthe notionsof chainsinan early
sectionbutstatedthat that formation, the “theoremof induction”,served only asabasisfor
inductionuponthe natural numbers.Aswasstatedinthe previoussectionof thiswork,Dedekind
referredtoinductionas VöllstandigeInduktioni
andfromthispointin WasSollen, itwas usedto
prove the majorityof the theoremsthatare to follow.Inparagraph126, the “theoremof the
definition by induction”is introduced;this wasthenusedtodefine addition,multiplicationand
exponentiationwiththeirsubsequenttheoremsandpropertiesprovenbyinduction.
i Beman translates this as complete induction.

44. History of Mathematical Induction & Recursion [B].
43
[2.3] Arithmetices principia and Was Sollen:
The authors of Arithmetices principia and Was Sollen bothhad the same objective inwriting
theirpublications;toestablishthatmore rigorousfoundationtomathematicsthatbothDedekind
and Peanothought mathematicsrequired atthe time.AlthoughDedekindandPeano attackedtheir
problemfromdifferentanglesandtakingdifferentapproaches,bothwere successfulinobtaining
theirobjective and theirideas were sostrongthatthey are still beingusedas a basisfor
mathematics today.
Dedekind’sopposite approachfromPeano wasinhow the natural numbers were defined;
havingfirstdefinedwhatitmeanstobe an infinitesetandsayingthatthe natural numbersare such
a set undera particularone-to-onefunction,whereas Peanobuiltupfromunity,usinghisaxioms.As
we have seenPeano’suse of induction enabledhimtoworkwiththe infinitefromafinite
perspective;itallowed Peanotorepresent the potentialinfinity,being finite ateachpointandnever
actuallyreachingan infinite totality.Incontrast, Dedekind’sdefinitionof the natural numbers isnot
free of the existence of the actual infinite. Atthe time thiswasa boldand unusual approachand
Peano’saxiomswouldhave beenmore readilyacceptedbythe mathematical community.
For Dedekind,the infinitewaseasierthanitscounterpart;the existence of aninfinite setandthe
definitionof the natural numbers,paragraphs 66 and72 respectively, were addressedbefore the
specificfinitecase of theorem81; 𝑛 isnot equal toits successor 𝑛′. To acceptthe existence of an
actual infinite setwaswhateveryothermathematician atthe time wasavoidingorwhose existence
was absurd to thembutit led to a newwayof lookingatthe natural numbers.
Thisnewoutlookonthe most simplestof all mathematical areashada huge bearingonthe
worksof others(those whodidn’trejectit) andthe proof of the existenceof aninfinitesethelpedin
mathematiciansacceptingthe workof GeorgCantorwhoat thistime hadalreadybegun topublish
hisworkson transfinite numbers.
We have seenthatDedekind’sdefinitionof the natural numbers,paragraph73, wasvery
wordyand itcame froma philosophical viewpoint,justlikethe ‘proof’of the existence of simple
infinite sets. Dedekindcouldhave definedthe natural numbersinamore mathematical structure
but thisapproachallowedDedekindtoabstractnumberfromanyneedof justificationof thought,
much like inthe same veinasPeano’suse of the inductionaxiom.Thismethodof abstractionwas
fundamental tobothPeanoandDedekindtocreate theirfoundations;itallowedbothmento
distance theirideasfromanyconcrete thing (Dinge) orfrom any justificationinnature andtheir
abstractionleadthemtogenerality. Theirrigorousfoundationswere laidoutandtodaytheystill
exist,togetherinfact,as the Dedekind-Peano Axioms.

45. History of Mathematical Induction & Recursion [B].
44
Dedekind’sacceptance of anactual infinitesetisworthconsideration.Before the theoremof
paragraph 66, thereexist infinite sets,any theoremthathad beenstatedwasalwaysfollowedbya
proof that stuck to mathematical reasoning.Howeverwiththistheorem,Dedekindbeganhisproof
with“My own realm of thoughts”whichimmediatelysuggeststhatthere were tobe no
mathematical reasoninginvolvedwiththisproof,andindeedthatwasthe case.Dedekindinstead
argueshiscase froma philosophical pointof view,butwhy?
Thiscouldbe because of the statusthat the infinite hadwithinmathematicsatthe time –
bannedbyAristotle inthe age of the great Greekmathematicians,noone daredtoacceptits
existence fromthenon,whetheritbe potentialoractual.An example of afamousmathematician
whowouldneveracceptthe infiniteisGalileoi
whobelievedinthe axiomsof Euclid’s Elementsand
inparticular– the wholeis greater than thepart.Therefore,toGalileo,tohave abijective
correspondence betweenasetanda propersubsetof that setwouldmeandenyingthe truthof
Euclid’saxiom.Dedekindtriedaddressingthisbytakingthis paradox andusingitasa definitionfor
the infinite,essentiallysaying,thatthe infiniteisthatwhichisnotinfinite.Thus,the infinite would
have differentpropertiestoitscomplementandvice-versa.
However,anyconsiderationof the infinite atthe time wouldhave beenmetwithsternobjection
by many.Most of the ‘justifications’thatwere usedtodothisusuallycame froma philosophical
pointof view.Thuswasthiswhy Dedekindfounditnecessarytoargue hispointinsucha manner?
Another,simplerreasonforthiscouldbe that Dedekindsimplybelievedittobe true,but couldnot
formulize amathematical proof tojustifyit.
Howeverthe questionarises,shouldtheorem66 be a mathematicaltheoremorshouldit
come underthe postulate/axiomtitleandjustbe acceptedastrue,in orderfor Dedekindto
continue hisworkmaintainingacontinuousmathematicaltruth?;foreverytheoremthatisbased
uponthat of 66 wouldbe basedona philosophicaljustification.
The structure of Dedekind’s WasSollen isdifferentfromthatof Peano’s.Dedekinddidnot
feel the needtodefine the usual arithmeticdefinitionsof additionandmultiplicationimmediately
afterdefiningthe natural numbers,insteaddefiningthe structure of the natural numbersandthe
orderingof itselements.One reasonforthiscouldhave beenbecause Dedekind’sdefinitionof the
natural numbersdependedonthe existence of infinite sets,soaneedtoinvestigatethe simple
infinite set 𝑁furthermayhave stoodinthe way.Anotherreasoncouldbe because Dedekinddidnot
feel the needtodoso as the needto define the structure of 𝑁 andthe orderof itselementswhere
more important.ThisallowedDedekindtodistinguishfurtherbetweenfinite andinfinitesets,use
i Galileo Galilei (1564-1642),Italy.

46. History of Mathematical Induction & Recursion [B].
45
definitionbyinduction,andtalkof the classesof infinitesets;all of thisbefore additionof the
natural numbers.Maybe there wasa hiddenagendaof tacklingthe actual infinite that Dedekind
wantedtodisguise behindthe natural numbers.
Whateverreasonthere wasforthe order of Dedekind’swork,itremainsthatDedekindhad
more material thathe wantedtopublishandshow toothers than simplydefiningthe natural
numbers.Thisextramaterial wasmainlyfocusedonthe actual infinite anditwasparallel tothe
workbeingdone byGeorg Cantorat the time.

47. History of Mathematical Induction & Recursion [B].
46
[2.4] Dedekind’s Interest in Infinity:
We have seenthatDedekindfoundsignificanceinthe infinite;fromthe infinite idealsof
Stetigkeit to the infinite systems of WasSollen. In hiswork,Dedekindtriedtojustifythe use of
infinity;byusinginfinite setsasabasisfor hisdefinitionof the natural numbers,Dedekindtriedto
make it difficultandcontradictorytodenythe existenceof suchsets.We know that Dedekindwas
seekingtoestablishsounderfoundationsformathematics,so toinclude infinitesetsinhis
foundations,Dedekindmusthave alsothoughtthatinfinite setswereanimportantpartof
mathematics.
Once the natural numberswere definedandthose theoremssimilartothe axiomsof Peano
stated,Dedekindproceededtodefinethe structure of the natural numbersunderthe greaterthan
and lessthan relations.AfterthisDedekindstated animportantresult,one whichCantorwouldfind
importantinhisownwork onwell-orderedsets:
¶96. “Theorem. In every part 𝑇of 𝑁 there existsone and only one leastnumber 𝑘, i.e. a
number 𝑘 which is less than every othernumbercontained in 𝑇.”
Thus,in anysubsetof the natural numbers,there isaminimal element,one smallerthan
everyotherelementinthe subset,accordingtothe orderon the natural numbers.Thiscan be
extendedtoanysetthatis similartothe natural numbersor a subsetof the natural numbers;soit
appliestoall simplyinfinite sets.Thiscoincideswiththe theoreminparagraph72, whichsaysthat
everysimplyinfinite sethasasubsetthatis similartothe setof natural numbers.Eventually
Dedekind proved thatall simplyinfinite setsare similartothe setof natural numbersand these
simplyinfinitesetscanbe groupedintodifferentclassescorrespondingtowhichsubsetof the
natural numberstheyare similarto.
Today,we wouldsaythat thismeansthat any simplyinfinitesetisone thatiscountable and
therefore amodel of the natural numbers.Inthe time of Dedekind,the natural numberswere
knownto be countable (theycountthemselves essentially),butothernumbersystemswere abitof
a grey areaat the time,especiallythe real numbers.Cantorbelievedthatthe real numberswere of a
differentsizethan the natural numbers,butthisfacedstiff oppositionfromKronecker,whosaidthat
the ideaof twodifferent(potentially) infinitesizeswasridiculous.However,tosaythat all simply
infinite setscanbe classedtogether,one couldworkoutthatthat wouldmean,since theydonot
have a leastelement,the setof real numberscouldnotbe putinto the same classas the natural
numbers.Elaboratingonthis,noteverysubsetof the real numberswouldhave aleastmember,take
the openinterval (0,1) andcompare it with [0, 1].

48. History of Mathematical Induction & Recursion [B].
47
The needforDedekindtodistinguishbetweenthe finite andthe infiniteisanessential partof
Was Sollen.The differencesbetweenthe twoare laidoutinsimplisticformsothat theycan be easily
followedandmore readilyaccepted.WhatDedekindtriedtoestablish,wastostopmathematicians
fromtacklingthe infinite fromafinite perspective.Asmathematicianswereapplyingpropertiesand
theoremsdefinedonthe finite,tothe infinite,theywere notgettingconsistencyor producingthe
resultsthatwere expected.Thisisevidentinthe case of Galileo andEuclid’saxiomabove andthis
treatmentof the infinite still happenedinthe 18th
century.Thus,there were some whowere
unwillingtoinvestigate andworkwiththe infinite.Butthe workof Dedekindtriedtocurbthisway of
thinkingandattract more mathematicians toacceptthe conceptof the actual infinite.
Anotherimportantaspectof WasSollen isDedekind’sfinal section.Itisentitled TheNumber
of Elements in a Finite System and itaddressesthe size of a finite set.The mostnotable paragraphin
thissectionis161 whichI shall give initsentirety:
¶161. “Definition.If a set 𝛴 is a finite system,then thereexists oneand only onesingle
number 𝑛 to which 𝑍 𝑛~ 𝛴, fora system 𝑍 𝑛.This number 𝑛 is called the number(Anzahl)
of elements in 𝛴; it showshowmany elementsarecontained in 𝛴. If numbersareused
to expressaccurately thisdeterminateproperty of finite systemsthey are called cardinal
numbers.If thereis a similar transformation 𝜓: 𝑍 𝑛 → 𝛴, then wecan say thatthe
elementsof 𝛴 are counted and setin order by 𝜓 in determinatemanner,and call 𝑎 𝑚 the
𝑚th element of 𝛴; if 𝑚 < 𝑛, then 𝑎 𝑚 is called the element following 𝑎 𝑚
i,and 𝑎 𝑛 is
called the last element.In this counting of theelements thereforethenumbers 𝑚
appearagain asordinalnumbers.”
Thus,the elementsof the finitesetcanbe indexedbythe natural numbersandif the order
of the elementsof the finite setare arrangedsuchthat theyfollow the orderof theirindex,thenwe
can say that theyfollowthe orderof the natural numbersandcan thus be calledordinal numbers,as
statedinDedekind’sdefinitionof the natural numbers.Conversely,the definitionsaysthat, if we
abstract fromthe orderof the elementsinaset,thenwe obtainthe cardinal numberof the set.
Dedekinddoesnotstate whenanumberisan ordinal number,butthe implicationisthatthisis
whenthe numberfollows the orderingof the numbersthatprecede itii
.Bothcardinal andordinal
numberswere the keyinterestsof cantorat the time,whowastryingto introduce bothtohis
transfinite settheory.The backingof Dedekindwouldhave beenamajorboostto theiracceptance
as the oppositionagainst Cantorandhistheorywasmounting.
i I think this should havesaid 𝑎 𝑛 is thenumber following 𝑎 𝑚 .
ii Thankfully Cantor gives a better definition of ordinal number.

49. History of Mathematical Induction & Recursion [B].
48
Cantor beganto workon infinitesetswhenthe needtoinvestigate furtherthe cardinalityof
infinite setsandthe numberof pointsinthe continuumi
became clear.ItappearsthatDedekindsaw
cardinalityasan importantaspectof everyset;fromthe remark at the endof WasSollen, Dedekind
statesthat the cardinalityof a setdoesnotchange undersimilarfunctions,thusthere canbe many
setsthat share the same cardinalityandbe collectedtogetherandtreatedasone andthe same.
However,Dedekinddoesnotinvestigate thisasit“doesnotlie in the line of this memoirto go
furtherinto their discussion”,noristhere any considerationof the cardinalityof infinitesets,surely
theyjusthave infinite cardinality anyway.
The newfoundationsof the infinite,andthe finite,wouldhave servedasabasisfor the work
of GeorgCantor,not for Cantorto buildhisworkuponit as he had alreadybeguntopublishworkon
histransfinite numbers,butforothermathematicianstoreference anydoubtsinthe workof Cantor.
i The real numbers, or equivalently,the open interval (0, 1) of the real numbers.