Table Of ContentSet Theory in Computer Science
A Gentle Introduction to Mathematical Modeling I
Jose´ Meseguer
UniversityofIllinoisatUrbana-Champaign
Urbana,IL61801,USA
(cid:13)c Jose´ Meseguer,2008–2010;allrightsreserved.
February28,2011
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Contents
1 Motivation 7
2 SetTheoryasanAxiomaticTheory 11
3 TheEmptySet,Extensionality,andSeparation 15
3.1 TheEmptySet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Extensionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 TheFailedAttemptofComprehension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.4 Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4 Pairing,Unions,Powersets,andInfinity 19
4.1 Pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.2 Unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.3 Powersets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.4 Infinity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5 CaseStudy: AComputableModelofHereditarilyFiniteSets 29
5.1 HF-SetsinMaude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.2 Terms,Equations,andTermRewriting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.3 Confluence,Termination,andSufficientCompleteness . . . . . . . . . . . . . . . . . . . 36
5.4 AComputableModelofHF-Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.5 HF-SetsasaUniverseforFinitaryMathematics . . . . . . . . . . . . . . . . . . . . . . . 43
5.6 HF-SetswithAtoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6 Relations,Functions,andFunctionSets 51
6.1 RelationsandFunctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6.2 Formula,Assignment,andLambdaNotations . . . . . . . . . . . . . . . . . . . . . . . . 52
6.3 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
6.4 ComposingRelationsandFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
6.5 AbstractProductsandDisjointUnions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.6 RelatingFunctionSets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
7 SimpleandPrimitiveRecursion,andthePeanoAxioms 65
7.1 SimpleRecursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
7.2 PrimitiveRecursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
7.3 ThePeanoAxioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
8 CaseStudy: ThePeanoLanguage 71
9 BinaryRelationsonaSet 73
9.1 DirectedandUndirectedGraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
9.2 TransitionSystemsandAutomata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
9.3 RelationHomomorphismsandSimulations . . . . . . . . . . . . . . . . . . . . . . . . . 76
9.4 Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
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9.5 SupsandInfs,CompletePosets,Lattices,andFixpoints . . . . . . . . . . . . . . . . . . . 81
9.6 EquivalenceRelationsandQuotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
9.7 ConstructingZandQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
10 CaseStudy: FixpointSemanticsofRecursiveFunctionsandLispy 91
10.1 RecursiveFunctionDefinitionsinaNutshell . . . . . . . . . . . . . . . . . . . . . . . . . 92
10.2 FixpointSemanticsofRecursiveFunctionDefinitions . . . . . . . . . . . . . . . . . . . . 93
10.3 TheLispyProgrammingLanguage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
11 SetsComeinDifferentSizes 97
11.1 Cantor’sTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
11.2 TheSchroeder-BernsteinTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
12 I-IndexedSets 99
12.1 I-IndexedSetsareSurjectiveFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
12.2 ConstructingI-IndexedSetsfromotherI-IndexedSets . . . . . . . . . . . . . . . . . . . 104
12.3 I-IndexedRelationsandFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
13 FromI-IndexedSetstoSets,andtheAxiomofChoice 107
13.1 SomeConstructionsAssociatingaSettoanI-IndexedSet . . . . . . . . . . . . . . . . . 107
13.2 TheAxiomofChoice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
14 Well-FoundedRelations,andWell-FoundedInductionandRecursion 117
14.1 Well-FoundedRelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
14.1.1 ConstructingWell-FoundedRelations . . . . . . . . . . . . . . . . . . . . . . . . 118
14.2 Well-FoundedInduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
14.3 Well-FoundedRecursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
14.3.1 ExamplesofWell-FoundedRecursion . . . . . . . . . . . . . . . . . . . . . . . . 120
14.3.2 Well-FoundedRecursiveDefinitions: StepFunctions . . . . . . . . . . . . . . . . 121
14.3.3 TheWell-FoundedRecursionTheorem . . . . . . . . . . . . . . . . . . . . . . . 123
15 CardinalNumbersandCardinalArithmetic 125
15.1 CardinalArithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
15.2 TheIntegersandtheRationalsareCountable . . . . . . . . . . . . . . . . . . . . . . . . 129
15.3 TheContinuumandtheContinuumHypothesis . . . . . . . . . . . . . . . . . . . . . . . 131
15.3.1 PeanoCurves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
15.3.2 TheContinuumHypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
16 Classes,IntensionalRelationsandFunctions,andReplacement 135
16.1 Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
16.1.1 MathematicalTheoremsareAssertionsaboutClasses . . . . . . . . . . . . . . . . 138
16.2 IntensionalRelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
16.3 IntensionalFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
16.3.1 TypingIntensionalFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
16.3.2 ComputingwithIntensionalFunctions . . . . . . . . . . . . . . . . . . . . . . . . 143
16.3.3 DependentandPolymorphicTypes . . . . . . . . . . . . . . . . . . . . . . . . . 144
16.4 TheAxiomofReplacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
17 CaseStudy: DependentandPolymorphicTypesinMaude 149
17.1 DependentTypesinMaude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
17.2 Polymorphic-and-DependentTypesinMaude . . . . . . . . . . . . . . . . . . . . . . . . 152
17.3 DefinabilityIssues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
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18 WellOrders,Ordinals,Cardinals,andTransfiniteConstructions 155
18.1 Well-OrderedSets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
18.2 Ordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
18.2.1 OrdinalsasTransitiveSets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
18.2.2 SuccessorandLimitOrdinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
18.2.3 OrdinalArithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
18.3 TransfiniteInduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
18.4 TransfiniteRecursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
18.4.1 α-Recursion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
18.4.2 SimpleIntensionalRecursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
18.4.3 TransfiniteRecursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
18.5 Well-Orderings,Choice,andCardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
18.5.1 Cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
18.5.2 MoreCardinalArithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
18.5.3 Regular,Singular,andInaccessibleCardinals . . . . . . . . . . . . . . . . . . . . 172
19 Well-FoundedSetsandTheAxiomofFoundation 173
19.1 Well-FoundedSetsfromtheTopDown . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
19.1.1 (cid:51)-Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
19.2 Well-FoundedSetsfromtheBottomUp . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
19.3 TheAxiomofFoundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
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Chapter 1
Motivation
“... we cannot improve the language of any science without at the same time improving the
science itself; neither can we, on the other hand, improve a science, without improving the
languageornomenclaturewhichbelongstoit.”
(Lavoisier,1790,quotedinGoldenfeldandWoese[23])
Ifoundtheinadequacyoflanguagetobeanobstacle;nomatterhowunwieldlytheexpressions
Iwasreadytoaccept,Iwaslessandlessable,astherelationsbecamemoreandmorecomplex,
to attain the precision that my purpose required. This deficiency led me to the idea of the
presentideography. ...IbelievethatIcanbestmaketherelationofmyideographytoordinary
languageclearifIcompareittothatwhichthemicroscopehastotheeye.Becauseoftherange
ofitspossibleusesandtheversatilitywithwhichitcanadapttothemostdiversecircumstances,
the eye is far superior to the microscope. Considered as an optical instrument, to be sure, it
exhibitsmanyimperfections,whichordinarilyremainunnoticedonlyonaccountofitsintimate
connection with our mental life. But, as soon as scientific goals demand great sharpness of
resolution, the eye proves to be insufficient. The microscope, on the other hand, is prefectly
suitedtopreciselysuchgoals,butthatisjustwhyitisuselessforallothers.
(Frege,1897,Begriffsschrift,in[52],5–6)
Language and thought are related in a deep way. Without any language it may become impossible to
conceive and express any thoughts. In ordinary life we use the different natural languages spoken on the
planet. Butnaturallanguage,althoughextremelyflexible,canbehighlyambiguous,anditisnotatallwell
suitedforscience. Imagine,forexample,thetaskofprofessionallydevelopingquantummechanics(itself
relying on very abstract concepts, such as those in the mathematical language of operators in a Hilbert
space)inordinaryEnglish. Suchataskwouldbevirtuallyimpossible;indeed,ridiculous: aspreposterous
as trying to build the Eiffel tower in the Sahara desert with blocks of vanilla ice cream. Even the task of
popularization,thatis,ofexplaininginformallyinordinaryEnglishwhatquantummechanicsis,ishighly
nontrivial,andmustofnecessityremaintoaconsiderableextentsuggestive,metaphorical,andfraughtwith
thepossibilityofgrossmisunderstandings.
Thepointisthatwithoutaprecisescientificlanguageitbecomesvirtuallyimpossible,oratleastenor-
mously burdensome and awkward, to think scientifically. This is particularly true in mathematics. One
of the crowning scientific achievements of the 20th century was the development of set theory as a pre-
ciselanguageforallofmathematics,thankstotheeffortsofCantor,Dedekind,Frege,Peano,Russelland
Whitehead,Zermelo,Fraenkel, Skolem,Hilbert,vonNeumann,Go¨del,Bernays,Cohen, andothers. This
achievementhasbeensoimportantanddefinitivethatitledDavidHilberttosay,alreadyin1925,that“no
one will drive us from the paradise which Cantor created for us” (see [52], 367–392, pg. 376). It was of
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course possible to think mathematically before set theory, but in a considerably more awkward and quite
restrictedway,becausethelevelsofgenerality,rigorandabstractionmadepossiblebysettheoryaremuch
greaterthanatanyotherprevioustime. Infact,manykeymathematicalconceptswenowtakeforgranted,
suchathoseofanabstractgrouporatopologicalspace,couldonlybeformulatedaftersettheory,precisely
becausethelanguageneededtoconceiveandarticulatethoseconceptswasnotavailablebefore.
Set theory is not really the only rigorous mathematical language. The languages of set theory and of
mathematicallogicweredevelopedtogether,sothat,asamathematicaldiscipline,settheoryisabranchof
mathematicallogic. Technically,asweshallseeshortly,wecanviewthelanguageofsettheoryasaspecial
sublanguageoffirst-orderlogic. Furthermore,othertheoriessuchascategorytheoryandintuitionistictype
theoryhavebeenproposedasalternativestosettheorytoexpressallofmathematics.
Therearevariouspreciselogicalformalismsotherthansettheorywhichareparticularlywell-suitedto
expressspecificconceptsinagivendomainofthought. Forexample,temporallogicisquitewell-suitedto
expresspropertiessatisfiedbythedynamicbehaviorofaconcurrentsystem;andbothequationallogicand
thelambdacalculusareverywellsuitedtodealwithfunctionsandfunctionalcomputation. However, set
theoryplaysaprivilegedroleasamathematicallanguageinwhichallthemathematicalstructuresweneed
in order to give a precise meaning to the models described by various other logical languages, and to the
satisfactionofformulasinsuchlanguages,canbedefined.
All this hasa direct bearing on thetask of formal software specificationand verification. Sucha task
would be meaningless, indeed utter nonsense and voodoo superstition, without the use of mathematical
models and mathematical logic. And it is virtually impossible, or extremely awkward, to even say what
needs to be said about such mathematical models and logical properties without a precise mathematical
language. Moreimportantly,itbecomesvirtuallyimpossibletothinkproperlywithouttheconceptualtools
provided by such a language. Either set theory or some comparable language become unavoidable: it is
partofwhatanywelleducatedcomputerscientistshouldbeconversantwith,liketheaironebreathes.
These notes include a review of basic set theory concepts that any well educated computer scientist
should already be familiar with. Although they go beyond reviewing basic knowledge in various ways,
nothingexceptbasicacquaintancewiththeuseoflogicalconnectivesandofuniversalandexistentialquan-
tificationinlogicisassumed: thepresentationisentirelyself-contained,andmanyexercisesareproposed
tohelpthereadersharpenhis/herunderstandingofthebasicconcepts. Theexercisesareanessentialpart
of these notes, both because they are used in proofs of quite a few theorems, and because by solving
problems in a mathematical theory one avoids having a superficial illusion of understanding, and gains
realunderstanding. Forthosealreadywell-versedinelementarysettheory, thesenotescanbereadrather
quickly. However, some topics such as well-founded relations, well-founded induction, well-founded re-
cursive functions, I-indexed sets, ordinals and cardinals, classes, and transfinite induction and recursion,
may be less familiar. Also, already familiar notions are here presented in a precise, axiomatic way. This
mayhelpevensomereadersalreadythoroughlyfamiliarwith“naive”settheorygainamoredetailedun-
derstandingofitasalogicallyaxiomatizedtheory. Becomingusedtoreasoncorrectlywithinanaxiomatic
theory—Euclideangeometryistheclassicalexample,andaxiomaticsettheoryfollowsthesameconcep-
tualpattern—isthebestwayIknowoflearningtothinkinaprecise, mathematicalway. Furthermore, a
number of useful connections between set theory and computer science are made explicit in these notes;
connectionsthatareusuallynotdevelopedinstandardpresentationsofsettheory.
Ishouldaddsomefinalremarksonthestyleofthesenotes. Therearethreeparticularstylisticfeatures
thatIwouldliketoexplain. First,thesenotestaketheformofanextendedconversationwiththereader,in
whichIproposeanddiscussvariousproblems,whytheymatter,andthrowoutideasonhowtosolvesuch
problems. ThisisbecauseIbelievethatscienceitselfisanongoingcriticaldialogue,andaskingquestions
in a probing way is the best way to understand anything. Second, I do not assume the proverbial mathe-
maticalmaturityonthepartofthereader, sincesuchmaturityispreciselythequoderatdemonstrandum,
andbringingitaboutisoneofthemaingoalsofthesenotes: Iamconvincedthatinthe21stcenturymath-
ematicalmaturityisvirtuallyimpossiblewithoutmasteringthelanguageofsettheory. Onthecontrary, I
assume the potential mathematical immaturity of some readers. This means that, particularly in the early
chapters, there is a generous amount of what might be called mathematical spoon feeding, hand holding,
andevenafewnurserytales. Thisdoesnotgoonforever,sinceateachstageIassumeasknownallthathas
beenalreadypresented,thatis,themasteryofthelanguagealreadycovered,sothatinmoreadvancedchap-
ters,althoughtheconversationalstyle,examples,andmotivationremain,thediscoursegraduallybecomes
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more mature. By the way, all this does not necessarily mean that the book is of no interest to mathemat-
ically mature readers, including professional mathematicians. What it does mean is that it should be an
easy read for them; one from which they may gain at least two things: (i) a more reflective, systematic
understandingof thesettheory theyuseevery day; and(ii)a goodunderstanding ofsomeof thewaysin
which mathematical models are used in computer science. The third stylistic feature I want to discuss is
thatthemindsetofcategorytheory,pervasiveinmodernmathematics,ispresenteverywhereinthesenotes,
but in Parts I and II this happens in a subliminal way. Categories and functors are only defined in Part
III;buttheyarepresentfromthebeginninglikeahiddenmusic. Andfunctorialconstructionsmakeearly
cameoappearancesinPartI(verymuchlikeAlfredHitchcockinhisownmovies)infourexercisesandin
§15. Indeed,§15,oncardinalsandcardinalarithmetic,isagreatbeneficiaryofthissubliminal,categorical
style; because much of basic cardinal arithmetic turns out to be an easy corollary of the functoriality of
theset-theoreticconstructionsusedtodefinearithmeticoperationsoncardinals. Thetempoofthishidden
categoricalmusicpicksupwhenuniversalalgebraandfirst-orderlogicarediscussedinPartII;buttherich
harvestfromthissubliminal,categoricalsowingisreallyreapedinPartIII.Inmyview,thisisagoodway
ofgentlypreparingthereader,witharichcollectionofexamples,tofullyappreciatethebeauty,power,and
conceptualeconomyofthelanguageofcategorytheory.
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