LargeCardinalsandDeterminacy
PeterKoellner
September12,2011
The developments of set theory in 1960’s led to an era of independence in
which many of the central questions were shown to be unresolvable on the
basis of the standard system of mathematics, ZFC. This is true of statements
from areas as diverse as analysis (“Are all projective sets Lebesgue
measurable?”), cardinalarithmetic(“DoesCantor’sContinuum Hypothesis
hold?”),combinatorics(“DoesSuslin’sHypotheseshold?”),andgrouptheory
(“IsthereaWhiteheadgroup?”).
These developments gave rise to two conflicting positions. The first
position—which we shall call pluralism—maintains that the independence
resultslargelyunderminetheenterpriseofsettheoryasanobjectiveenterprise.
Onthisview,althoughtherearepractical reasonsthatonemightgive
infavourofonesetofaxiomsoveranother—say,thatitismoreusefulfora
giventask—,therearenotheoretical reasonsthatcanbegiven;and,moreover, this
either implies or is a consequence of the fact—depending on the variant of
the view, in particular, whether it places realism before reason, or
conversely—that there is no objective mathematical realm at this level.
Thesecondposition—whichweshallcallnon-pluralism—maintainsthatthe
independence results merely indicate the paucity ofour standard resources
forjustifyingmathematicalstatements. Onthisview,theoreticalreasonscan
begivenfornewaxiomsand—again,dependingonthevariantoftheview—
thiseitherimpliesorisaconsequence ofthefactthatthereisanobjective
mathematicalrealmatthislevel.
The theoretical reasons for new axioms that the non-pluralist gives are
quite different that the theoretical reasons that are customarily at play in
mathematics,inpartbecause,havingsteppedintotherealmofindependence,
amoresubtle formofjustification isrequired, one thatrelies more heavily
onsophisticatedmathematicalmachinery. Thedisputebetweenthepluralist
1
andthe non-pluralistis thus onethatlies atthe intersection ofphilosophy and
mathematics—on the one hand, it is intimately connected with some
ofthecentralquestions ofphilosophyofmathematics, suchasthequestion of
realism and the nature of justification; on the other hand, it addresses
thesequestionsinamannerthatissensitivetoagreatdealofsophisticated
mathematics.
The debate between the pluralist and the non-pluralist is really a hierarchy
of debates. At the one extreme there is pluralism with regard to all of
mathematics and at the other extreme there is non-pluralism with regard to
all of mathematics. The intermediate positions (which are much more
common) involve embracing non-pluralism for certain domains (say
first-orderarithmetic),whileadvocatingpluralismwithregardtoothers(say
thosewherethenotionofthefullpowersetentersthepicture). Fortunately, the
mathematical systems thatarise in practice can be arrangedin a
wellfoundedhierarchy—rangingfromweaksystemslikeRobinson’sarithmeticQ,
upthroughPeanoarithmeticPA,thesubsystemsofsecond-orderarithmetic, the
subsystems of set theory, and the hierarchy of large cardinal axioms. One
can thus arrange the hybrid non-pluralist/pluralist positions and the
correspondingdebatesinawell-foundedhierarchy,wherealongthewayone
embraces moreandmorenon-pluralism. The challenge
forthosewithnonpluralist tendencies is to make the case that non-pluralism
extends further andfurtherintotheupperreachesofhighermathematics.
Thechallengefor those with pluralist tendencies is to draw the line in a
principled manner andmake thecasethatnoargumentfornon-pluralismwill
succeed forthe domainsofmathematicsbeyondthatline.
1It is sensible to approach the search for new axioms and the question of
pluralism in a stepwise fashion, seeking first axioms that resolve certain
low-levelquestionsandmakingthecasefornon-pluralismatthatlevel,and
thenproceedingupwardtoquestionsofgreatercomplexityandthequestion
ofpluralismathigherlevels. Thepresententryfocusesonthisdebateasit
takesplaceinthesettingofclassicaldescriptivesettheory. Therearethree
reasonsforthischoice.
First,apopularviewembracesnon-pluralismforfirstorderarithmeticbutrejectsthes
earchfornewaxiomsforfullsecond-order
arithmeticandsettheoryandembracespluralismattheselevels.Theques1Thisviewthatonlynaturalnumbers(andthingsreducibletothem)haverealmathematicalexistence
hasalongtradition,goingbackatleasttotheGermanmathematician Martin Ohm (1792–2872).
Other notable figures in this tradition include Leopold
Kronecker(1823–1891),HermannWeyl(1885–1955),and,morerecently,SolomonFeferman.
2
tions ofdescriptive set theory are importantin this connection since many
ofthemarestatementsofsecond-orderarithmeticwhichareindependentof
thestandardaxioms, ZFC. Itthus represents acriticaljuncture inthe
debatebetweenthepluralistandnon-pluralist. Second,herethemathematical
landscape with regard to the implications ofnew axioms has largely stabilized
and we now have the resources required to address the issue between the
pluralist and the non-pluralist. (In the case of questions of third- and
higher-order arithmetic the matter still awaits further mathematical
developments. Thattopicistreatedintheentry“TheContinuumHypothesis”.)
Finally, the case for new axioms at this level involves so-called “extrinsic
justifications”(somethingwe shalldescribe below)and,moreover, provides
thestrongestcurrentexampleofsuchjustifications.
Hereisanoverviewofthepresententry: Section1providesfurtherphilosophical
motivation and sharpens the question of pluralism by briefly
discussingtheinterpretability hierarchy, theincompleteness phenomenon, and
the nature of axioms. Section 2 describes the central notions and results of
classical descriptive set theory along with the independence results
concerningthequestions thatremainedopenandwereultimatelyshowntobe
unresolvableonthebasisofthestandardaxioms,ZFC. Section3describes the
two main approaches to new axioms—axioms of definable determinacy
andlargecardinalaxioms—anddiscussestheimplicationsoftheseaxiomsfor
theundecided questions ofclassical descriptive settheory. Section 4treats of
the intimate relationship between the two approaches and outlines the
casethathasbeenmadeforaxiomsofdefinabledeterminacyandlargecardinal
axioms. Section 5 revisits the debate between the pluralist and the
non-pluralistinlightoftheforegoingmathematicaldevelopments.
1 PhilosophicalMotivation
Oneway tosharpen the question ofpluralism involves appeal
totheinterpretabilityhierarchy.
Inthissectionwewill(1)introducetheinterpretability hierarchy, (2) briefly discuss
the incompleteness phenomenon, (3) sharpen
thequestionofpluralism,and(4)bringoutthephilosophicalissuesinvolved
bybrieflydiscussing thenatureofaxiomsandthenatureofjustificationin
mathematics.
SeeFerreir´os(2007),pp.11–13forfurtherdiscussionontheearlyhistory.
3
1.1 Interpretability Hierarchy
SupposeT andT2 arerecursivelyenumerableaxiomsystems. WesaythatT 1
1
(
T
isint
erpr
etabl
einT
1
T2
tothelanguageofT2
1
2
1
1
1
T2
1
andweshallwriteT1
<T2
=T2 whenbothT1
2
andT2
1
01
1
)when,roughlysp
eaking,thereisatranslation
tfromthelanguageofTsuchthat,foreachsentenc
eϕ ofthelanguageofT,ifT1ϕthenT2t(ϕ).
WeshallwriteTwhen TandT2T T T.
Inthelattercase,T1andT2aresaidtobemutually
interpretable. The equivalence class of all
theories mutually interpretable with T is
called the interpretability degree of T. The
interpretability hierarchy is the collection
ofalltheoriesorderedundertherelation .
Thereisausefulcharacterizationoftherelatio
n thatapplieswhenitis
restrictedtothetypeoftheoriesthatwewillbecon
sidering. Todescribethis
weneedafewmorenotions. AtheoryT
isreflexiveprovideditproves the
consistencyofeachofitsfinitefragments,itisS0
1-completeprovideditproves eachtrueS0
1-statement,anditisS0 1-statement.
Let‘T-sound provideditdoesnotproveafalse
S1⊆Π0 1T2’beshorthand forthestatement
thatallΠsentences
provableinT1areprovableinT2. The following
characterization
ofinterpretabilityiscentraltothetheoryofinterpre
tability: IfTisreflexive andS0 1-complete,then
T1 T2iff T1⊆Π0 1T22. This result is due to Orey
and H´ajek.1It follows from this result and the
secondincompletenesstheoremthatforanythe
oryTmeetingthesetwoconditions, the theory
T+Con(T) is strictly stronger than T, that is, T
< T+Con(T).
Moreover,itfollowsfromthearithmetizedcomple
tenesstheoremthattheT+¬Con(T)isinterpretab
leinT,hence,T=T+¬Con(T).
It turns out that the interpretability order is
01
3
exceedingly complex. For
example,throughtheusesofmetamathematical
codingtechniques,onecan
showthatforanytwotheoriesT1andT2suchthatT
thereisathird theoryT suchthatT1<T<T21<T2.
AndforanytheoryT,onecanshowthat
therearetheories T1andT2such thatT1>T
andT>T andyet neither
T T2norT2 T1(thatis,T1andT22areaboveTandar
eincomparablein theinterpretabilityorder).
Thus,theorderonthedegreesofinterpretability
isneitherwell-foundednorlinearlyordered.
2ForfurtherdetailsseeTheorem6onp.103ofLindstr¨om(2
003)andFact3.2ofVisser (1998).
3
S
e
e
F
e
f
e
r
m
a
n
(
1
9
6
0
)
.
4
Nevertheless, whenonerestrictstotheoriesthat“ariseinnature”(that is, the
kind of formal systems that one encounters in a mathematical text as
opposed to examples of the type above, which are metamathematically
manufactured by logicians to have the deviant properties in question) the
interpretabilityorderingisquitesimple: Therearenodescendingchainsand there
are no incomparable elements. In other words, it is a well-ordering. This is
quite remarkable: If one takes any two “natural theories”—even
fromcompletelydifferentdomains,sayonefromanalysisandanotherfrom
combinatorics—then generally it turns out that one is interpretable in the
otherandthatthetwolineupalongawell-foundedpaththroughtheinterpretabilityhie
rarchy.
Thiswellfoundedpaththroughtheinterpretabilityhierarchyhasacanonicalseq
uencethroughitwhichenablesonetoclimbthehierarchyinaprincipledmanner.
AtthebaseonehastheveryweaktheoryQandoneclimbs
upwardbyadding“closureprinciples”. Thisstartswithprinciplesasserting that
certain fast-growing functions—exponentiation, super-exponentiation,
etc.—aretotalanditproceedsupwardthroughsetexistenceprinciples,passingthro
ughthelayersofsecond-orderarithmetic,thesubsystemsofsettheory,andthehier
archyoflargecardinalaxioms.
Ofparticularinterestwithregardtothequestionofpluralismarethose
pointsalongthewell-foundedsequencethatcorrespondtocertainlimitative
conceptionsofthenatureofmathematics, suchasthedegreemarkedbyQ
(whichcorrespondstostrictfinitisminthesensearticulatedbyNelson),the
degreemarkedbyPRA(whichcorrespondstofinitisminthesensearticulated
byTait),andthedegreemarkedbyATR0(whichcorrespondstopredicativism
inthesensearticulatedbyFeferman). Ineachcase,onedrawsthelineata
certainpointinthehierarchy,maintainingnon-pluralismbelowandpluralism
above.
Further Reading: For a more detailed account of the interpretability hierarchy
see the entry “Independence andLargeCardinals”. Foreven further
detailseeLindstr¨om(2003)andVisser(1998).
1.2 IndependencePhenomenon
We noted above that the second incompleteness theorem provides us with a
case where the statement, namely, Con(T), leads to a jump in the
interpretability hierarchy whileits negationdoes not. There arealsonatural
5
examplesofthisphenomenon—forexample,largecardinalaxioms. Whena
statementϕisindependentofatheoryTinthismanner—thatis,invirtue
ofthefactthatT <T+ϕ—weshallrefertothekindofindependence involved as
vertical independence. Thus, thesecond incompleteness theorem
demonstrated that Con(PA) is vertically independent of PA (assuming, of
course, thatPAisconsistent) andthatthelargecardinalaxioms“Thereis
astronglyinaccessiblecardinal”isverticallyindependentofZFC(assuming
thatZFC+“Thereisastronglyinaccessiblecardinal”isconsistent).
In contrast to vertical independence there is the kind of independence
thatariseswhenasentenceϕisindependentofatheoryTbecause
T=T+ϕ and T=T+¬ϕ.
SuchasentenceϕiscalledanOreysentencewithrespecttoT. Weshallrefer
tothiskindofindependence ashorizontal independence.
Usingmetamathematicaltechniques onecanconstructsuch sentences.
Butagain,asinthe
caseofverticalindependence,therearealsonaturalnon-metamathematical
examples—forexample,PU(thestatementthattheprojectivesetshavethe
uniformization property, a property we shall discuss below) is a statement
of(schematic)second-orderarithmeticthatisanOreysentencewithrespect to
ZFC, and CH is a statement of third-order arithmetic that is an Orey
sentencewithrespecttoZFC.
1.3 TheQuestion of Pluralism
With these pieces in place we can now begin to sharpen the question of
pluralism.
As a first step we will restrict our attention to theories which are
formulatedinacommonlanguageandourlanguageofchoice willbethelanguage
of set theory. The reason for restricting to a common language is
thatotherwisetherewillbetheorieswhicharemerelynotationalvariantsof one
another. And the reason for focusing on the language of set theory is
thatitisauniversallanguageinthesensethatanysystemofmathematics is
mutually interpretable with a theory formulated in the language of set theory;
for example, PA is mutually interpretable with ZFC-Infinity. We arethus
setting aside the question ofpluralism as itmight arise in amore
metaphysicalsettingwhereoneconsidersquestionssuchaswhethernatural
numbersare“really”setsorwhetherinsteadtherearetwodistinctdomains
6
ofobjects—naturalnumbersandthefiniteordinalsofsettheory. Ourfocus isonthe
question ofpluralism asitarises inresponse tothe independence phenomenon
and the issues at play in this discussion are invariant across different
stances on the above metaphysical question. With regard to our
questionthereisnolossofgeneralityintherestrictiontothelanguageofset
theorysince theproblem offinding new axiomsforPAtranslates (through
thenaturalinterpretation ofnumber theoryintoset theory)into theproblem
offinding new axioms forZFC-Infinity, andconversely. Nevertheless,
inwhatfollowsweshall,largelyoutofconvenienceandfamiliarity,continue
tospeakoftheoriesthatareformulatedinotherlanguages—suchasQand
PA—butthereadershouldalways understand ustobestrictly speakingof
thecorrespondingtheoryinthelanguageofsettheory.
Thequestionofpluralisminitsmostgeneralformulationisthequestion
ofwhether there is acorrect path through the hierarchy ofinterpretability or
whether instead a multitude of mutually incompatible theories must be
admitted as equally legitimate from the pointof view oftheoretical reason and
mathematical truth. There are really two problems that must be addressed.
Thefirstisthequestionofhowfarthehierarchyofinterpretability extends, that is,
which degrees are non-trivial in that the theories within them are consistent.
This is the problem of consistency (or non-triviality).
Thesecondisthequestionofwhether,foragiven(non-trivial)degreeofinterpretabili
ty,therearetheoreticalreasonsthatcanbegivenforonetheory overanother.
Thisistheproblemofselection. Ifonemanagedtosolveboth
problems—determining the extent of the non-trivial degrees and locating
thecorrectpaththroughthem—thenonewouldhavecompletelysolvedthe
problem ofmathematical truth. There are, however, two important things
tostressaboutthis. First,becauseoftheincompletenesstheorems,anysuch
solutionwouldhavetobe“schematic”concerningthefirstpart(verticalindependen
ce). Second, thesolution tothe fullproblem ofselection even for
thedegreeofZFCisavery difficultonesinceitinvolvesresolvingCHanda
multitudeofotherindependentstatementsthatarisealreadyatthislevel.
Becauseofthedifficulty ofthesecondproblemitisusefultosplititup
intoitsownhierarchybyclassifyingstatementsintermsoftheircomplexity.
Weshalldiscussthehierarchyofstatementcomplexityinmoredetailinthe
nextsection. Butforthemomentsufficeittosaythat(roughlyspeaking)one
startswithquestionsofsecond-orderarithmetic(stratifiedaccordingtotheir
complexity) and then one moves on to questions of third-order arithmetic
(stratifiedaccordingtotheircomplexity),andsoon. Thus,onefirsttriesto
7
4solvetheproblemofselectionatthelevelof,say,statementsofsecond-order
arithmeticandthenonemovesontotheproblemofselectionforstatements
ofthird-orderarithmetic,andsoon. Thequestionofpluralismthatwewill
beaddressingislocatedinthehierarchyasfollows: Ittakesanon-skeptical
stancetowardZFCandseeksaxiomsthatsolvetheproblemofselectionfor
thelevelofsecond-orderarithmeticandsomewhatbeyond.
Letusnowgivesomeexamplesofthevariouspluralist/non-pluralistpositions.
At one extreme there is radical pluralism which maintains that all non-trivial
theories in the interpretability hierarchy are on a par and that there are no
theoretical reasons that one can give for any theory over another. This form
of pluralism is difficult to articulate since, for example, assuming that Q is
consistent the theory Q+¬Con(Q) is consistent. But it is hard to see how the
pluralist can admit this theory as a legitimate candidate since it contains a
principle—namely, ¬Con(Q)—that is in conflictwiththebackgroundassumption
thatthepluralistemploys inarguing that the theory is a legitimate candidate.
For this reason most people are
non-pluralistsaboutsomedomainsbutremainpluralistsaboutothers.
5Thelimitativeviewsthatwementionedearlier—strictfinitism,finitism,
andpredicativism—are examples ofsuch hybrid views. Forexample, strict
finitists(ononeconstrual)acceptQandcertaintheoriesthataremutually
interpretable with Q, thereby adopting non-pluralism for this domain. In
addition, the strict finitist takes these theories to exhaust the domain of
objectivemathematicsandadoptsapluraliststancetowardtheremainder.
Inasimilarfashion,finitismandpredicativismprovideuswithexamplesof
hybridnon-pluralist/pluralistpositions.
Predicativism is of particular interest for our present purposes since it
grantstheframeworksofthenaturalnumbersbutrejectsmostofsettheory and, in
fact, most of second-order arithmetic. The predicativist is a non4Weshouldnotethattherearesomeviewsthatacceptthestandardhierarchyoflarge
cardinalaxioms—andso climb the canonicalpaththroughthe well-foundedstem ofthe
interpretabilityhierarchy“alltheway”—andyetarepluralistaboutstatementsofthirdorderarithmetic
,inparticularCH. Soonceonehassolvedtheproblemofconsistency—
eveninthestrongsensewhereoneactuallyadoptsthetheoriesinthecanonicalsequence—
thereisstillplentyofroomforpluralisminapproachingtheproblemofselection.
5It should be noted that some strict finitists—like Nelson himself—think that exponentiationis
nottotalandsothink thatmostofthe degreesabovethatofQ aretrivial (inconsistent).
Suchastrictfinitistwillnotbeapluralistabout,say,PRAandthetheories
extendingitforthesimplereasonthatsuchastrictfinitistwillthinkthatthesetheories areinconsistent.
8
pluralist about first-order arithmetic and thinks that there is an objective fact
of the matter concerning every statement formulated in the language of
first-order arithmetic. And the predicativist accepts as much of secondorder
arithmetic and set theory that can be simulated in a theory such as ATR 0.
Butthepredicativistremainsapluralistaboutwhatliesabove. The
predicativistwilloftengrantthatthereisutilityinemployingaxiomaticsystems of set
theory but will maintain that the issue of truth does not arise
atthatlevelsincethereisnounivocalunderlyingconceptionofset.
Forexample,thepredicativistwillthinkthatthereisanobjectivechoicebetween
PAandPA+¬Con(PA)butnotthinkthatthereisanobjectivechoicebetweenPA 2+P
UandPA2+¬PU(whereherePAis(schematic)second-order
arithmetic)orbetweenZFCandZFC+“Therearenostronglyinaccessible
cardinals”. Insofarasthepredicativistthinksthattheindependenceresults
insettheoryconcerningsecond-orderarithmeticareareasonforendorsing
pluralism about bothZFC andPA22the non-pluralist case we shall present
belowisdirectlyengagedwiththepredicativist. Butweshouldstressthatfor
thisdiscussionweshallbepresupposingZFCandsothemostdirectbrand
ofnon-pluralism/pluralism thatwe shallbeengagedwith isonethattakes
anon-pluraliststancetowardZFCbutapluraliststancetowardstatements
likePUwhichareformallyunresolvableonthebasisofZFC. Thisisafairly
commonview. See,forexample,Shelah(2003).
Further Reading: For more on strict finitism see Nelson (1986); for more
finitismseeTait(1981);formoreonpredicativismseeFeferman(1964)and
Feferman(2005).
1.4 TheNatureof Axioms
Theabovecharacterizationofthequestion ofpluralismmakes itclearthat it is
closely connected with issues concerning the nature of justification in
mathematics.
Themosttraditionalformofjustificationinmathematicsinvolvesjustifyingastateme
ntbyprovingitfromtheaxioms. Thisthenraises thequestion:
Whatisthesourceofthejustificationoftheaxioms?
The radical pluralist maintains that there is nothing that justifies the
axioms;rathertheyaresimplypostulatesthatoneadoptshypotheticallyfor a given
purpose. This view faces the difficulty raised above. Most would maintain
thataxioms have aprivileged status and thatthis status is more
thanmerelyhypothetical.
9
1.4.1 The Evidentiary Order A popular view is that axioms are self-evident
truths concerning their do6main. The difficulty with this view is that the notion of self-evidence is a
subjectivenotion. Thereisconsequentlymuchunresolvabledisagreementas
towhatistocountasself-evidentandifwerestrictourselvestotheintersectionofthest
atementsthatall mathematicianswouldregardasself-evident
thentheresultwillbequitelimitedinreach,perhapscoincidingwithQor
slightlymore.Markovhadasimilarcomplaintwiththeemploymentofthe
relatednotionofbeing“intuitivelyclear”:
Icanonnoway agreetotaking‘intuitively clear’asacriterion
oftruthinmathematics,forthiscriterionwouldmeanthecompletetriump
hofsubjectivismandwouldleadtoabreakwiththe understanding of
science as a form of social activity. (Markov (1962))
Forthesereasonswewillsetasidethenotionofself-evidence. But there is a
related notion that we shall employ, namely, the notion
ofonestatement beingmore evident thananother. Incontrast tothe case
ofself-evidence, thereisgenerallymuchmoreagreementonthequestionof
whatismoreevidentthanwhat.
7Toillustratethisitisbesttogivesomeexamples: (1)Overthebasetheory EFA, the
Finite Ramsey Theorem is equivalent to the totality of
superexponentiation.See Friedman (2011), Section 0.5. (2) over the base
theory RCA0,theHilbertBasisTheoremisequivalenttothestatementthatωωis
well-ordered. See Friedman(2011),Section 0.6D.(3)Over the base theory
ACA,Goodstein’sTheoremandtheHydraTheoremareeachequivalentto theS 0
1-soundness ofPA. SeeFriedman(2011),Sections 0.8Band0.8E.(4) Over the
base theory RCA, Kruskal’s Tree Theorem is equivalent to the
statementthatθΩw0iswell-ordered. SeeFriedman(2011),Section0.9B.(5)
OverthebasetheoryACA,theExoticCase(ofBooleanRelationTheory)
isequivalenttotheS0 1-soundnessofSMAH. SeeFriedman(2011),Sections
0.14C.
6Nelsonisonemathematicianwhodoesnotacceptthetotalityofexponentiationand
thisaloneservestoruleoutmostsystemsbeyondthedegreeofQ.
7This theoryis obtainedfromQbyaddingS0-inductionandthe statementasserting
thatexponentiationistotal.
10
first and
there is general agreement on this fact. For example, the Hilbert Basis
Theorem is far from immediate. It is the sort of thing that we set
outtoprovefromthingsthataremoreevident. Incontrast, thestatement
thatωiswell-foundedissomethingthatbecomesclearbyreflectingonthe
conceptsinvolved. Itisnotthesortofthingthatwesetouttoprovefrom
somethingmoreevident. ThisiswhatliesbehindthefactthattheHilbert Basis
Theorem is called a theorem and not an axiom. As one climbs up through the
other examples, the asymmetry becomes even more dramatic.
Forexample,theHydraTheoremandKruskal’sTheoremareactuallyquite
surprisingandcounter-intuitive. Thesearedefinitelythesortofstatements
thatrequireproofonthebasisofstatementsthataremoreevident.
We shall organize the statements in each interpretability degree by the
relation ofbeing more evident than and we shall speak of the collection of
allstatementssoorderedastheevidentiaryorder.
Afewwordsofcautionarenecessaryconcerningthisordering. First,in saying
that X is more evident than Y we do not intend that X (or Y) is self-evident
orevenevident. Tounderscore thispointwewilloftenusethe
expressions“XispriortoY intheevidentiaryorder”and“Xisevidentially prior toY”.
Second,eveninthecasewhereXisminimalintheevidentiary
order(withinagiveninterpretabilitydegree)werefrainfromsayingthatX
isself-evident; indeed ifthedegree issufficiently highthen thiswill notbe the
case. Third, the evidentiary order is to be understood as an epistemic
orderandnot,say,theorderofconceptualcomplexity.
One might raise skeptical worries about the evidentiary order. For
example, perhaps judgments of the form “Xis prior to Y in the evidentiary
order”arereallyjustreflectionsofourmathematicaltraining. Thebestway
toallaysuchconcernsandtesttherobustnessofthenotionistoexamineexamples.
Compare,forexample,thestatementoftheHydraTheoremandthe
statementthat0iswell-orderedwithrespecttoprimitiverecursiverelations (a weak
(semi-constructive) form of the statement that 0is well-ordered).
Ifyoudescribetherelevantnotionstoamathematicianorlaypersonwhois not
familiar with them then you will find the following: There will be no
expectationthatHerculeswinstheHydra. Infact,ifyoudirecttheperson
tooneofthecomputerprograms(availableonline)thatsimulatestheHydra
youwillfindthattheywillbeastounded athowfasttheHydragrowsand
ifyouassurethemthatiftheykeptonplaying—foraperiodofmoreyears
thanthereareatomsinthephysicaluniverse(ascurrentlyestimated)—the
11
ωIneachoftheabovecases,thesecondstatementismoreevidentthanthe
tideswilleventuallyturn,theywillmakegainsandeventuallycuttheHydra down to
size, you will find that they will look at you in amazement. This is something
that needs proof. Now if you describe to them the ordinals
allthewayuptoandyoudescribethenotionofawell-orderingandask them whether
00is well-founded you will find that the answer rests solely onwhether they
have graspedtheobjectinquestion. Tocometotheconclusion that 0is
well-founded one does not search for a proof from some
otherstatementwhichismoreevident. Ratheroneworksdirectlywiththe
conceptsinvolved. Thestatementis,asitwere,restsonthelegitimacyofthe
underlyingconceptsandonecomestoacceptitbygraspingthoseconcepts.
Itisherethatthebuckstopsintheprocessofseekingjustification. Andthis
iswhatwemeaninsayingthatthestatement“0iswell-ordered”isminimal
intheevidentiaryorder(ofthedegreeoftheHydraTheorem).
8Inseekingaxiomsfortheorieswithinagiveninterpretabilitydegreewe
seekaxiomsthatareaslowaspossibleintheevidentiaryorder. Ultimately one
would like to find axioms that are minimal in the evidentiary order. But
questions remain: (1) Can we be assured of finding minimal points that are
secure? The concern is that as one climbs up the interpretability degrees the
security of the minimal points will weaken to the point where
therewouldarguablybelittlesecurityleft. Toseethisconsiderstatements
oftheform“aiswell-ordered”whereaistheproof-theoreticordinalofthe
interpretabilitydegree. Asoneclimbsthedegreesonepassesthrough 0,G,
andθΩw0. Thisbarelyscratchesthesurfaceandyetalreadythecorresponding
statementshavebecomelessevident. Thisleadstothesecondquestion. (2)
Isfindingaminimalpointintheevidentiaryorder(ofagiveninterpretability
degree)themostonecouldhopefororarethereother,moresubtlewaysof
accumulatingevidenceforaxioms,waysthatmightfurtherbuttresseventhe
minimalpointsintheevidentiaryorder(ofagiveninterpretabilitydegree)?
1.4.2 Intrinsicversus ExtrinsicJustifications There isadiscussion
ofG¨odelthathasbearingoneachofthese questions,
namely, his discussion of intrinsic and extrinsic justifications in his classic
paperG¨odel(1947)(expandedasG¨odel(1964)).
In introducing the notionof an intrinsic justification G¨odel gives as an
examplecertainlargecardinalaxiomsthatarejustbeyondthereachofZFC:
8Notethatweareindividuatingourtheoriesbytheiraxioms.
12
[The] axioms of set theory [ZFC] by no means form a system
closed in itself, but, quite on the contrary, the very concept of
setonwhichtheyarebasedsuggeststheirextensionbynewaxioms
which assert the existence of still further iterations of the
operation“setof”(260).
Asexamples hementions theaxioms assertingtheexistence ofinaccessible
cardinalsandMahlocardinals:
These axioms show clearly, not only that the axiomatic system
ofsettheoryasusedtodayisincomplete,butalsothatitcanbe
supplemented without arbitrariness by new axioms which only
unfoldthecontentoftheconceptofsetasexplainedabove. (260–
261,myemphasis)
Since G¨odel later refers to such axioms as having “intrinsic necessity” we
shall accordingly speak of such axioms being intrinsically justified on the
basisoftheiterativeconceptofset.
Thenotionofextrinsicjustificationisintroducedasfollows: [E]ven
disregarding the intrinsic necessity of some new axiom,
and even in case it has no intrinsic necessity at all, a probable
decisionaboutitstruthispossiblealsoinanotherway, namely,
inductivelybystudyingits“success”. (261)
Here by “success” G¨odel means “fruitfulness in consequences, inparticular
“verifiable”consequences”. Inafamouspassagehewrites:
There might exist axioms so abundant in their verifiable
consequences,sheddingsomuchlightuponawholefield,andyielding
such powerful methods for solving problems (and even solving
them constructively, as far as that is possible) that, no matter
whetherornottheyareintrinsically necessary, theywouldhave
tobeacceptedatleastinthesamesenseasanywell-established
physicaltheory. (261)
Let us now consider some examples. Consider first the conception of
natural number that underlies the theory of Peano Arithmetic (PA). This
conception ofnaturalnumber notonlyjustifies mathematical inductionfor
thelanguageofPAbutalsoforanymathematicalextensionofthelanguage
13
9of
PA that is meaningful.10For example, if we extend the language of PA by
adding the Tarski truth predicate and we extend the axioms of PA by
addingtheTarskitruthaxioms,then,onthebasisoftheconceptionofnaturalnumber,
wearejustifiedinacceptinginstancesofmathematicalinduction involving the
truth predicate. In the resulting theory, one can prove Con(PA).
Thisprocesscanthenbeiterated.Incontrast,theΠ0 1-statement
Con(ZFC+“Thereareω-manyWoodincardinals”) (a statement to which
weshallreturnbelow)isundoubtedlynot intrinsicallyjustifiedonthebasis of the
conception of natural number; rather its justification flows from an
intricatenetworkoftheoremsincontemporarysettheory,asweshallargue below.
Considernexttheiterativeconceptionofset. Asinthecaseofarithmetic,
thisconception(arguably)intrinsicallyjustifiesinstancesofReplacementand
Comprehension for certain extensions of the language of set theory. But
there are richer principles that are (arguably) intrinsically justified on the
basisofthisconception,namely,theset-theoreticreflectionprinciples. These
principlesassert(roughly)thatanypropertythatholdsofV holdsofsome
initialsegment Va. These principles yield inaccessible andMahlo cardinals
(and more) and quite likely underlie G¨odel’s claims in the above passage.
Somehavearguedthatsuchreflectionprinciplesareintrinsicallyjustifiedon
thebasisoftheiterativeconceptionofsetand,moreover,thattheyexhaust
theprinciplesthatcanbeintrinsicallyjustifiedonthatconception. See,for
example,Tait(2001),reprintedinTait(2005b),pp.283–284.
Thereisaviewaccordingtowhichintrinsicallyjustifiedaxiomsareminimalpoints
intheevidentiaryorderand,moreover,thatthatisthemostthat onecanhopefor.
Onsuchaview,ifonegrantsthattheaboveprinciplesof
settheoryareintrinsicallyjustifiedonthebasisoftheiterativeconceptionof
set,thenalthoughonewouldhaveaxiomsthatfarsurpassedthereachesof
predicativismandevenZFC,limitationswouldstillremainandpresumably
aproponentofsuchaviewwouldembracepluralismfortheremainder. For
example,if,asTait(2001)argues,intrinsicjustificationsareconfinedtothe
9NoticethatinspeakingofX
beingintrinsicallyjustifiedonthebasisofY,X stands
forastatementandY standsforaconception. Therelationisthusquitedifferentthanthe
usualrelationofjustifiedonthebasisof whichholdsbetweenstatementsandstatements. The
presentnotionbears comparisonwith the evidentiaryordersince in the casewhere onehas
reacheda minimalpoint—suchasthe case of“0is well-ordered”one looksnot
tootherstatementsforepistemicsupportbutrathertotheunderlyingconception.
10FormoreonthissubjectseeFeferman(1991)andthereferencestherein.
14
hierarchy of reflection principles investigated in Tait (2005a), then by the
results of Koellner (2009a), such a view would have to embrace pluralism
withregardtoPU.
11Extrinsicjustificationsprovideuswiththehopeofsecuringaxiomsthat have
much greater reach. For they provide us with an additional way to secure
axioms, onethatproves tobevery rich. Onthisview onecanhope
formuchmorethanaminimalpointintheevidentiaryorder—onecanhope to garner
evidence for a proposition examining its inter-relationships with
otherstatementsthatarelowintheevidentiaryordering. Letusdescribein
verybroadbrushstrokeshowthismightwork. Tobeginwith,letussaythat
astatementis“intrinsicallyplausible”ifitis“low”intheevidentiaryorder
butthesortofthingthatcouldbeoverturnedbyotherevidence.Oneway to garner
support for a proposition is to show that it leads to intrinsically
plausibleconsequences. Ifonethenshowsthatitistheonly statementthat
hastheseintrinsicallyplausibleconsequencesthenthatstrengthensthecase by
ruling out possible incompatible competitors. If one shows that there
areotherintrinsicallyplausiblestatementsthatactuallyimplytheaxiomin
questionthenthatagainstrengthensthecasebymappingoverthedegreeof
intrinsicplausibilityattachedtoonestatementovertoanother.
Onthispicturethegeneralstructureofthecasefornewaxiomsisthis:
First,foreachinterpretabilitydegreeoneisolatestheminimalpointsinthe
evidentiaryorderandoneorganizestheentiredegreeunderthedegreeofone
statementbeingmoreevidentthananother. Asoneclimbstheinterpretability
degrees the minimal points generally become less evident and so one seeks
additional support. Second, one then examines the interconnections among
the statements in the interpretability degree, in particular, among the
statements which are low in the evidentiary order. The hope is that a
seriesofdeepmathematicaltheoremswillrevealstructuralconnectionsthat
providesmoreandmoreevidential supportforthestatements. Shouldthis
bethecasethenthelossinevidenceattachedtotheminimalpointsasone climbs
theinterpretability hierarchy wouldbecompensatedforbythegain
instructuralconnectionsthatarerevealedasoneascendstoricherdomains
ofmathematics.
All of this is rather abstract at this stage. It is now time to turn to
11We
intend to use this term in the same sense as Parsons. See Parsons (2000) and
chapter9ofParsons(2008)foramoredetaileddiscussionofthenotionandsomeofthe
themeswetouchuponhere.
15
a concrete mathematical case where such structural connections have been
found. Thequestion,ofcourse,willbewhetherthemathematicsmakesfor
aconvincingextrinsiccasefornewaxioms. Andthequestionofpluralismat
thislevel—wherewearepresupposingZFCandtryingtosolvetheproblem
ofselectionforsecond-orderarithmetic—willrestonthenatureofthecase.
FurtherReading:
ForfurtherdiscussionofthenatureofaxiomsandjustificationinmathematicsseePa
rsons(2008)andMaddy(2011).
2 ClassicalDescriptiveSetTheory
Descriptive set theory is the study ofthe structural properties ofdefinable sets
of real numbers. In this section we shall (1) provide a brief historical
overview,(2)introducethebasicnotionsofdescriptivesettheory(explaining
whatwemeanby“definablesets”and“structuralproperties”),(3)describe the
central classical results, and (4) discuss the limitative results that ultimately
explained why the classical descriptive set theorists had run into
difficultieswhenattemptingtogeneralizetheirresults.
2.1 Historical Overview
Onereasonforconcentratingondefinablesetsofrealsisthatitisgenerally
easiertogaininsightintodefinablesetsofrealsthanarbitrarysetsofreals and,
furthermore, there is hope thatthis insight will shed light on the latter. For
example, in one of the earliest results in the subject, Cantor and
Bendixsonshowedthateveryclosedsetiseithercountableorhasthesizeof
thecontinuum;inotherwords,theclosedsetssatisfyCH.
Themainoriginsofthesubjectlieintheearly20thcenturyinthework of the French
analysts Borel, Baire, and Lebesgue. The subject grew into an independent
discipline through the work of the Russian mathematician Luzin (who had
learned of the subject as a student while in Paris) and was developed by his
students, most notably Suslin. The end of this early period of descriptive set
theory was marked by a major theorem proved by the Japanese
mathematician Kondˆo in 1937. After nearly 40 years of
development,progresscametoanabrupthalt. Theexplanationforthiswas
eventuallyprovidedbytheindependenceresultsofG¨odelandCohen(in1938
and1963). For,usingthismachinery,itwasshownthattheopenquestions
16
that the early descriptive set theorists were trying to answer could not be
resolved on the basis ofthe standard axioms ofmathematics, ZFC. It is a
testamenttotheingenuityoftheearlydescriptivesettheoriststhattheyhad
developedtheirsubjectrightuptotheedgeofundecidability.
The independence results of G¨odel and Cohen thus showed that if one
wished to develop descriptive set theory further then one must invoke
additional axioms. There were two main approaches, each of which led to a
rebirth of descriptive set theory. The first approach was to invoke axioms of
definable determinacy. The axiom of determinacy (AD) was introduced by
Mycielski and Steinhaus in1962anditwas quickly seen toyield generalizations
of the results of classical descriptive theory. Unfortunately, AD contradicts AC
and for this reason it was never really proposed as a new axiom. However,
Solovay and Takeuti pointed out that there is a natural suchsubuniverse,
namely,L(R),inwhichADcouldhold,consistently with assuming AC in the full
universe. This led toa hierarchy ofaxioms of
definabledeterminacythatunderwentintensiveinvestigationinthe1970sand
1980s.
ThemainsourcesforthisperiodareMoschovakis(1980)andthevolumesoftheCab
alseminarduring1977–1985. Thesecondapproachwasto
invokelargecardinalaxioms. ThisapproachwasinitiatedbySolovayin1965.
Finally,inthelate1960sSolovayconjecturedthatlargecardinalaxiomsactually
imply axioms of definable determinacy. If true, this would open the way
foraunification ofthe two approaches. This hope was realized inthe
workofMartin,Steel,andWoodinfrom1984onward.
FurtherReading: Formoreonthehistoryofclassicaldescriptivesettheory
seeKanamori(1995).
2.2 Basic Notions
2.2.1 Hierarchies of Definability
Indescriptivesettheoryitisconvenienttoworkwiththe“logician’sreals”,
that is, Baire space: ωω12. This is the set of all infinite sequences of
naturalnumbers (with the product topology(taking ωas discrete)).In what
12ωThis
structure is homeomorphic to the irrationals (as standardly construed). The
reasonforworkingwithωinsteadofRisthatωωishomeomorphictoitsfiniteproducts, (ω)nωω, in
contrast to the situation with R. This enables one to concentrate on proving resultsforω,
andsimply notethat theylift (throughthe naturalhomeomorphisms)to thefiniteproducts.
17
follows,whenwewrite‘R’weshallalwaysmean‘ωω’. Weareinterested insets
ofrealsthatare“definable”or“builtupfrom
below by simple operations.” We shall discuss three central regions ofthe
hierarchyofdefinability—theBorel setsofreals,theprojective setsofreals,
andthesetsofrealsinL(R).
ωTheBorelsetsofreals(orsetsofk-tuplesofreals)areobtainedbystarting withthe
closed subsets ofω(or(ωω)kforsome k<ω)andclosing under the operations of
countable union and countable intersection. This can be
describedlevelbylevelasfollows: Letk<ω. Atthebaselevel,letS~ 0ωconsist
oftheopensubsetsof(ω)kandletΠ~ 011consistoftheclosedsubsetsof(ωω)k.
Foreachordinalasuchthat0<a<ω1,recursivelydefineS~ 0atoconsistof
thesetsthatarecountableunionsofsetsappearinginsomeΠ~ 0,forß<a, anddefine
Π~ 0aßtoconsist ofthesets thatarecountable intersections ofsets appearingin
some S~ 0ß13, forß<a. The Borel sets of(k-tuples of)reals are
thesetsappearinginthishierarchy.In1905Lebesgueemployed Cantor’s
notionofauniversal setandtechnique ofdiagonalizationtoshow thatthe
hierarchyofBorelsetsisaproper hierarchy(thatis,newsetsappearateach level).
ωThe projective sets of reals (or sets of k-tuples of reals) are obtained
bystartingwiththeclosedsubsets of(ω)kωanditeratingtheoperationsof
complementation and projection. For A⊆ (ω)kω, the complement of Ais
simply(ω)k-A. ForA⊆(ωω)k+1,theprojection ofAis
p[A]={x1,...,xk∈(ωω)k|∃yx1,...,xk,y∈A}. (Thinkofthecasewherek+1=3.
HereAisasubsetofthree-dimensional
spaceandp[A]istheresultof“projecting”Aalongthethirdaxisontothe plane
spanned by the first two axes.) We can now define the hierarchy of
projectivesetsasfollows: Atthebaselevel,letS~ 1ωconsistoftheopensubsets
of(ω)kandletΠ~ 100consist oftheclosed subsets of(ωω)k. Foreach nsuch that 0 <
n < ω, recursively define Π~ 1to consist of the complements of setsinS~
1n,anddefineS~ 1n+1ntoconsistoftheprojectionsofsetsinΠ~ 1n. The
projectivesetsof(k-tuplesof)realsarethesetsappearinginthishierarchy.
Thishierarchyisalsoaproperhierarchy,ascanbeseenusinguniversalsets
anddiagonalization.
It is useful to also introduce the ∆~ 1nsets. A set of (k-tuples of) reals is ∆~ 1niff
it is both S~ 1nand Π~ 1n. In 1917, Suslin showed that the Borel sets
13It
is straightforwardto see that this collection is the least collection containing the
closedsetsandclosedundertheoperationsofcomplementationandcountableunion.
18
areprecisely the∆~ 11sets. Thus, theprojective hierarchy extends theBorel
hierarchy.
Itmightseemoddtorefertotheabovehierarchiesashierarchies ofdefinability.
However, asthe notationindicates, theabove hierarchies canbe recast interms
ofdefinability. Forexample, theprojective sets areexactly
thesetswhicharedefinable(withparameters)insecond-orderarithmetic. In
thenotationS~ 1n,thesuperscriptindicatesthatweareallowingquantification over
objects of second-order (that is, sets of integers (or, equivalently, real
numbers)), ‘S’ indicates thatthe starting quantifier is existential, the
subscriptindicatesthattherearen-alternationsofquantifiers,and‘~’indicates
thatweareallowingrealparameters. Inthecasewhererealparametersare
prohibited,thecorrespondingpointclassesaredenotedS1 n,Π1 n,and∆1 n14. The
formerhierarchy(whererealparametersareallowed)iscalledaboldface
hierarchy, while the latter (where real parameters are prohibited) is called a
lightface hierarchy. Inasimilarmanneronecandefinealightfaceversionof the
Borel hierarchy.Thus, we really have two hierarchies of definability:
thelightfacehierarchyandtheboldfacehierarchy.
Theprojectivehierarchycanbeextendedintothetransfiniteasfollows: Foraset
X,letDef(X)consist ofthesubsets ofXthataredefinable over
XusingparametersfromX. Thisisthedefinablepowerset operation. The
hierarchy L(R) is the result of starting with R and iterating the definable
powersetoperationalongtheordinals;moreprecisely,weletL0(R)=V, La+1(R) =
Def(La(R)), Lλ(R) = a<λLaω+1(R) for limit ordinals λ, and, finally,
L(R)=
La(R).
a∈On
ωWeareinterestedinthesetsof(k-tuplesof)realsappearinginthishierarchy,
that
is, P((ω)k)nLaω(R) for a ∈ On. Notice that in the definition of
theprojectivesets,complementationcorrespondstonegationandprojection
corresponds to existential quantification over real numbers. It follows that
theprojectivesetsof(k-tuplesof)realsareexactlyP((ω)kω(R). Thus, the hierarchy
of sets P((ω)k)nLa)nL1(R) (for a∈ On) does indeed yield a
transfiniteextensionoftheprojectivesets.
Onecanpassbeyondtheabovehierarchyofsetsofrealsbyconsidering
14 We
shall not discuss the details but let us note that the result is that the ∆0 1
sets
coincidewiththerecursivesetsandthelightfaceversionoftheBorelhierarchycoincides
withthehyperarithmeticalhierarchy.
19
richer notionsofdefinability. Weshall say littleaboutthese stages, except
tonotethattherearenaturallimitstothelightfaceandboldfacehierarchy, namely
that of being definable from ordinal parameters (OD) and that of
beingdefinablefromordinalparametersandrealnumbers(OD(R)). Tosee
thatthenotionofordinaldefinability
isalimitingcaseofthelightfacehierarchynoticefirstthatanynotionofdefinabilitywhi
chdoesnotrenderallof
theordinalsdefinablecanbetranscended(ascanbeseenbyconsideringthe
leastordinalwhichisnotdefinableaccordingtothenotion)andsecondthat
thenotionofordinaldefinability
cannotbesotranscended(sincebyreflectionODisordinaldefinable).
(Similarconsiderationsapplytotheboldface hierarchy.) It is for this reason that
G¨odel proposed the notion of ordinal
definabilityasacandidateforan“absolute”notionofdefinability.
Asalimitingcaseofthelightfaceandboldfacehierarchywethereforehavethesets
ofrealsthatareODandthesetsofrealsthatareOD(R),respectively.
2.2.2 Regularity Properties
Therearemanypropertiessharedby“well-behaved”setsofreals. Theseare
calledregularity properties. Weshallconcentrateonthethreemostcentral
regularity properties—the perfect set property, the property of Baire, and
Lebesguemeasurability.
A set of reals is perfect iff it is nonempty, closed and contains no
isolatedpoints. Thekeyfeatureofthisforourpurposesisthatifasetofreals
containsaperfectsubsetthenitmusthavethesamesizeasthecontinuum
(i.e.thereals). Asetofrealsissaidtohavetheperfectsetproperty iffitis either
countable orcontains aperfectsubset. The interest ofthis property is that the
sets of reals with the perfect set property satisfy CH. In early work,
CantorandBendixson established (in1883)thatallclosed sets have the perfect
set property. Thus, there is no closed counter-example to the
continuumhypothesis. Later,in1903,YoungshowedthattheΠ~ 02setshave
theperfectsetproperty. Acentralquestionwashowfarthispropertypropagates.
DoalloftheBorelsetsofrealshavetheperfectsetproperty? Doall
oftheprojectivesetsofrealshavetheperfectsetproperty?
A set of reals is nowhere dense iff its closure under limits contains no
opensets. A set ofreals is meager iffitis the countable union ofnowhere
densesets. Finally,asetofrealsAhasthepropertyofBaireiffitis“almost
open”inthesensethatthereisanopensetOsuchthattheregionwhereO
andAdonotoverlap(thatis,thesymmetricdifference(O-A)∪(A-O))
20
ismeager. ItisstraightforwardtoshowthatallBorelsetshavetheproperty ofBaire.
DoalloftheprojectivesetshavethepropertyofBaire?
ThepropertyofLebesguemeasurabilityisfamiliarfromanalysisandhere
weshallbemorebrief. Tobeginwith,oneintroducesaprobabilitymeasure
oversetsofreals. Asetofrealsisnull iffithasmeasurezero. Asetofreals
AisLebesguemeasurable iffitis“almostBorel”inthesensethatthereisa
BorelsetBsuchthattheregionwhereBandAdonotoverlapisnull. The
interestofthisnotioncanperhapsbebestbroughtoutbyrecallingatheorem
ofBanachandTarski. BanachandTarskishowed thatitispossible tocut
theunitsphereintofinitelymanypieces,andshiftthosepiecesaround(using only
translations and rotations) to bring them back together into a sphere
oftwicethesize. ThisrequiresakeyapplicationoftheAxiomofChoice. It
isimmediatethatsuchpiecescannotbeLebesguemeasurable. Thequestion is
“How complex must such pieces be?” and this amounts to the question
“Which sets are Lebesgue measurable?” It is clear thatthe Borel sets are
Lebesguemeasurable. AretheprojectivesetsLebesguemeasurable?
InallthreecasesitisstraightforwardtousetheAxiomofChoicetoshow
thattherearesetsofrealswithouttheregularityproperty. Thequestionis
whetheralldefinable setsofrealshavetheseproperties.
2.2.3 Structural Properties
Therearemanystructuralpropertiesconnectedwithsetsofreals—theuniformizationproperty,theseparationproperty,theprewellorderingproperty,
thebasisproperty,theexistenceofscales, Suslincardinals,theexistence of
homogeneous representations, etc. We shall concentrate on just one such
property—theuniformizationproperty.
Let Aand Bbe subsets of the plane (ωω)2ω. Auniformizes Biff A⊆ B and for all
x ∈ ω, there exists y such that x,y ∈ B iff there is a unique ysuch that x,y ∈ A.
In other words, Ayields a choice function for B. Using AC it is clear that every
set B has a uniformizing set A. However,
thequestionofwhetheruniformizationsexistbecomesinteresting
whenoneimposesadefinabilityconstraint. LetGbeacollectionof(k-tuples of) sets
of reals. Such a class is called a pointclass. The pointclass G has
uniformization property (Unif(G)) iff every subset of the plane that is in G
admitsauniformizationthatisalsoinG.
21
2.3 Classical Results
Duringtheearly20thcenturythereweremanyadvancesbytheFrenchand
Russiananalystsinestablishingthatdefinablesetshadtheaboveregularity and
structural properties. We shall limit our discussion to the crowning
achievements:
Theorem 2.1 (Luzin, Suslin, 1917). Assume ZFC. The S~ 11sets have the
perfectsetproperty,thepropertyofBaire,andareLebesguemeasurable.
Notice that the property of Baire and Lebesgue measurability have the
featurethatiftheyholdofAthentheyholdofthecomplementofA. However,
theperfectsetpropertydoesnothave thisfeature. Itremainedopen whether all Π~
11sets have the perfect set property and whether all S ~ 12sets
havethepropertyofBaireandareLebesguemeasurable.
Theorem 2.2(Kondˆo,1937). AssumeZFC. ThenUnif(S~ 1)holds. It remained
open whether all sets of reals in Π~ 12, S~ 132, etc. have the
uniformizationproperty.
2.4 Limitative Results
Itproveddifficulttoextendtheaboveresultstohigherpointclasses,somuch so that
in 1925 Luzin said that “one will never know” whether all of the
projectivesetsofrealshavetheregularityproperties.
Wenowknowthereasonforthedifficulties—theearlyanalystswereworkinginZ
FCandtheabovequestionscannotberesolvedonthebasisofZFC alone. It is a
testament to the power of the early analysts that they
obtainedthestrongestresults thatwere withinthereachoftheirbackground
assumptions.
The independence of the above statements follows from the dual
techniques of inner model theory (invented by G¨odel in 1938)and outer
model theory (inventedbyCohenin1963).
ThefirsthalfwasestablishedbyG¨odelandAddison: Theorem 2.3 (G¨odel).
Assume ZFC+V=L. ThenthereareS~ 1setsthat
donothavethepropertyofBaireandarenotLebesguemeasurableandthere areΠ~
11setsthatdonothavetheperfectsetproperty. 2
22
Theorem 2.4 (Addison,1959). AssumeZFC+V=L. Thenforalln 2, Unif(S~ 1n).
Itfollowsthat(assumingthatZFCisconsistent),ZFCcannotprovethatS~ 12
sets have the property ofBaire and areLebesgue measurable, thatΠ~ 11sets
havetheperfectsetproperty,andthatUnif(S~ 1n)failsforn 2. Itremained
openwhetherZFCcouldestablishthenegations.
ThesecondhalfwasestablishedbyMartin,SolovayandLevy: Theorem 2.5
(Solovay, 1965). Assume ZFC and that there is a strongly
inaccessiblecardinal. Thenthereisaforcingextensioninwhichallprojective sets
have the perfect set property, the property of Baire and are Lebesgue
measurable.
The strongly inaccessible cardinal is necessary for the perfect set property
(even for Π~ 1sets (Solovay)) and Lebesgue measurability (Shelah) but not
forthepropertyofBaire(Shelah). However,MartinandSolovayshowedin
1970thatifZFCisconsistentthensoisZFCalongwiththestatementthat allS 1
21setshavethepropertyofBaireandareLebesguemeasurable.
Theorem2.6(Levy,1965).AssumeZFC. Thenthereisaforcingextension
inwhichUnif(Π~ 12)fails.
Infact,Levynotedthatthisisthecaseintheoriginalmodelintroducedby
Cohen. Puttingeverythingtogetherwehavethefollowing: (1)IfZFCisconsistent, then ZFC cannot determine whether all S~ 12sets have the property of
BaireandareLebesguemeasurable. (2)IfZFC+“Thereisastronglyinaccessible
cardinal”is consistent, then ZFCcannotdetermine whether allΠ~ 11
setshavetheperfectsetproperty. (3)IfZFCisconsistent,thenZFCcannot
determine the pattern of the uniformization property; more precisely, it is
consistentwithZFCthattheuniformizationpropertyhaseitherthepattern
... S1S~ 12S4
~1
~1
Π~
S~
Π~
Π
12
13
11
~
Π~
14
13
Figure1: UniformizationunderV=L.
23
orthepattern
S~
S~
S~
12
13
14
...
Π
~
14
S~
11
Π~
Π~
12
13
Π~ 11
Figure2: UniformizationintheCohenmodel.
FurtherReading:
Formoreonthehistoryofclassicaldescriptivesettheory
seeKanamori(1995). Formoreonthemathematicsofclassical
descriptive set theory see sections 12and 13 of Kanamori
(2003), Kechris (1995), and Moschovakis(1980).
3 DeterminacyandLarge Cardinals
TheabovelimitativeresultsshowthatZFClackssufficientpowe
rtoresolve many questions about simple definable sets of
reals—it cannot determine whetherS~
12setshavetheregularitypropertiesanditcannotdeterminethe
patternoftheuniformizationpropertybeyondS~ 12.
Thisplacedmathematiciansinaninterestingpredicament.
Inordertoresolvethesequestionsthey
mustexpandtheirresourcesbeyondZFC.
Buttherearemanywaysofdoing this and it is unclear at the
outset whether there is a correct direction of expansion.
Thereweretwomaincandidatesfornewaxiomsthatunderw
entintensive investigationinthe1970sand1980s:
Axiomsofdefinabledeterminacyand large cardinal axioms.
In this section we will (1) introduce the basic
machineryofdeterminacyanddescribetheimplicationsofaxio
msofdefinable
determinacyforclassicaldescriptivesettheoryand(2)describe
theimplicationsoflargecardinalaxiomsforclassicaldescriptive
settheoryandbriefly
touchontheconnectionbetweenthetwoapproaches.
24
3.1 Determinacy
For a set of reals A⊆ ωωconsider the game where two players take turns
playingnaturalnumbers:
I x(0) x(2) x(4) ... II x(1) x(3) ...
At the end of a round of this game the two players will have produced a real
x, obtained through “interleaving” their plays. We say that Player I wins the
roundifx∈A; otherwise Player IIwins the round. The set Ais
saidtobedetermined ifoneoftheplayers hasa“winningstrategy”inthe associated
game, that is, a strategy which ensures that the player wins a
roundregardlessofhowtheotherplayerplays. TheAxiomofDeterminacy
(AD)isthestatementthatevery setofrealsisdetermined. Thisaxiomwas
introducedbyMycielskiandSteinhausin1962.
It is straightforward to see that very simple sets are determined. For
example, if Ais the set of all reals then clearly I has a winning strategy; if Ais
empty then clearly II has a winning strategy; and if Ais countable
thenIIhasawinningstrategy(by“diagonalizing”). Thismightleadoneto
expectthatallsetsofrealsaredetermined. However,itisstraightforwardto
usetheAxiomofChoice(AC)toconstructanon-determinedset(bylisting
allwinningstrategiesand“diagonalizing”acrossthem). ForthisreasonAD
wasneverreallyconsideredasaseriouscandidateforanewaxiom. However,
there is an interesting class ofrelated axioms thatare consistent with AC,
namely,theaxiomsofdefinabledeterminacy. Theseaxiomsextendtheabove
patternbyasserting thatallsets ofrealsatagivenlevel ofcomplexity are
determined, notableexamples being∆~ 11L(R)-determinacy (allBorelsetsofreals
aredetermined),PD(allprojectivesetsofrealsaredetermined)andAD
(allsetsofrealsinL(R)aredetermined).
Oneissueiswhetherthesearereallynewaxiomsorwhethertheyfollow
fromZFC. Oneoftheearliestresultsinthesubject(duetoGaleandStewart
in1953)isthat(inZFC)allΠ~ 01(thatis,closed)gamesaredetermined. This result
was extended to the next few levels of the Borel hierarchy. Finally,
in1974,Martinprovedthelandmarkresultthat∆~ 1-determinacyisprovable in ZFC.
It turns out that this result is close to optimal—as one climbs
thehierarchyofdefinability,shortlyafter∆~ 111onearrivesataxiomsthatfall
outsidetheprovenanceofZFC. Forexample,thisistrueofΠ~ 1L(R)1-determinacy,
PD and AD. Thus, we have here a hierarchy of axioms (including PD
25
andADL(R))whicharegenuinecandidatesfornewaxioms.
ForapointclassGof(k-tuplesof)setsofreals,letDet(G)bethestatement
thatallsetsofrealsinGaredetermined. Inwhatfollowsweshallassumethat
Ghascertainminimalclosureproperties;moreprecisely,weshallassumethat
Gisanadequate pointclassinthesenseofMoschovakis(1980). Theprecise
detailsofthisnotionarenotimportantforourpurposes;sufficeittosaythat
allofthepointclassesintroducedthusfarareadequate.
Theorem3.1(Banach,Mazur,Oxtoby,1957).AssumeZFC+Det(G). Then
thesetsofrealsinGhavethepropertyofBaire.
Theorem3.2(Mycielski-Swierczkowski,1964).AssumeZFC+Det(G). Then
thesetsofrealsinGareLebesguemeasurable.
Theorem 3.3 (Davis, 1964). Assume ZFC+Det(G). Then the sets in G
havetheperfectsetproperty.
L(R)Inparticular,ifoneassumesPDthenallprojectivesetsofrealshavethe
regularitypropertiesandifoneassumesADthenallofthesetsofreals
inL(R)havetheregularityproperties.
RInfactbymodifyingtheaboveproofsonecanobtaintheregularityproperties for a
larger pointclass, namely, ∃G. This is the pointclass obtained by taking
projections of sets in G. So, for example, if G is ∆~ 11Rthen ∃G is S~ 11. It
follows that assuming ∆~ 1-determinacy (which is provable in ZFC) one gets
that the S~ 11115sets of reals have the regularity properties.In this
sensedeterminacy liesattheheartoftheregularitypropertiesandmaybe
consideredtheirtruesource.
Theaboveresultsshowthataxiomsofdefinabledeterminacyextendthe
resultsonregularitypropertiesthatcanbeprovedinZFC. ZFCcanestablish thatall
S~ 11sets of reals have the regularity properties and∆~ 11-determinacy (a
theorem of ZFC) lies at the heart of this result. The extension of ∆~
1L(R)1determinacy tostronger formsofdefinable determinacy (PD, AD, etc.)
generalizestheseresultsbyextendingtheregularitypropertiestothecorrespondin
gpointclasses(projective,inL(R),etc.)
Inasimilarfashion,axiomsofdefinable determinacy extendtheresults
ontheuniformizationproperty.
). ThenUnif(Π~
Theorem3.4(Moschovakis,1971).AssumeZFC+Det(∆~ 12n 12n+1
~ 12n+2).
15Seetheexercisesin6.GofMoschovakis(1980).
26
) and Unif(S
Thus,underPDtheuniformizationpropertyhasthefollowingpattern:
S~
S~
S~
...
12
13
14
S~
Π
11
~
14
Π~
Π~
12
13
Π~
11
Figure3: UniformizationunderPD.
L(R)Thispatternrecursathigherlevelsunderstrongerformsofdeterminacy.
For example, under ADthe pattern recurs throughout the transfinite
extensionoftheprojectivesetswithinL(R). ForallofL(R)onehas:
Theorem3.5(Martin-Steel,1983). AssumeZFC+ADL(R). TheninL(R), S~
21hastheuniformizationproperty. (Here S ~ 21is defined just as (the logical
version) of S~ 11except that now the
initialquantifierrangesoversetsofrealsandnotjustreals(asisindicated
bythesuperscript).)
Moreover,theuniformizationpropertyisnotanisolatedcase. Thesame
patternholdsforotherproperties,suchastheprewellorderingpropertyand
thescaleproperty.
3.2 Large Cardinals
Thesecondapproachtonewaxioms—whichraninparalleltotheapproach
using definable determinacy—was toinvoke largecardinal axioms. In
fact, itwas graduallyshown (asSolovayhadconjectured
inthelate1960s)that
largecardinalaxiomsactuallyimplyformsofdefinabledeterminacy.
Theresultthatunderscoredthesignificanceoflargecardinalaxiomsfor the
structure theory of simply definable sets of reals is the following early
resultofSolovay:
Theorem3.6(Solovay,1965). AssumeZFCandthatthereisameasurable
cardinal. Thenall S~ 12setsofrealshavetheperfectsetproperty,theproperty
ofBaire,andareLebesguemeasurable.
27
Given thatΠ~ -determinacy implies the regularity properties forS~
11
12
n<ω n
n+1
κn(j).(Notice that j(κω(j)) = κω
~ 11
i
0
n(j)).
Let κw
set
s of reals (see the discussion after Theorems 3.1–3.3), it was
reasonable to suspect that a measurable cardinal actually implied Π~
11-determinacy. This wasestablishedbyMartin:
Theorem 3.7 (Martin,1970). AssumeZFCandthatthereisameasurable
cardinal. Then Π-determinacyholds.
Sincedefinabledeterminacylayattheheartofthestructuretheory,much
of the energy was focused on trying to show that it followed from large
cardinalaxioms. Thenextmajoradvancewasmadein1978.
Theorem 3.8 (Martin,1978). AssumeZFCandthat thereisanon-trivial
elementaryembeddingj:Vλ+1→Vλ+1. Then Π~ 12-determinacyholds. (Martin
actually used a somewhat weaker assumption but it is somewhat
technicalanditwouldnotbeilluminatingtostateithere.) In early 1979,
Woodin extended this by showing that PD follows from
much strongerlargecardinalhypotheses. Todescribe this,
letusintroduce some terminology: Suppose j : Vλ+1→ Vis a non-trivial
elementary embedding.
Letκ0λ+1(j)bethecriticalpointofjandforeachnletκ(j)= j(κ(j) = sup,...,j(j).) A
sequenceofembeddingsjisann-foldstrongranktorankembedding
sequence ifthereexistsaλsuchthat
(1) foreachi=n, j
:Vλ+1 →Vλ+1
isanon-trivialelementaryembeddingand
(2) foreachi<n,
κω(ji)<κω(ji+1). Theorem 3.9 (Woodin,1979). AssumeZF.
Foreachn<ω,ifthereisan
n-foldstrongranktorankembeddingsequencethenΠ~
1n+2-determinacyholds.
SoPDfollowsfromtheassumptionthatforeachn,thereisann-foldstrong
ranktorankembeddingsequence. The interesting thing about these large
cardinal hypotheses is that for
n>0theyareinconsistentwithAC(byaresultofKunen). Moreover,there
isnowreasontobelievethatforn>0theyareoutrightinconsistent(inZF).
28
SeeWoodin(2011). Shouldthisturnouttobethecaseitwouldberather
interestingsinceitwouldbeacasewhereonederivedstructuretheoryfrom
aninconsistencywithoutawarenessoftheinconsistency.
Fortunately,inDecemberof1983Woodinbothstrengthenedtheconclusionand
reducedthelargecardinalhypothesis toonethatisnotknown to
beinconsistentwithAC:
Theorem3.10(Woodin,1984).AssumeZFCandthatthereisanon-trivial
elementaryembeddingj:L(Vλ+1)→L(VL(R)λ+1)withcriticalpointlessthanλ.
ThenADholds. Infact,WoodingotmuchmoredefinabledeterminacythanADL(R).
Still,thelargecardinalassumptionwasthoughttobetoostrong. Atthis
point the episode of history described earlier—the one involving Solovay’s
theoremof1965andMartin’stheoremof1970—repeateditself. Forin1984
Woodinshowed thatifthereisasupercompactcardinalthenallprojective
setshavetheregularityproperties. InjointworkwithShelah,thehypothesis
wassignificantlyreduced.
Theorem 3.11 (Shelah-Woodin, 1984). Assume ZFC and that there are
nWoodin cardinals with a measurable cardinal above them all. Then the S ~
1n+2have the perfect set property, the property of Baire, and are Lebesgue
measurable.
As in the earlier episode, this suggested that the above large cardinal
hypothesisactuallyimpliedΠ~ 1n+1-determinacy. Thiswassoonestablishedby
alandmarkresultofMartinandSteel.
Theorem 3.12 (Martin-Steel, 1985). Assume ZFC and that there are n
Woodin cardinals with a measurable cardinal above them all. Then Π~
1n+1determinacyholds.
Finally, in combination with Woodin’s work on the stationary tower, the
followingresultwasestablished.
L(R)Theorem
3.13 (Martin, Steel, andWoodin,1985). Assume ZFC andthat
there are ωmany Woodin cardinals with a measurable cardinal above them
all. ThenADholds. Inthiswaythetwoapproaches
tonewaxioms—viaaxiomsofdefinable
determinacyandvialargecardinalaxioms—wereshowntobecloselyrelated.
29
In fact, in subsequent work it was shown that the connections are much
deeper. But tofully appreciate this we must first introduce the machinery
ofinnermodeltheory.
FurtherReading: Formoreondeterminacyseesections27–32ofKanamori
(2003)andMoschovakis(1980). Forfurtherdetailsconcerningthestructure
theoryprovided bydeterminacy see thevolumes oftheCabalSeminarand
Jackson(2010).
4 TheCasefor DefinableDeterminacy
Wearenowinapositiontodiscussaspectsofthecasethathasbeenmade
foraxiomsofdefinabledeterminacy.
4.1 Basic Notions
Letusstressfromthebeginningthatbecauseoftheincompletenesstheorems the
case for any new axiom is going to have to be delicate. The standard
wayofestablishingapropositioninmathematicsisbyprovingitwithinsome
acceptedsetofaxioms.
However,theaxiomsimplicitlyacceptedbymathematiciansineveryday
practiceseem tobeexhausted by thoseofZFC,and
weareheredealingwithquestionsandnewaxiomsthatarebeyondthereach
ofZFC. Given the nature of the case, it seems unlikely therefore that one
willhaveaknock-downargumentfornewaxioms. Insteaditseemsthatthe
mostthatonecanhopeforistoaccumulateevidenceuntileventuallythecase
becomescompelling. Theotheralternativeisthatthecaseremainsdivided
betweentwoequallycompellingbutmutuallyinconsistentapproaches.
Let us first say something very general about how such a case might
proceed. Some propositions have a certain degree of intrinsic plausibility. For
example, the statement that there is no paradoxical decomposition of the
sphere using pieces that are S~ 12is intrinsically plausible, although it is
notprovableinZFC. Suchinitialsupportforapropositionisnotdefinitive. For
example, the statement that there are no space filling curves and the
statement that there is no paradoxical decomposition of the sphere using any
pieces whatsoever might have once seemed intrinsically plausible. But in
each case these statements were overturned by theorems proved against
the backdrop of systems that had much more than this degree of initial
30
plausibility in their favour. However, althoughsuch intrinsic plausibility is
notdefinitive,itneverthelesshassomeweight.
Moreover, theweightcanaccumulate. Supposewehavetwostatements from
conceptually distinct domains that are intrinsically plausible—for example,
one might concern the existence of (inner models of)large cardinal
axiomsandtheothermightconcerntheregularitypropertiesofsimply
definablesets. Ifitisshownthatthefirstimpliesthesecondthenthesecond inherits
the initial plausibility attached to the first. Should the two
statementsultimatelybeshowntoimplyoneanotherthiswouldstrengthenthe
caseforeach. The hopeis thatthroughtheaccumulation ofmathematical
results—theoremsprovedinfirst-orderlogic,whicharemattersuponwhich
everyone can agree—various independent intrinsically plausible statements
will come to reinforce each other to the point where the case as a whole
becomescompelling.
Somehavemaintainedthatthisisactuallythesituationwithaxiomsof definable
determinacy and large cardinal axioms. In the remainder of this
sectionweshallpresentthecaseinthestrongestwayweknowhow.
FurtherReading: Formoreonthenotionofintrinsicplausibility andhowit
figuresinthestructureofreason,seeChapter9ofParsons(2008).
4.2 StructureTheory
Fordefiniteness,whenwespeakofthe“structuretheory”(atagivenlevel)we
shallbereferringtotheregularitypropertiesintroducedabove—theperfect
setproperty,andthepropertyofBaire,andLebesguemeasurability—along
withtheuniformizationproperty. Thereisactuallymuchmore(asindicated
above)tothestructuretheoryandwhatweshallhavetosaygeneralizes. We shall
concentrate on two levels of definable determinacy—∆~ 1L(R)1-determinacy
and AD. Again, this is just for definiteness and what we shall have to
saygeneralizestocertainhigherformsofdefinabledeterminacy.
The axiom ∆~ 1-determinacy (equivalently, Borel-determinacy) is a
theoremofZFCandliesattheheartofthestructuretheoryofBorelsets(and, in fact,
the S~ 111-sets and S~ 12sets). One virtue of ADL(R)is that it lifts this
entirestructuretheoryfromtheBorelsetstothesetsinL(R). Summarizing
resultsfromabovewehave:
Theorem4.1. AssumeZFC+ADL(R). TheneverysetofrealsinL(R)has
31
the perfect set property, the property of Baire, is Lebesgue measurable, and
S2 1-uniformizationholdsinL(R). Many descriptive set theorists take these
consequences to be intrinsiL(R)cally plausible. Moreover, they have found the particular pattern of the
uniformization property provided by ADto be a plausible extension of the
pattern initiated inZFC. Forthis lastpoint consider the first ωmany levels
and compare the case of V=Lwith that of PD. In ZFC, the
uniformizationpropertypropagatesfromΠ~ 11toS~ 12. Itisnaturaltoexpectitto
continuetozig-zagbackandforth. However,underV=Litstabilizes:
... S1S~ 12S4
~1
~1
Π~
S~
Π~
Π
12
13
11
~
Π~
14
13
Figure4: UniformizationunderV=L.
Incontrast,PDprovideswhatappearstobethecorrectextrapolation:
S~
S~
S~
...
12
13
14
Π
S~
11
Π~
Π~
12
13
~
14
Π~
11
Figure5: UniformizationunderPD.
Again,this“periodicityphenomenon”occursathigherlevelsinL(R).
OnemightgrantthattheseareindeedfruitfulconsequencesofADL(R)L(R)and as
such are evidence that ADL(R)holds. However, one might be concerned that
perhaps there are axioms that share these fruitful consequences and
yetareincompatiblewithAD. Asaresponse, onemightnotethatthis
objectionisneverraisedintheanalogouscaseofphysicswhereamultitude
ofincompatibletheoriescanagreeintheirempiricalconsequences. Butthere
isamuchstrongerresponsethatonecanmake. Forincontrasttothecaseof
physics—whereonecouldneverhopetoshowthattherewasonlyonetheory
32
L(R)with
the desired empirical consequences—in set theory this is actually the
case; more precisely, any theory that has these fruitful consequences—that
is,whichliftsthestructuretheorytothelevelofL(R)—must implyAD.
Thisremarkablefactisembodiedinthefollowingtheorem:
Theorem4.2(Woodin).AssumethatallsetsinL(R)areLebesguemeasurable,have
thepropertyof Baire,andthat S2 1L(R)-uniformizationholdsinL(R). ThenAD.
Summary: Thereisa“goodtheory”(onethatliftsthestructuretheory)
L(R)andallgoodtheoriesimplyAD.
4.3 Large Cardinals
Largecardinalaxiomsaretaken(bythosewhostudythem)tohavecertain degrees
of intrinsic plausibility. As we noted above, large cardinal axioms
implyformsofdefinabledeterminacyandsothelatterinherittheconsiderationsinfav
ouroftheformer.
L(R)Theorem4.3(Martin,Steel,andWoodin).AssumeZFCandthatthereare
ωmanyWoodincardinalswithameasurableabovethemall. ThenAD. In fact, it
turns out that the existence of certain inner models of large
cardinalaxiomsissufficient toprovedefinabledeterminacy.
Sinceitis(arguably)evenmoreplausiblethattherearesuchinnermodels,this(argu
ably) strengthens the case. So definable determinacy is a
fruitfulconsequence of theexistenceofsuchinnermodels.
16However,justasintheprevioussubsection,onewouldlikesomeassurance
thattherearenotincompatibleaxiomsthatsharethisfruitfulconsequence.
Remarkably,thisturnsouttobethecase—axiomsofdefinabledeterminacy
areactuallyequivalent toaxiomsassertingtheexistence ofinnermodelsof large
cardinals. We discuss what is known about this connection, starting
withalowlevelofboldfacedefinabledeterminacyandproceedingupward. It
shouldbeemphasized thatourconcernhereisnotmerelywithconsistency
strengthbutratherwithoutrightequivalence(overZFC).
16Innermodeltheoryisaveryadvancedsubbranchofsettheoryandwewillnotbeable
to describe it
in anydetail here. Suffice it to say thatthe goalofinner modeltheory is to find“L-like”models
thatcancontainlargecardinals. Inthe results belowMis the
canonicalinnermodelforω-manyWoodincardinalsandM# ωωisthe“sharp”ofthismodel.
ForamoredetaileddiscussionoftheseresultsseeKoellner&Woodin(2010).
33
Theorem 4.4(Woodin). Thefollowingareequivalent: (1) ∆~ 12ω-determinacy. (2)
For all x ∈ ω, there is an inner model M such that x ∈ M and
M|=“ThereisaWoodincardinal”.
Theorem 4.5(Woodin). Thefollowingareequivalent: (1) PD(Schematic). (2)
For every n <ω, there is a fine-structural, countably iterable inner
modelMsuchthatM|=“TherearenWoodincardinals”. Theorem
4.6(Woodin). Thefollowingareequivalent:
L(R)(1)
AD. (2) InL(R),foreverysetSofordinals,thereisaninnermodelMandan
a<ωL(R) 1suchthatS∈MandM|=“aisaWoodincardinal”. Theorem 4.7(Woodin).
Thefollowingareequivalent:
#
andR#
ω
L(R)(1)
AD exists. (2) Mexistsandiscountablyiterable. Theorem
4.8(Woodin). Thefollowingareequivalent:
B |=ADL(R)
(1) ForallB,V
34
#ω
. (2) Mexistsandisfullyiterable. Summary: Large
cardinal axioms are sufficient to prove definable determinacyandinnermodelsoflargecardinalaxiomsarenecessarytoprove
definable determinacy. In fact, through a series of deep results, these two
conceptuallydistinctapproachestonewaxiomsareultimatelyseentobeat
rootequivalent.
4.4 Generic Absoluteness
L(R)Largecardinalaxiomshavethefruitfulconsequencethattheyimplyaxioms
ofdefinabledeterminacyandhenceilluminatethestructuretheoryofdefinablesets
ofreals. Infact,ADappearstobe“effectivelycomplete”with
regardtoquestionsconcerningL(R). Letustrytomakethispreciseinterms
ofgenericabsoluteness.
Themotivatingresultinthisareaisthefollowing: Theorem 4.9 (Shoenfield).
AssumeZFC. SupposeϕisaS1 2sentenceand BisacompleteBooleanalgebra.
Then
B
V |=ϕ iff V
12
12
17
L(R)V
VB
L(R)
L(R)
L(R)
L(R).
17
35
|=ϕ.
Inotherwords,theS-theorycannotbealteredbythemethodofoutermodels.
Insuchasituationwesaythatϕisgenericallyabsolute. Sinceourmain
methodofestablishingindependence(withoutleavinganinterpretabilitydegree)is
themethodofoutermodels, thisexplains why therearenoknown Sstatements
thatlieintheinterpretability degreeofZFCandareknown tobeindependentofZFC.
A fruitful consequence of large cardinal axioms is that they place us in
exactlythissituationwithregardtoamuchlargerclassofstatements.
Theorem 4.10 (Woodin). AssumeZFCandthatthereisaproperclassof Woodin
cardinals. Suppose ϕis a sentence and B is a complete Boolean algebra.
Then
|=ϕ iff L(R)|=ϕ. In this situation one, of course, has AD. Again, one might be
concerned that one can have this fruitful consequence without having AD.
Thisisaddressedbythefollowingtheorem:
Theorem 4.11 (Woodin). AssumeZFCandthatthereisaproperclassof
stronglyinaccessiblecardinals. SupposethatthetheoryofL(R)isgenerically
absolute. ThenAD. Summary: There is a “good theory” (one that freezes the
theory of
L(R)) and (granting a proper class of strongly inaccessible cardinals) all
goodtheoriesimplyAD
SeeLarson(2004)foranaccountofthisresult.
4.5 Overlapping Consensus
WehavediscussedtworecoveryresultsforADL(R). First,ADL(R)L(R)isimplied
byitsstructuralconsequences(Theorem4.2). Second,ADL(R)isrecoveredby
thegenericabsolutenessthatfollowsfromlargecardinalsthatimplyAD
(Theorem4.11). Theserecovery resultsarenotisolatedoccurrences.
Thetechniqueused
L(R)to establish them is known as the core model induction. This technique
hasbeenusedtoshowthatmanyotherstrongtheoriesimplyAD. This
evenholdsfortheoriesthatareincompatiblewithoneanother. Forexample,
considerthefollowing:
Theorem4.12. (Woodin) AssumeZFC+thereisanω1-denseidealonωL(R).
ThenAD. Theorem 4.13. (Steel) AssumeZFC+PFA. ThenADL(R)1. These two
theories are incompatible. There are many other examples. In
L(R)fact, the examples are so ubiquitous and the technique is so general that
thereisreasontobelievethatADliesintheoverlappingconsensusofall
sufficientlystrong,naturaltheories.
Summary: ADL(R)L(R)liesintheoverlappingconsensusofnumerousstrong
theoriesandsoitinheritstheconsiderationsofeach.
Moreover,thereisreasontobelievethatADliesintheoverlappingconsensusofall
sufficiently strong,“natural”theories.
4.6 Beyond L(R)
Much ofthe above case can be pushed even beyond L(R). To explain this
wewillneedtointroducethenotionofauniversallyBaire setofreals.
V[G]Foracardinald,asetA⊆Risd-universallyBaireifforallpartialorders
Pofcardinalityd,thereexisttreesSandTonω×λ(forsomeλ)suchthat A= p[T] and,
if G⊆ P is V-generic, then p[T]= RV[G]V[G]-p[S]8. A
setA⊆RisuniversallyBaireifitisd-universally Baireforalld. Thekey
pointisthatuniversallyBairesetshavecanonicalinterpretationsingeneric
extensionsV[G]. LetGbethecollectionofsetsofrealsthatareuniversally Baire.
This collection has strong closure properties. For example, Woodin
showedthatifthereisaproperclassofWoodincardinalsandA∈G+then
(1)L(A,R)|=ADand(2)P(R)nL(A,R)⊆G8. HereAD+8isa(potential)
36
strengtheningofADdesignedformodelsoftheformL(P(R)). SeeWoodin (1999).
Forκaninfinite regularcardinal, let H(κ)be theset ofallsets xsuch
thatthecardinalityofthetransitiveclosureofxislessthanκ.
8Theorem
4.14. (Woodin) Supposethere is a proper classof Woodin
cardinalsandA∈G. SupposeG⊆PisV-generic. Then
(H(ω1),∈,A)V≺(H(ω1)V[G],∈,AG). Thatis,wehave genericabsoluteness
for“projective-in-A”whereAisuniversallyBaire. Infact,onehas“S2 1(G8)-genericabsoluteness”: Theorem 4.15.
(Woodin) Supposethere is a proper classof Woodin car8dinalsandletϕbeasentenceoftheform
∃A∈G(H(ω1),∈,A)|=ψ. SupposeG⊆PisV-generic. Then
V |=ϕ iff V[G]|=ϕ. Stronger large cardinal axioms imply that many sets of
reals beyond L(R)
areuniversallyBaire. Letuscallasetabsolutely ∆2 1 ifthereareS2 1
sets of reals are universally Baire.18
21
19
21
18
37
19
formulas
which definecomplementary sets ofrealsinallgeneric extensions. Woodin
showed thatifthere isaproperclassofmeasurable Woodincardinalsthen all
absolutely ∆This is one precise
senseinwhichCHwasanunfortunatechoiceofatestcasefortheprogram for large
cardinals—large cardinal axioms effectively settle all questions of
complexitystrictlybelow(intheabovesense)thatofCH.
4.7 Summary
Eachoftheaboveargumentshastwoparts. Inthefirstpart,onearguesthat
thereisa“good”theory(onehavingcertainfruitfulconsequences). Inmany
Under the same hypothesis one has that all of the “provably-∆” sets of reals are
universallyBaire.
Onemightworrythatwhatisreallygoingonhereisthatlargecardinalaxiomsthrow
awrenchintotheforcingmachinery. Butthisisnotso. Underlargecardinalassumptions one has
more generic extensions. What is really going on is that large cardinal axioms
generatetreesthatarerobustandactasoraclesfortruth.
casesthistheoryexplicitlyincludes(orjustis)ADL(R)L(R). Inthesecondpart,
oneshowsthatallsuchtheoriesincludeAD.
Wewanttostressthissecondpart—therecoverytheorems. Withoutthe
recoverytheoremsonecouldalwayswonderwhetherthereareincompatible
axiomswiththesamefruitfulconsequences. Incontrasttothecaseofphysics,
whereonecouldneverhopetoshowthatthedatalogicallyimpliesthetheory, in the
case of the search for new axioms this is in fact possible. It is the
recoverytheoremsthatsealthecase.
L(R)Perhapsthestrongestargumentinfavourofaxiomsofdefinabledeterminacyisth
elast. Fortheresultsshowthattherearedeepconnectionsbetween
approachestonewaxiomsfromconceptuallydistinctdomainsandthat,remarkably
,ADispartofthepictureprovidedbyeach. Inshort,ADL(R)
liesintheoverlappingconsensusofallapproachesthatinvolveclimbingthe
hierarchyofinterpretabilityalonganaturalpath.
FurtherReading: FormoreonthetopicsofthissectionseeKoellner(2006),
Koellner (2009b), Maddy (1988a), Maddy (1988b), Martin (1998), Steel
(2000), Woodin (2001a), Woodin (2001b), Woodin (2005b), and Woodin
(2005a).
5 PhilosophicalDiscussion
L(R)Letusreturntothequestionofpluralism.
Dotheaboveresultscollectively
provideuswithastrongextrinsicjustificationofADandtherebysecure
non-pluralismatthislevel?
L(R)The pluralist will certainly accept the above results—after all, they are
mathematicaltheorems—butwillresisttheclaimthattheyprovideuswith
theoretical reasons for adopting ADL(R). The pluralist may grant that the
results are quite interesting and that they show that ADL(R)has a certain
appeal—since, for example, it renders L(R) a paradise for analysts—but
theywillviewthisasapracticalreasonforadoptingADL(R),notatheoretical reason.
Onthisviewtherearemanyroadsthatonecantake—V=L,AD, etc.—and that while
each has its advantages with respect to certain aims
thereisno“factofthematter”astowhichisthecorrectone,anymorethan there is a
“fact of the matter” as to which of two notations is the correct notation.
Atthislevel,accordingtothepluralist,notheoreticalreasonsare at play and there
is no issue of objective mathematical truth. This is the
38
viewofFeferman(1999),Kanamori(2003),Shelah(2003),andmanyother
philosophersandmathematicians.
L(R)Thenon-pluralistregardstheaboveresultsascollectivelyprovidingvery
strongtheoreticalreasonsforthinkingthatADholdsandthattheother,
incompatiblealternativesfail. ThisistheviewofMartin(1998),Steel(2000),
Woodin(2005a),manyphilosophersandmathematicians,andmostdescriptiveset
theoristsandinnermodeltheorists.
Aswementionedintheintroduction,somepluralistsandnon-pluralists take
realism to be prior to reason while others take reason to be prior to realism. It
is important to keep these two versions separate and address
themindividually.
Let us start with those who take realism to be prior to reason. To fix
ideasconsiderthepluralistwhoacceptsrealismaboutthenaturalnumbers
butnotaboutarbitrarysubsetsofnaturalnumbersandthen,onthisbasis,
concludesthattheoreticalreasonscanbeofferedinthecaseofnewaxioms
innumbertheorybutnotinthecaseofanalysisorsettheory. Whenpressed on the
source of this asymmetrical stance concerning realism in the two
domainsthepluralistusuallyrespondsbyresortingtotheintuitionthatthe
naturalnumbersare“clear”whilethenotionofanarbitrarysubsetofnatural
numbersis“inherentlyvague”. Tothisthenon-pluralistmightrespondthat
thenotionofclarityisnotclear.
Intuitionsofclarity,likeintuitionsofselfevidence,arenotoriouslyvagueandsubjecti
veandhenceaweakpointupon whichtorestone’scase.
Letusnowturntothosewhotakereasontobepriortorealism. People in this
category take objectivity to be the hallmark of realism and they come to their
conclusions concerning realism about a given domain only
afteronehasagoodunderstandingofwhatkindoftheoreticalreasonshave
tractioninthatdomain. Apluralistwhothinksthattheoreticaljustification
inmathematicsmustultimatelytracebacktoaxiomswhichareself-evident will be
unmoved by the above extrinsic case. Likewise for a pluralist who thinks that
theoretical justification in mathematics must ultimately trace back to axioms
that are extrinsically justified. The case we have given is clearly an extrinsic
one. The issue, then, comes down to the legitimacy of extrinsicjustifications.
Wearecertainlynotinapositiontoresolvethisdebatehere. Butitwill be helpful
to draw a parallel with the debate between the instrumentalist
andnon-instrumentalistinphysics. Anextremeformofinstrumentalismwas
advancedbytheneo-KantianHansVaihingerinhisbookThePhilosophyof
39
As If (1911). According to this view, the “hard data” of physics consists
inthedataofourimmediatesenseimpressions. Uponthisbasiswearefree to
construct—in common sense and in physics—a “reality”, but the result
isnotgenuineknowledge—rather,itisausefulfictionformovingaroundin everyday
life and making predictions in physics. In a similar fashion, the
extremepluralisttakesthe“harddata”toconsistofthetheoremsthathave
actuallybeenestablishedandregardseverythingelseasausefulfiction.
20In the physical case the non-instrumentalist sees theoretical reason at
playatamuchhigher-level,farbeyondthedata. Forexample,inthetimeof
Copernicus,whenthePtolemaicandCopernicantheorywereobservationally
indistinguishable,theoreticalreasonscouldstillbegivenfortheCopernican theory
over the Ptolemaic theory.L(R)Similarly in the set theoretic case the
non-pluralistseestheoreticalreasonatplayathigherlevel,beyondthetheorems
and the intrinsically plausible statements. Just as in astronomy the
non-instrumentalistfindsevidenceofahigher-levelstructureintheconstellation of
connections, likewise in the case of set theory the non-pluralist takes the
constellation of connections in the interpretability hierarchy—in
particular,theabovetheoremsconcerningAD—asprovidingevidenceof
structureofahigherlevel. There is, ofcourse, much moretobesaid. But
wehopetohavebroughttheissuestothefore.
It should be mentioned that the non-pluralist about ADL(R)is open to
thepossibility thatpluralismholdsathigherlevels, sayatthelevelofCH.
Foratreatmentofthatsubjectseetheentry“TheContinuumHypothesis”.
References
Feferman,S.(1960). Arithmetizationofmetamathematics
inageneralsetting,FundamentaMathematicae49:35–92.
Feferman, S. (1964). Systems of predicative analysis, Journal of Symbolic
Logic29:1–39.
Feferman,S.(1991).Reflectingonincompleteness,JournalofSymbolicLogic
56:1–49.
20Consider,forexample,thereasonsgiveninKepler’s“DefenseofUrsus”andEpitome
ofCopernican Astronomy.
40
Feferman,S.(1999).
Doesmathematicsneednewaxioms?,AmericanMathematicalMonthly106:99–1
11.
Feferman, S. (2005). Predicativity, in S. Shapiro (ed.), The Oxford
HandbookofPhilosophyofMathematicsandLogic,OxfordUniversityPress,
pp.590–624.
Ferreir´os, J. (2007). Labyrinth of Thought: A History of Set Theory and
ItsRoleinModernMathematics,secondrevisededn,Birkh¨auserVerlag AG.
Foreman, M. & Kanamori, A. (2010). Handbook of Set Theory,
SpringerVerlag.
Friedman,H.M.(2011). BooleanRelationTheory,AssociationofSymbolic Logic.
Forthcomming.
G¨odel, K. (1947). What is Cantor’s continuum problem?, (G¨odel 1990),
OxfordUniversityPress,pp.176–187.
G¨odel, K. (1964). What is Cantor’s continuum problem?, (G¨odel 1990),
OxfordUniversityPress,pp.254–270.
G¨odel,K.(1990).
CollectedWorks,VolumeII:Publications1938–1974,OxfordUniversityPress,Ne
wYorkandOxford.
Jackson,S.(2010). Structuralconsequences ofAD,(Foreman&Kanamori
2010),Springer.
Kanamori, A. (1995). The emergence of descriptive set theory, in J. Hintikka
(ed.), From Dedekind to G¨odel: Essays on the Development of
theFoundationsofMathematics,Vol.251ofSyntheseLibrary,Kluwer,
Dordrecht,pp.241–262.
Kanamori, A. (2003). The Higher Infinite: Large Cardinals in Set Theory from
their Beginnings, Springer Monographs in Mathematics, second
edn,Springer,Berlin.
Kechris,A.S.(1995).ClassicalDescriptiveSetTheory,Vol.156ofGraduate
TextsinMathematics,Springer-Verlag,NewYork.
41
Koellner,P.(2006). Onthequestionofabsoluteundecidability,Philosophia
Mathematica 14(2): 153–188. Revised and reprinted with a new
postscriptinKurtG¨odel: EssaysforhisCentennial,editedbySolomon
Feferman,CharlesParsons,andStephenG.Simpson.LectureNotesin
Logic,33.AssociationofSymbolicLogic,2009.
Koellner, P. (2009a). On reflection principles, Annals of Pure and Applied
Logic157(2–3):206–219.KurtG¨odelCentenaryResearchPrizeFellowships.
Koellner, P.(2009b). Truthinmathematics: The question ofpluralism, in
O.Bueno&Ø.Linnebo(eds),NewWavesinPhilosophyofMathematics,NewWave
sinPhilosophy,PalgraveMacmillan,pp.80–116.
Koellner, P. & Woodin, W. H. (2010). Large cardinals from determinacy,
(Foreman&Kanamori2010),Springer.
Larson,P.B.(2004).TheStationaryTower: NotesonaCoursebyW.Hugh Woodin,
Vol. 32 of University Lecture Series, American Mathematical Society.
Lindstr¨om,P.(2003). AspectsofIncompleteness,Vol.10ofLectureNotesin
Logic,secondedn,AssociationofSymbolicLogic.
Maddy, P. (1988a). Believing the axioms I, Journal of Symbolic Logic
53:481–511.
Maddy, P. (1988b). Believing the axioms II, Journal of Symbolic Logic
53:736–764.
Maddy,P.(2011). DefendingtheAxioms,OxfordUniversityPress. Markov, A. A.
(1962). On constructive mathematics, Trudy MatematicheskogoInstitutaImeniV.A.Steklova67(8–14).TranslatedinAmerican
MathematicalSocietyTranslations: Series2,98,1–9.
Martin,D.(1998).Mathematicalevidence,inH.G.Dales&G.Oliveri(eds),
TruthinMathematics,ClarendonPress,pp.215–231.
Martin, D. A. & Steel, J. R. (1989). A proof of projective determinacy,
JournaloftheAmericanMathematicalSociety2(1):71–125.
42
Moschovakis,Y.N.(1980). DescriptiveSetTheory,StudiesinLogicandthe
FoundationsofMathematics,North-HollandPub.Co.
Nelson, E. (1986). Predicative Arithmetic, number 32 in Princeton
MathematicalNotes,PrincetonUniversityPress.
Parsons,C.(2000). Reasonandintuition,Synthese125:299–315.
Parsons,C.(2008). MathematicalThoughtanditsObjects,CambridgeUniversityPress. Shelah, S. (2003). Logical dreams, Bulletin of the
American Mathematical
Society40(2):203–228. Steel,J.(2000).
Mathematicsneedsnewaxioms,BulletinofSymbolicLogic
6(4):422–433.
Tait,W.W.(1981).Finitism,JournalofPhilosophy78:524–556.Reprinted
in(Tait2005b). Tait, W.W. (2001). G¨odel’s unpublished papers on
foundations ofmathematics,PhilosophiaMathematica9:87–126. Tait,W.W.(2005a).
Constructingcardinalsfrombelow,(Tait2005b),OxfordUniversityPress,pp.133–154.
Tait,W.W.(2005b).TheProvenanceofPureReason: EssaysinthePhilosophy of Mathematics and Its History, Logic and Computation in
Philosophy,OxfordUniversityPress.
Visser, A.(1998). Anoverview ofinterpretability logic, Advancesin modal
logic,Vol.1(Berlin,1996),Vol.87ofCSLILectureNotes,CSLIPubl.,
Stanford,CA,pp.307–359.
Woodin, W. H. (1999). The Axiom of Determinacy, Forcing Axioms, and the
Nonstationary Ideal, Vol. 1 of de Gruyter Series in Logic and its
Applications,deGruyter,Berlin.
Woodin, W. H. (2001a). The continuum hypothesis, part I, Notices of the
AmericanMathematicalSociety48(6):567–576.
43
Woodin,W.H.(2001b). Thecontinuumhypothesis, partII,Noticesof the
AmericanMathematicalSociety48(7):681–690.
Woodin,W.H.(2005a).Thecontinuumhypothesis,inR.Cori,A.Razborov,
S.Todorˆcevi´c&C.Wood(eds),LogicColloquium2000,Vol.19ofLectureNotesin
Logic,AssociationofSymbolicLogic,pp.143–197.
Woodin,W.H.(2005b). SettheoryafterRussell: thejourneybacktoEden, in G.
Link (ed.), One Hundred Years Of Russell’s Paradox:
Mathematics,Logic,Philosophy,Vol.6ofdeGruyter SeriesinLogicandIts
Applications,WalterDeGruyterInc,pp.29–47.
Woodin,W.H.(2011).SuitableextendermodelsI,JournalofMathematical
Logic11(1–2):101–339.
44