Investigations of Subsystems of Second Order Arithmetic and Set

77 Pages·2010·0.45 MB·English

Investigations of Subsystems of Second Order
Arithmetic and Set Theory in Strength between
Π1-CA and ∆1-CA + BI: Part I
1 2
MichaelRathjen
DepartmentofPureMathematics
UniversityofLeeds
LeedsLS29JT,England
Abstract This paper is the first of a series of two. It contains proof–
theoretic investigations on subtheories of second order arithmetic and set
theory.Amongtheprinciplesonwhichthesetheoriesarebasedonefindsau-
tonomouslyiteratedpositiveandmonotoneinductivedefinitions,Π1 trans-
1
finite recursion, ∆1 transfinite recursion, transfinitely iterated Π1 depen-
2 1
dent choices, extended Bar rules for provably definable well-orderings as
well as their set-theoretic counterparts which are based on extensions of
Kripke-Platek set theory. This first part introduces all the principles and
theories. Itprovideslowerboundsfortheirstrengthmeasuredintermsof
the amount of transfinite induction they achieve to prove. In other words,
itdetermineslowerboundsfortheirproof-theoreticordinalswhichareex-
pressedbymeansofordinalrepresentationsystems. Thesecondpartofthe
paperwillbeconcernedwithordinalanalysis. Itwillshowthatthelower
bounds established in the present paper are indeed sharp, thereby provid-
ingtheproof-theoreticordinals. Alltheresultswereobtainedmorethen20
yearsago(inGerman)intheauthor’sPhDthesis[43]buthaveneverbeen
publishedbefore,thoughthethesisreceivedareview(MR91m#03062). I
thinkitishightimeitgotpublished.
1 Introduction
To set the stage for the following, a very brief history of ordinal-theoretic proof
theoryfromthetimeafterGentzen’sdeathuntiltheearly1980sreadsasfollows:
Inthe1950’sprooftheoryflourishedinthehandsofSchu¨tte. In[59]heintroduced
aninfinitarysystemforfirstordernumbertheorywiththeso-calledω-rule,which
had already been proposed by Hilbert [23]. Ordinals were assigned as lengths to
derivations and via cut-elimination he re-obtained Gentzen’s ordinal analysis for
number theory in a particularly transparent way. Further, Schu¨tte extended his
2 MichaelRathjen
approachtosystemsoframifiedanalysisandbroughtthistechniquetoperfection
in his monograph “Beweistheorie” [60]. Independently, in 1964 Feferman [13]
andSchu¨tte[61],[62]determinedtheordinalboundΓ fortheoriesofautonomous
0
ramifiedprogressions.
AmajorbreakthroughwasmadebyTakeutiin1967, whoforthefirsttimeob-
tainedanordinalanalysisofastrongfragmentofsecondorderarithmetic. In[69]
he gave an ordinal analysis of Π1 comprehension, extended in 1973 to ∆1 com-
1 2
prehension in [70]. For this Takeuti returned to Gentzen’s method of assigning
ordinals (ordinal diagrams, to be precise) to purported derivations of the empty
sequent(inconsistency).
Thenextwaveofresults,whichconcernedtheoriesofiteratedinductivedefini-
tions, wereobtainedbyBuchholz, Pohlers, andSieginthelate1970’s(see[10]).
Takeuti’s methods of reducing derivations of the empty sequent (“the inconsis-
tency”)wereextremelydifficulttofollow,andthereforeamoreperspicuoustreat-
ment was to be hoped for. Since the use of the infinitary ω-rule had greatly fa-
cilitated the ordinal analysis of number theory, new infinitary rules were sought.
In 1977 (see [5]) Buchholz introduced such rules, dubbed Ω-rules to stress the
analogy. They led to a proof-theoretic treatment of a wide variety of systems, as
exemplified in the monograph [11] by Buchholz and Schu¨tte. Yet simpler infini-
taryruleswereputforwardafewyearslaterbyPohlers, leadingtothemethodof
local predicativity, which proved to be a very versatile tool (see [40–42]). With
theworkofJa¨gerandPohlers(see[28,29,33])theforumofordinalanalysisthen
switchedfromtherealmofsecond-orderarithmetictosettheory, shapingwhatis
now called admissible proof theory, after the models of Kripke-Platek set theory,
KP. Theirworkculminatedintheanalysisofthesystemwith∆1 comprehension
2
plusBarinduction,(BI),[33].Inessence,admissibleprooftheoryisagatheringof
cut-eliminationtechniquesforinfinitarycalculioframifiedsettheorywithΣand/or
Π reflectionrules1thatlenditselftoordinalanalysesoftheoriesoftheformKP+
2
“therearexmanyadmissibles”orKP+“therearemanyadmissibles”. Byway
of illustration, the subsystem of analysis with ∆1 comprehension and Bar induc-
2
tioncanbecouchedinsuchterms,foritisnaturallyinterpretableinthesettheory
KPi:=KP+∀y∃z(y∈z∧zisadmissible)(cf.[33]).
Theinvestigationsofthispaperfocus,asfarassubsystemsofsecondorderarith-
metic are concerned, on theories whose strength strictly lies in between that of
∆1-CAand∆1-CA+(BI). ∆1-CAisactuallynotmuchstrongerthanΠ1-CA,
2 2 2 1
the difference being that the latter theory allows one to carry out iterated hyper-
jumpsoflength< ω whiletheformerallowsonetocarryoutiteratedhyperjumps
1Recallthatthesalientfeatureofadmissiblesetsisthattheyaremodelsof∆0collectionandthat∆0
collectionisequivalenttoΣreflectiononthebasisoftheotheraxiomsofKP(see[3]). Further-
more,admissiblesetsoftheformLαalsosatisfyΠ2reflection.
InvestigationsofSubsystemsofSecondOrderArithmeticandSetTheory 3
oflength<ε . Thejumpfrom∆1-CAto∆1-CA+(BI)isindeedenormous. By
0 2 2
comparison,eventheascentfromΠ1-CAto∆1-CA+BR(withBRreferringto
1 2
theBarrule)isratherbenign. Togetanappreciationforthedifferenceonemight
also point out that all hitherto investigated subsystems of second order arithmetic
in the range from Π1-CA to ∆1-CA+BR can be reduced (as far as strength
1 0 2
isconcerned)tofirstordertheoriesofiteratedinductivedefinitions.2 Thetheories
investigatedherearebeyondthatlevel. Amongtheprinciplesonwhichthesethe-
ories are based one finds autonomously iterated positive and monotone inductive
definitions,Π1 transfiniterecursion,∆1 transfiniterecursion,transfinitelyiterated
1 2
Π1dependentchoices,extendedBarrulesforprovablydefinablewell-orderingsas
1
well as their set-theoretic counterparts which are based on extensions of Kripke-
Platek set theory. This first part introduces all the principles and theories. It pro-
videslowerboundsfortheirstrengthmeasuredbytheamountofprovabletransfi-
niteinduction. Inotherwords,itdetermineslowerboundsfortheirproof-theoretic
ordinalswhichareexpressedbymeansofordinalrepresentationsystems. Thesec-
ondpartofthepaperwillbeconcernedwithordinalanalysis. Itwillshowthatthe
lowerboundsestablishedinthepresentpaperareindeedsharp,therebyproviding
theproof-theoreticordinals. Alltheresultswereobtainedmorethen20yearsago
(inGerman)intheauthor’sPhDthesis[43]buthaveneverbeenpublishedbefore,
though the thesis received a review (MR 91m#03062). I always thought that the
results in my thesis were worth publishing but in the past I never seemed to have
enough time to sit down for six weeks and type the entire PhD thesis again. The
thesis was produced by the now obsolete word processing system “Signum” and
it was also written in German. Over the past 20 years or so academic life has
changedinthattime,e.g. fordoingresearch,hasbecomealuxurygood. Iwould
liketothankAndreasWeiermannfornudgingmeagainandagaintopublishit.
Zueignung
Denkreatu¨rlichenFreundenBobby,HonkyTonk,SchnuffiundMarlenegewidmet.
Outline of the paper
In the following I give a brief outline of the contents of this paper. It is roughly
dividedintotwochapters. Thefirstchapter,entitled“THEORIES”,introducesthe
backgroundandpresentsalltheprinciplesandtheoriestobeconsidered. Italsoes-
tablishesinterrelationshipsbetweenvarioustheories. Thesecondchapter,entitled
2Sincethisthesiswaswritten,though,ordinal-theoreticprooftheoryhasprogressedwaybeyondthe
strengthofsuchtheoriesandcannowanalyzesystemsbasedonΠ1comprehension(cf.[53,54,56,
2
57]).
4 MichaelRathjen
“WELL-ORDERINGPROOFS”,introducesanordinalrepresentationsystemand
establishes lower bounds for the proof-theoretic ordinals of most of the theories
considered.
Section 2 carefully defines the basic theory of arithmetical comprehension,
ACA , whichformsthebasisforallsubsystemsofsecondorderarithmetic, and
0
alsothebasicsettheoryBTwhichformsthebasisofallsettheories. Whilesuch
attentiontodetailwillnotmatterthatmuchforthepresentpaperitwillcertainlybe
ofimportancetoitssequelwhichfeaturesproofanalysesofinfinitarycalculi. Sec-
tion 3 introduces second order theories of iterated inductive definitions. Systems
investigated in the literature before used to be first order theories with the induc-
tivelydefinedsetsbeingcapturedviaadditionalpredicatesanditerationsrestricted
toarithmeticalwell-orderings.Goingtosecondordertheoriesallowsonetoformal-
ize iterations along arbitrary well-orderings and also to address the more general
scenarioofmonotoneinductivedefinitions. Section4comparesthetheoriesofthe
foregoingsectionwiththeoriesoftransfiniteΠ1 comprehension. Insection5itis
1
shown that theories of iterated inductive definitions can be canonically translated
intosettheoriesofiteratedadmissibility. Thistranslationexploitsthestructurethe-
oryofΣ -inductivedefinitionsonadmissiblesetsoriginatinginGandy’sTheorem
+
(cf.[3,VI]). Section6featuresiterationsbasedonstrongeroperationssuchas∆1
2
comprehensionandΣ1 dependentchoices. Section7dealswiththeirset-theoretic
2
counterpartswhicharetobefoundincertainformsofΣrecursion.
In order to approach the strength of ∆1-CA+(BI) it is natural to restrict the
2
schema (BI) to specific syntactic complexity classes of formulae, (F-BI). An
alternativeconsistsindirectingtheattentiontothewell-orderingoverwhichtrans-
finiteinductionisallowedinthatonerequiresthemtobeprovablywell-orderedor
parameter-free. This will be the topic of section 8. Particular rules and schemata
consideredincludetheruleBR(impl-Σ1)andtheschemaBI(impl-Σ1):
2 2
∃!X(WO(X) ∧ G[X])
(BR(impl-Σ1))
2 ∀X(WO(X) ∧ G[X]→TI(X,H))
whereG[U]isaΣ1 formula(withoutadditionalparameters),H(a)isanarbitrary
2
L formula,WO(X)expressesthatX isawell-ordering,andTI(X,H)expresses
2
theinstanceoftransfiniteinductionalongX withtheformulaH(a).
(BI(impl-Σ1)) ∃!X(WO(X)∧G[X])→∀X(WO(X)∧G[X]→TI(X,H))
2
whereG[U]isaΣ1 formula(withoutadditionalparameters)andH(a)isanarbi-
2
traryL formula.
2
The rule BR(impl-Σ1) is, on the basis of ∆1-CA, much stronger than the
2 2
rule BR whereas BR(impl-Σ1) is still much weaker than (BI). The difference
2
InvestigationsofSubsystemsofSecondOrderArithmeticandSetTheory 5
in strength between (BI) and BR(impl-Σ1) is of course owed to the fact that the
2
firstisarulewhilethesecondisaschema. Butonecansaysomethingmoreillu-
minativeaboutit. Asitturnsout,BR(impl-Σ1)andBI(impl-Σ1)areofthesame
2 2
strength(onthebasisof∆1-CA),inactualitythetheories∆1-CA+BR(impl-Σ1)
2 2 2
and∆1-CA+BI(impl-Σ1)provethesameΠ1 statements. Thusthemaindiffer-
2 2 1
encebetweenBR(impl-Σ1)and(BI)istobefoundinthepremissofBI(impl-Σ1)
2 2
requiringthewell-orderingtobedescribableviaaΣ1formulawithoutparameters.
2
Section 8 also considers set-theoretic versions of (BR(impl-Σ1) and
2
(BI(impl-Σ1))whichcanbeviewedasformalcounterpartsofthenotionofagood
2
Σ definition of an ordinal/set known from the theory of admissible sets (cf. [3,
1
II.5.13]).
Withthenextsectionweenterthesecondchapterofthispaper. Sections9and
10developanordinalrepresentationsystemOT(Φ)whichwillbesufficientunto
thetaskofexpressingtheproof-theoreticordinalsofalltheforegoingtheories.
Section 11 introduces the technical basis for well-ordering proofs. By a well-
orderingproofinagiventheoryT wemeanaproofformalizableinT whichshows
thatacertainordinalrepresentationsystem(orasubsetofit)iswell-ordered. The
notionofadistinguishedset(ofordinals)(inGerman: ausgezeichneteMenge)will
be central to carrying out well ordering proofs in the various subtheories of sec-
ondorderarithmeticintroducedinearliersections. Atheoryofdistinguishedsets
developedforthispurposeemergedintheworksofBuchholzandPohlers[4,6,7].
Theremainingsections12-15areconcernedwithwell-orderingproofsformost
of the theories featuring in this paper. The lower bounds for the proof-theoretic
ordinalsoftheoriesestablishedinthisarticleturnouttobesharp. Proofsofupper
bounds, though, will be dealt with in the second part of this paper which will be
devoted to ordinal analysis. The final section of this paper provides a list of all
theoriesandtheirproof-theoreticordinals.
I. THEORIES
2 The formal set-up
Thissectionintroducesthelanguagesofsecondorderarithmeticandsettheorywith
thenaturalnumbersasurelements. Moreover,acollectionoftheories,comprehen-
sionandinductionprinciplesformalizedintheselanguageswillbeintroduced.Our
presentationofsecondorderarithmeticisequivalenttothosefoundinthestandard
literature (e.g. [10,65]). The same applies to set theory with urelements, where
6 MichaelRathjen
we follow the standard reference [3]. Slight deviations are of a purely technical
nature,onepeculiaritybeingthatwedefineformulaeinsuchawaythatnegations
occur only in front of prime formulae, another being that function symbols will
be avoided. Instead, we axiomatize number theory by means of relation symbols
representingtheirgraphs.
2.1 The language L
2
ThevocabularyofL consistsoffreenumbervariablesa ,a ,a ,...,boundnum-
2 0 1 2
bervariables x ,x ,x ,..., free set variables U ,U ,U ,..., bound setvariables
0 1 2 0 1 2
X ,X ,X ,...,thelogicalconstants¬,∧,∨,∀,∃,theconstants(numerals)n¯ for
0 1 2
eachn∈N,a1-placerelationsymbolP,three2-placerelationsymbols∈,≡,SUC
andtwo3-placerelationsymbolsADD,MULT.Inaddition,L hasauxiliarysym-
2
bolssuchasparenthesesandcommas.Theintendedinterpretationofthesesymbols
isthefollowing:
1. Number variables range over natural numbers while set variables range over
setsofnaturalnumbers.
2. Theconstantn¯denotesthenthnaturalnumber.
3. Pstandsforanarbitrarysetofnaturalnumbers.
4. ∈denotestheelementhoodrelationbetweennaturalnumbersandsetsofnatu-
ralnumbers.
5. ≡denotestheidentityrelationbetweennaturalnumbers.
6. SUC, ADD, and MULT denote the graphs of the numerical functions n (cid:55)→
n+1,(n,m)(cid:55)→n+m,and(n,m)(cid:55)→n·m,respectively.
ThetermsofL arethefreenumbervariablesandtheconstantsn¯. Assyntactical
2
wealsousea,b,c,d,eforfreenumbervariables,R,S,U,V forfreesetvariables,
u,v,w,x,y,z,i,jforboundnumbervariables,W,X,Y,Zforboundsetvariables,
r,s,tforterms,andA,B,C,D,F,G,H forformulaeofL . IfEisanexpression,
2
τ ,...,τ aredistinctprimitivesymbolsandσ ,...,σ arearbitraryexpressions,
1 n 1 n
then by E(τ ,...,τ | σ ,...,σ ) we mean the expression obtained from A by
1 n 1 n
writing σ in place of τ at each occurrence of τ . If A is a formula of the form
i i i
B(τ ,...,τ | σ ,...,σ ) then this fact will also be expressed (less accurately)
1 n 1 n
bywritingBasB(τ ,...,τ )andAasB(σ ,...,σ ).
1 n 1 n
InvestigationsofSubsystemsofSecondOrderArithmeticandSetTheory 7
Definition 2.1. The atomic formulae of L are of the form (s ≡ t), (s ∈ U),
2
SUC(s,t),P(s),ADD(s,t,r),andMULT(s,t,r).
The L -formulae are defined inductively as follows: If A is an atomic formula
2
thenAand¬AareL -formulae. IfAandBareL -formulaethensoare(A∧B)
2 2
and(A∨B). IfF(a)isanL -formulainwhichtheboundnumbervariablexdoes
2
notoccurthen∀xF(x)and∃xF(x)areL -formulae. IfG(U)isanL -formula
2 2
in which the bound set variable X does not occur then ∀XG(X) and ∃XG(X)
areL -formulae.
2
The negation, ¬A, ofa non–atomicformula A isdefined tobetheformula ob-
tainedfromAby
(i)putting¬infrontanyatomicsubformula,
(ii)replacing∧,∨,∀x,∃x,∀X,∃X by∨,∧,∃x,∀x,∃X,∀X,respectively,and
(iii)droppingdoublenegations.
Asusual,(A→B)abbreviates(¬A∨B)and(A↔B)standsfortheformula
((A → B) ∧ (B → A)). Outer most parentheses will usually be dropped. We
writes (cid:54)= tfor¬(s ≡ t)ands ∈/ U for¬(s ∈ U). Toavoidparenthesiswealso
adopt the conventions that ¬ binds more strongly than the other connectives and
that∧,∨bindmorestronglythan→and↔.
WealsousethefollowingabbreviationswithQ∈{∀,∃}:
Qx ...x F(x ,...,x ) := Qx ...Qx F(x ,...,x ),
1 n 1 n 1 n 1 n
QX ...,X F(X ,...,X ) := QX ...QX F(X ,...,X ),
1 n 1 n 1 n 1 n
and∀x∃!yH(x,y) := ∀x∃yH(x,y) ∧ ∀xyz(H(x,y) ∧ H(x,z)→y ≡z).
Definition 2.2. The formula class Π1 (as well as Σ1) consists of all arithmetical
0 0
L -formulae,i.e.,allformulaewhichdonotcontainsetquantifiers.
2
If F(U) is a Σ1-formula (Π1-formula) then ∀XF(X) (∃XF(X)) is a Π1
n n n+1
(Σ1 )formula.
n+1
2.2 The theory ACA
0
As a base for all theories in the language L we use the theory ACA which in
2 0
additiontotheusualnumber-theoreticaxiomshastheaxiomschemaofarithmetical
comprehensionandaninductionaxiomforsets. Aswewillsubjectthesetheories
toproof-theoretictreatmentweshallpresenttheaxiomatizationofACA inmore
0
detailthanwouldotherwisebenecessary.
Definition2.3. ThemathematicalaxiomsofACA arethefollowing:
0
(i) Equalityaxioms
(G1) ∀x(x≡x).
8 MichaelRathjen
(G2) ∀xy(x≡y →[F(x)↔F(y)])forF(a)inΠ1.
0
(G3) n¯ ≡n¯.
(G4) n¯ (cid:54)≡m¯ ifn,maredifferentnaturalnumbers.
(i) AxiomsforSUC,ADD,MULT.
(SUC1) ∀x∃!ySUC(x,y).
(SUC2) ∀y[y ≡¯0 ∨ ∃xSUC(x,y)].
(SUC3) ∀xyz(SUC(x,z) ∧ SUC(y,z)→x≡y).
(SUC4) SUC(n¯,n+1).
(SUC5) ¬SUC(n¯,m¯)ifn+1(cid:54)=m.
(ADD1) ∀xy∃!zADD(x,y,z).
(ADD2) ∀xADD(x,¯0,x).
(ADD3) ∀uvwxy[ADD(u,v,w) ∧ SUC(v,x) ∧ SUC(w,y) → ADD(u,x,y)].
(ADD4) ADD(n¯,m¯,n+m).
(ADD5) ¬ADD(n¯,m¯,k¯)ifn+m(cid:54)=k.
(MULT1) ∀xy∃!zMULT(x,y,z).
(MULT2) ∀xMULT(x,¯0,¯0).
(MULT3) ∀uvwxy[MULT(u,v,w) ∧ SUC(v,x) ∧ ADD(w,u,y) →
MULT(u,x,y)].
(MULT4) MULT(n¯,m¯,n·m).
(MULT5) ¬MULT(n¯,m¯,k¯)ifn·m(cid:54)=k.
(iii) InductionAxiom
(Ind) ∀X[¯0∈X∧∀xy[SUC(y,x)∧y ∈X →x∈X] → ∀x(x∈X)].
(iv) ArithmeticalComprehension
(ACA) ∃X∀y[y ∈X ↔F(y)]
whereF(a)isΠ1andX doesnotoccurinF(a).
0
InvestigationsofSubsystemsofSecondOrderArithmeticandSetTheory 9
AslogicalrulesandaxiomsforeverytheoryformulatedinthelanguageofL we
2
choosethefollowing:
(L1) AllformulaeofL thatarevalidinpropositionallogic.
2
(L2) The number quantifier axioms ∀xF(x) → F(t) and F(t) → ∃xF(x) for
everyL -formulaF(a)inwhichxdoesnotoccurandeverytermt.
2
(L3) Thesetquantifieraxioms∀XH(X) → H(U)andH(U) → ∃XF(X)for
everyL -formulaH(V)inwhichX doesnotoccurandsetvariableU.
2
(L4) Modusponens: FromAandA→BdeduceB.
(L5) From A → F(a) deduce A → ∀xF(x) and from F(a) → A deduce
∃xF(x) → A providing the free number variable a does not occur in the
conclusionandxdoesnotoccurinF(a).
(L6) From A → H(U) deduce A → ∀XH(X) and from H(U) → A deduce
∃XF(X) → A providing the free set variable U does not occur in the con-
clusionandX doesnotoccurinF(U).
WewriteT (cid:96) AwhenT isatheoryinthelanguageofL andAcanbededuced
2
from T using the axioms of T and any combination of the preceding axioms and
rulesofACA .
0
ByACAwedenotethetheoryACA augmentedbytheschemeofinductionfor
0
allL -formulae:
2
(IND) F(¯0) ∧ ∀xy[SUC(y,x) ∧ F(y)→F(x)] → ∀xF(x)
whereF(a)isanarbitraryformulaofL .
2
ThesublanguageofL withoutsetvariableswillbedenotedbyL .
2 1
2.3 The languages L∗ and L∗(M)
L∗ willbethelanguageofsettheorywiththenaturalnumbersasurelements. L∗
comprisesL andinadditionhasaconstantNforthesetofnaturalnumbers,a1-
1
placepredicatesymbolSetfortheclassofsets,anda1-placepredicatesymbolAd
fortheclassofadmissiblesets. TheintendedinterpretationsofL andL∗ diverge
1
withrespecttothescopesofthequantifiers∀xand∃xwhichinthecaseofL∗ are
viewed as ranging over all sets and urelements. Moreover, L∗ has also bounded
quantifiers(∀x ∈ t)and(∃x ∈ t)whichwillbetreatedasquantifiersintheirown
right.
10 MichaelRathjen
WewillalsohaveuseforanextendedlanguageL∗(M)whichhasaconstantM,
intendedtodenotethesmallestadmissibleset.
ThetermsofL∗(L∗(M))consistofthefreevariablesandtheconstantsn¯andN
(andM).
The atomic formulae of L∗ (L∗(M)) consists of all strings of symbols of the
forms(s≡t),(s∈t),P(t),SUC(s,t),ADD(s,t,r),MULT(s,t,r),Ad(s),and
Set(s),wheres,t,rarearbitrarytermsofL∗(L∗(M)).
Definition2.4. L∗-formulaeareinductivelydefinedasfollows:
1. Aand¬AareL∗-formulaewheneverAisanatomicL∗-formula.
2. IfAandBareL∗-formulaesoare(A∧B)and(A∨B).
3. IfF(a)isanL∗-formulainwhichxdoesnotappearandtisanL∗ termthen
∀xF(x),∃xF(x),(∀x∈t)F(x),and(∃x∈t)F(x)areL∗-formulae.
L∗(M)-formulaearedefinedinasimilarvein. Thenegation,¬A,ofaformulaAis
definedasinDefinition2.1,butextendedbytheclauses¬(∀x∈t)F(x) := (∃x∈
t)¬F(x)and¬(∃x∈t)F(x) := (∀x∈t)¬F(x)fortheboundedquantifiers.
Definition 2.5 (Translating L into L∗). Let U∗ := a , X∗ := x , a∗ :=
2 i 2·i i 2·i+2 i
a ,x∗ :=x ,andn¯∗ :=n¯.
2·i+1 i 2·i+1
ToeveryL -formulaAweassignanL∗-formulaA∗ asfollows: Replaceevery
2
free variable X in A by X∗. Replace number quantifiers ∀x...x... and ∃x...x...
by (∀x∗ ∈ N)...x∗... and (∃x∗ ∈ N)...x∗..., respectively. Replace set quanti-
fiers ∀X...X... and ∃X...X... by ∀X∗[Set(X∗) ∧ X∗ ⊆ N → ...X∗...] and
∃X∗[Set(X∗) ∧ X∗ ⊆ N ∧ ...X∗...], respectively, where X∗ ⊆ N stands for
∀u[u ∈ X∗ → u ∈ N]. The translation A (cid:55)→ A∗ provides an embedding of L
2
into L∗, preserving the intended interpretations. In what follows we view L as
2
sublanguageofL∗,formallyfixedbythenaturaltranslation ∗.
2.4 Syntactic classifications
Definition2.6. The∆ formulaearethesmallestcollectionofL∗ formulaecon-
0
taining all quantifier-free formulae closed under ¬,∧,∨ and bounded quantifica-
tion. Spelled out in detail the last closure clause means that if F(a) is ∆ , t is
0
a term and x is a bound variable not occurring in F(a) then (∃x ∈ t)F(x) and
(∀x∈t)F(x)are∆ .
0
L∗formulaewhichare∆ oroftheform∃xF(x)withF(a)∆ aresaidtobeΣ .
0 0 1
Dually,aformulaisΠ ifitisthenegationofaΣ formula.
1 1