The Baire Category Theorem in Weak Subsystems of Second-Order Arithmetic Author(s): Douglas K. Brown and Stephen G. Simpson Reviewed work(s): Source: The Journal of Symbolic Logic, Vol. 58, No. 2 (Jun., 1993), pp. 557-578 Published by: Association for Symbolic Logic Stable URL: http://www.jstor.org/stable/2275219 . Accessed: 05/04/2012 09:29
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http://www.jstor.org THE JOURNAL OF SYMBOLIC LoGic Volume58, Number2, June 1993
THE BAIRE CATEGORY THEOREM IN WEAK SUBSYSTEMS OF SECOND-ORDER ARITHMETIC
DOUGLAS K. BROWN AND STEPHEN G. SIMPSON
Abstract.Working within weak subsystemsof second-orderarithmetic Z2 we considertwo versionsof theBaire Category theorem which are notequivalent over the base systemRCAo. We showthat one version (B.C.T.I) is provablein RCAo whilethe second version (B.C.T.II) requiresa strongersystem. We introduce two new subsystemsof Z2, whichwe call RCA' and WKL', and show that RCA' sufficesto prove B.C.T.II. Some model theoryof WKL' and its importancein viewof Hilbert'sprogram is discussed,as well as applicationsof our resultsto functionalanalysis.
?0. Introduction.This paper consistsof some of thematerial contained in [2], whichis concernedwith the development of the basic definitionsand theoremsof functionalanalysis within second-order arithmetic, Z2. Such studiestake place withina broaderprogram initiated by Friedmanand carriedforward by Friedman, Simpson,and others.The goal of thisprogram is to examinethe Main Question: Whichset existenceaxioms are neededto provethe theoremsof "ordinarymathe- matics?"An expositionof themeaning of "ordinarymathematics" can be foundin [22, 21, 2] -for the purposesof thispaper it sufficesto note that the theoryof completeseparable metric spaces is an exampleof ordinarymathematics. The languageof second-orderarithmetic is a two sortedlanguage with number variablesi, j] k,m, n, ... and set variablesX, Y,Z..... Numericalterms are builtup as usual fromnumber variables, constant symbols 0 and 1,and thebinary operations of addition(+) and multiplication(.). Atomicformulas are t1 = t2, t1 < t2, and t1 e X wheret1 and t2 are numericalterms. Formulas are builtup as usual from atomic formulas by means of propositional connective A, V, -, , -+, number quantifiersVn and 3n, and set quantifiersVX and ]X. The formalsystem Z2 includes the orderedsemiring axioms for N, +, *,0, 1, < as well as the inductionaxiom
(OeX A Vn(neX-*n + 1 eX))-*Vn(neX) and the comprehension scheme
3XVn(n E X *-(p(n)), where (p(n) is any formula in which X does not occur freely.
ReceivedSeptember 16, 1991. Researchof thesecond author was partiallysupported by NSF grantDMS-8701481.
?0 1993, Association for Symbolic Logic 0022-481 2/93/5802-0009/$03.20
557 558 DOUGLAS K. BROWN AND STEPHEN G. SIMPSON
In thecourse of studyingthe Main Questionit has been notedthat a greatdeal of ordinarymathematics may actuallybe done in variousweak subsystemsof Z2 [1,2,8,22]. In this paper we are primarilyconcerned with the followingthree subsystems: RCAo. Here the acronymRCA stands for recursivecomprehension axiom. Roughlyspeaking, the axioms of RCAo are onlystrong enough to provethe exis- tenceof recursivesets (though they do not ruleout the existenceof nonrecursive sets).As weak as thissystem is, it is strongenough to provesome of theelementary factsabout countablealgebraic structures [8] and continuousfunctions of a real variable [21, 22]. The axioms of RCAo consistof the orderedsemiring axioms togetherwith the schemes of Z1?induction and A ?comprehension. WKLo. This systemconsists of RCAo plus a furtheraxiom known as Weak Konig'slemma which states that every infinite {0, 1}-treehas a path.This systemis veryweak fromthe viewpoint of mathematicallogic in thatthe first-order part of WKLo is the same as that of RCAO, viz., Z1?induction (this resultis due to Harrington;for a proofsee [21]). Furthermore,WKLo is conservativeover Prim- itiveRecursive Arithmetic (PRA) withrespect to Ho sentences[21]. On the other hand, fromthe mathematicalpoint of view,WKLo is verypowerful. It is strong enoughto prove a greatmany theorems of ordinarymathematics which are not recursivelytrue and hencenot provablein RCAO. Includedin thiscategory are the Heine-Borelcovering lemma [2, 6, 21], theprime ideal theoremfor countable com- mutativerings [8], the maximumprinciple for continuous functions on a closed bounded interval[21], and the local existencetheorem for solutions of ordinary differentialequations [22]. The above remarkshave importantimplications in the foundationsof mathe- maticsvis 'a vis Hilbert'sprogram. Tait [24] has made a strongcase forthe identi- ficationof Hilbert'snotion of finitismwith the formal system PRA and pointedout thatthe primary concern of thisfinitism is theprovability of certainHo sentences. Thus a fullrealization of Hilbert'sdesire to justifythe use of infinitisticmathe- maticswould consistof developinga formalinfinitistic system whose consistency was provablein PRA, as it would thenfollow that any Io sentenceprovable in thisinfinitistic system was in factprovable in PRA, i.e.,finitistically. The infinite objectsof thelarger system would thenbe justifiableas devicesto be used to prove theoremsabout noninfiniteobjects and have theseresults be finitisticallyaccept- able. Of course Gddel's work dashed any hopes for such a full realizationof Hilbert'sprogram, but partialrealizations are made possible by considering,not systemswhose consistencyis provablein PRA, but ones whichare conservative over PRA with respectto I7H sentences.The fact noted above that WKLo is conservativeover PRA withrespect to I7Hsentences thus gives a slightlystronger resultand indicatesthat the theoremsof ordinarymathematics mentioned above providepartial realizations of Hilbert'sProgram in thissense. For a fullerexposi- tionof thistheme see [19]. ACAo. The axioms of ACAo are the same as those of Z2 expectthat the com- prehensionscheme is restrictedto arithmeticalformulas N(n) in whichX does not occur freely.ACAo permitsa smooththeory of sequentialconvergence [5, 6, 22] and isolates the same portionof mathematicalpractice which was identifiedas THE BAIRE CATEGORY THEOREM 559
"predicativeanalysis" by Weylin Das Kontinuum[25]. For moredetails on all of thesesystems see Simpson[21]. Investigationsinto the Main Question have also revealedthe followingMain Theme:very often, if a theoremof ordinarymathematics is provedfrom the "right" set existenceaxioms, the statementof thattheorem will be provablyequivalent to thoseaxioms over some weak base system(for us thisis RCAO).This themeis known as ReverseMathematics and is exhibitedin such worksas [2, 3, 7, 8, 9, 21, 26]. This typeof "reversal",proving that a set of axioms followsfrom the statementof the theorem,together with the more usual proofof the theoremfrom the same set of axiomsprovides the precise knowledge that these axioms are in some senseneces- saryto prove a theoremof ordinarymathematics. In such a case we have a very completeanswer to the Main Question. In [2] our primaryaim was to examinesome of the fundamentaltheorems of functionalanalysis on separable spaces in the contextof ReverseMathematics. Amongthese theorems were the Banach-Steinhaustheorem and the Open Map- pingand Closed Graphtheorems. In standardtexts on real analysisthese theorems are usuallyproved using the Baire Categorytheorem. In the settingof weak sub- systemsof Z2, however,the version of theBaire Category theorem needed to prove theBanach-Steinhaus theorem and theversion needed to provethe Open Mapping and Closed Graph theoremsare not thesame. This resultis due to two notionsof a closed subsetof a completeseparable metric space whichare not equivalentin weak subsystems.These notions are discussedin detail in [2] and [3] and are summarizedin ? 1 below,along withthe necessary technical definitions and results about completeseparable metric spaces and theirtopology in weak subsystemsof Z2- In ?2 we presentthe two versionsof the Baire Categorytheorem referred to above and show thatthe first version is easilyproved in RCAo. In ?3 we consider theaxiomatic strength needed to provethe second version of thetheorem. In ?4 we introduce two new subsystems of Z2, which we call RCA0 and WKL0, and show thatthe systemRCA' sufficesto provethe second versionof the Baire Category theorem.In ?5 we considerthe Open Mappingand Closed Graphtheorems. Finally, in ?6, we considersome model theoryof WKL' and show thatWKL' is conser- vativeover PRA withrespect to HO sentences.
?1. Metricspaces. WithinRCAo we definea (code fora) completeseparable metricspace to consistof a set A c N togetherwith a functiond: A x A -k R such thatfor all a, b,c E A: (i) d(a, a) = 0, (ii) d(a,b) = d(b,a), (iii) d(a, c) < d(a,b) + d(b,c). Now let(A, d) be a code fora completeseparable metric space, as above. We define, again withinRCAO, a pointin thecompletion A to be a functionf: N -+ A such that VnVi[d(f(n),f(n + i)) < 2]. The idea here is that (A,d) is a code forthe completeseparable metricspace A consistingof all such points.For example,R = Q underthe usual psuedometric. Of course A does not formallyexist withinRCAo. A point f: N -+ A will be 560 DOUGLAS K. BROWN AND STEPHEN G. SIMPSON denoted by x =
d(
KCnk: k e N> = d(an+3 bn+3)
We willsometimes use d(x,y)n to denotecn, n. Whereno confusionwill result, d will be used to denote both d and d. We embed A into A by identifyingthe element a e A with the point xa e A defined by xa =
00 A = U (Ao x x Am)= {
d(
== fa. if i < m, i ci otherwise and J if i < =bi n, otherwise. = {c, We can then prove within RCAo the following facts: (i) A is a complete separable metric space; (ii) the points of A can be identifiedwith the sequences
d(
00 A = H Ai. i=o THE BAIRE CATEGORY THEOREM 561
In particular,we have withinRCAo the Cantorspace 00 2N= {0,1}N = H{O,1} i=o and theBaire space
NN= HN. i=O The pointsof theCantor space and theBaire space can be identifiedwith functions f: N -+ {0, 1} and f: N -+ N respectively. WithinRCAo we definea code foran openball in A, a completeseparable metric space withcode (A,d), to be an orderedpair (x,8) wherex E A and 8 E R' (the positivereals). We say thata pointy E A is an elementof theball (x,8) ifd(x, y) < e. An open ball is a basic open set if it is of the form(a, r) wherea E A and r E Q+ (the positiverationals). A code foran open set U is a sequence of basic open sets <(an,re): n E N>. We say thata pointx E A belongsto U if thereis a basic open set (a, r) E U such thatx is an elementof (a, r).We willdenote an elementx of an open ball (y,8) by writingx E (y,8) and a pointx belongingto an open set U by writing xe U. One naturaldefinition of a closed setis thatit is thecomplement of an open set. Thus we definea (code for a) closed set C to be a sequence of basic open sets <(an,rn): n E N> and say that a point x in a completeseparable metricspace A belongsto C ifd(an, x) > rnfor all n E N. Note thata code fora closed set may also be regardedas a code forthe open set whichis its complement.It is theneasy to prove,over RCAO, such standard results as thecountable union of open setsis open and the countableintersection of closed sets is closed [3]. However,since every closed subsetof a completeseparable metric space is itselfa completeseparable metricspace, a secondnatural definition of a closed set is thatit is theclosure of a countableset of points.We thereforedefine a (code fora) separablyclosed set S to consistof a sequenceS =
(ii) If C is a closedsubset of a completeseparable metric space thenC is a separably closedset. PROOF. See [3]. The system ll-CAO is Ill comprehension(i.e., the compre- hensionscheme is restrictedto IH1formulas) and is muchstronger than the three systemsmentioned above. Thus foran arbitrarycomplete separable metricspace the equivalenceof the two notionsof closed set requires,and is equivalentto, HI1-CA0.In the settingof compactspaces thisequivalence can be provedin theweaker system ACAo [3]. It also followsthat, in termsof separablyopen and closed sets,the standardresults on countable unions and intersectionsreferred to above require,and again are equivalentto, Hll-CA0 over RCAo [3]. Let A and B be completeseparable metric spaces withcodes A and B, respec- tively.Within RCAo we definea (code fora) continuouspartial function from A to B to be a functionP: N - A x Q+ x B x Q+ such thatfor all m,n E N, a, a' E A, b,b' e B and r,rt, sst eQ: (i) ?(m) = (a, r,b, s) A ?(n) = (a', r',b', s') -+ d(b,b') < s + St; (ii) ?P(m)= (a, r,b, s) A (b,s) < (b',st) -k[i(k) = (a, r,b', s')]; (iii) ?(m) = (a, r,b, s) A (a', r') < (a, r) > ]k[0(k) = (a', r',b, s)]. Here (a, r) < (b,s) means r < s - d(a, b). We write(a, r,b, s) E dP if d>(n)= (a, r,b, s) forsome n E N. The idea here is that ( encodes a continuouspartial function 1 fromA to B. Intuitively,(a, r,b, s) E ( is a piece of informationto the effectthat d(o(x), b) < s wheneverd(x, a) < r. A pointx E A is said to belongto thedomain of 0 if,for all E > 0, thereexists a (a, r,b, s) E qPsuch that d(x, a) < r and s < e.If x E A is in thedomain of 4, we define+(x) to be thepoint y e B such thatd(y, b) < s for all (a, r,b, s) E ( withd(x, a) < r. We can prove,within RCAO, that y existsby using thecode ( and the u-operator.Note thaty = +(x) is unique up to theequality of pointsin a completeseparable metric space as definedabove. We definea (code fora) separableBanach space to be a set A c N togetherwith operations+: A x A - A, A x A -A, and *:Q x A - A and a distinguished element0 E A such that
lXII =
wherem E N is theleast such that((Ilaoll)o + 1q01+ 2)2-m< 1. Thus A enjoysthe usual propertiesof a normedvector space overR. Let A and B be separableBanach spaces. We definea continuouslinear opera- tor fromA to B to be a totallydefined continuous function 4: A -+ B such that 0(/x + fly)= oc4(x)+ fl,(y) forall x,y E A and a, ftE R. We definea (code fora) boundedlinear operator from A to B to be a functionF: A >+B such that: (i) F(q1a1 + q2a2) = q1F(a1) + q2F(a2) forall q1,q2 E Q and a,, a2 E A; (ii) thereexists a real numbera such thatIIF(a)II < aIIaII forall a E A. For F and a as above and x =
IIy- F(an)II < 2-n,,. Thus IIF(x)II< aIIxII for all x E A. We writeF: A -+ B to denotethis state of affairs. If a E R is such that IIF(x)II< aIIxII forall x E A, we writeIIFiI < a. Specializing to thecase B = R we obtaina boundedlinear functional on A. We have thefollow- ing standardresult from Banach space theoryrelating the two typesof operators definedabove. THEOREM 1.4 (RCAO). Givena continuouslinear operator 0: A -+ B, thereexists a boundedlinear operator F: A -> B such thatF(x) = +(x) for all x E A. The con- versealso holds. PROOF. See [4]. Specializingthe definitions above to thecontext of separableBanach spaces,we definea closedlinear subspace C of a separableBanach space A to be a closedsubset C of A whichis also a linearsubset of A. On theother hand we definea separably closed linearsubspace S of A to be a separablyclosed subset of A whichis also linear.
THEOREM 1.5(RCAO). SupposeA is a separableBanach space andS =
?2. The Baire categorytheorem. In standard metricspace theorythe Baire Categorytheorem can be statedas follows:Given a completemetric space X and a sequence
<
= nCmn = {x IVm(IITm(x)ll< n)}. Cn mEN
Then Cn is closed [2]. Claim A = UN Cn Indeed, fix x e A. By hypothesis we have SUPNII Tn(X)II< cx, say SUPNIIITn(X)Il < m. Then for all n e N, x c Cn m and hence THE BAIRE CATEGORY THEOREM 565
x E 2neN Cn m = Cm. Since x was arbitrary,it followsthat A = UNCQ. Now, from Corollary2.2 it followsthat not all of the Cn'sare nowheredense; i.e.,for some no e N. Cnocontains a basic open set (aO,ro). Fix x E A such thatIlxii # 0, and con- siderao + rOx/21x11. We have
(ao x r= + 2llXlI 2llxl1 so (aO + rox/211xI) E (aO,ro). Also, for any n E N, ro I T ro I1x11 || | ( ao? 2-lXox~ ) 211IIT(x) = 2lXll x) *a< +n +
Then, since ao and ao + rox/211xIIboth belong to (aO,ro) which in turn is a mem- ber of Cn0I)In = {x IVmI Tm(x)II< ro}, it follows that (rO/2I1x In(x)II < 2no. Thus, for all n E N and x E A, IIlTn(x)lI< (4no/ro)IIxII(we assumed IIxHI= 0, but clearly this also holds if IlxII = 0). Taking any M E N with M ? 4no/ro completes the proof of the theorem. Thus we see that another important theorem of ordinary mathematics can be proved in the weak base system RCAo. However, other interestingconsequences of the Baire Category theorem require the strongerversion B.C.T.II, which we turn to now.
?3. B.C.T.II. We now consider the axiomatic strengthneeded to prove B.C.T.II. As an initial approximation we have the following. THEOREM 3.1 (ACAO). B.C.T.II holds for any complete separable metric space A.
PROOF. Let <
= M U lXmk M Eco 566 DOUGLAS K. BROWN AND STEPHEN G. SIMPSON
Fix m q w).We claim
Rec U {Xm,k: k e col, meco
whereeach
?4. The systemsRCAo and WKLo. In thissection we introducea subsystem of Z2 whichsuffices to prove B.C.T.II; we call the systemRCAo. The axioms of RCAo are those of RCAo plus the followingscheme (*): let U, T range over 2 X rangeover 2N, and let p be any arithmeticformula; then
(*) VnVu3Tz(z D A p(n,z)) -* 3XVn3k(p(n,X[k])). The idea hereis thatgiven a sequence of arithmeticallydefined dense subsetsof 2
The idea hereis that7t(a) liststhe numberof zeros occurringbetween successive ones and the lengthof 7t(a) is the numberof ones occurringin a. For example,if a = <010011 > then 7t(a) = < 1,2, 0>. THEOREM4.1 (RCA+). Let C*, z* range over N VnVu*3]*(z*D U* A f(n,z*)). Thenthere exists an x E NN suchthat Vni(t(n, x[i])). PROOF. For n E N and - E 2 We claim that VnVo3z(zD ff A f(n,p(n, z)). Indeed, fixn E N and a E 2 x = K4(y[i]): i E N> and note thatx existswithin RCAo (giveny). By note (3) above, x E NN and, by definition, (p(n, Y [in ] )=> 0! (n, 7 ( y[in] ) ThereforeVn3io/(n, x[i]), as desired. We now show thatB.C.T.II holds forany completeseparable metric space A. Given any such space, let A be thecode forA and (p(n,r*) _=3iVkVy*[y* D -* d((T*Y*),xnk) 2 2i] 568 DOUGLAS K. BROWN AND STEPHEN G. SIMPSON We claim thatVnVu*3r*[E* D* A p(n,*)]. Indeed,fix n and u*. Let lh(,r*^ z_= 7r(T0* z*) = Therefore d(7r(zo*X*),xk) ? r -2- 2 r/2 ? 2-i and so p(n,z*) holds and the claim follows. Now apply Theorem 4.1 to obtain y e NN such that Vn3ky(n,y[k]) and let x = lim~ ir z(zoy[i]). Then: (1) x e A: For i < j we have d( d(7yr])(T *A Yri])) * ? d(ir(zoy[i]), 1(0 y[j])io + i) + d(7t(To y[i])io + i,t(zo y[j])i0 + D) + d(7(To*^Y[j])io + j, (z*y[j])) < 2-('o+') + 2-(io+i) + 2-(io+j) < 3 2-('O+') < 2- (sinceio ? 2). That x e A now follows fromthe completeness of A [2]. (2) x e On for all n e N. Fix n e N and let kn be such that p(n,y[kn]). Then 3inVj? knVkd(7r(T*Ay[]], Xnk) ? 2-in. Choose j > max(kn, in). Then forall k we have 2-',, <- d(7r(T*Ayj < d(t(zr*Ay[E],x) + d(xXn,k) < 2 - j + d(x, Xn,k). Thus d(x,Xn,k)> 2 - 2-' > O, so xe O?n,as claimed. THE BAIRECATEGORY THEOREM 569 (3) x e (ao, ro): We have d(x,ao) < d(x,7t(z*y[io])) + d(t(zr*y[iO]),aO) < 2`0 + 2`o (since t(z4o*y[i0])begins with a sequenceof ao's of lengthio) < ro. This completesthe proof of thetheorem. Thus we see that B.C.T.II holds in RCA+. In factRCA+ provesthe following strongerresult: if <<(afl,k,rflk): k e N>: n e N> is an arithmeticallydefined sequence of sequencesof basic open sets in a completeseparable metric space A such that foreach n the open set Un= <(afnk,rnfk):keN> is dense,then nN Un is dense (the resultsof thissection only need a Z3 sequence <<(an,k, rnfk): k E N>: n E N>). Ap- parentlythis is the strongestversion of the Baire Categorytheorem provable in RCA+. ?5. The Open Mappingand Closed Graph theorems. We now applyTheorem 4.2. In whatfollows let A and B be separableBanach spaces. For n e N, let Sn = {a I 11all< 2}. Note thatif T: A -+ B then T(Sn) = (O,ro) c T(S1) -bobc 2T(S1) c T(So). More generally,by linearity,(0, r0/2n) c T(Sn) for any n e N. We claim that if IIYII< r0/2then there is an x e A such that IlxII< 1 and T(x) = y. Indeed,suppose y e (0,r0/2). Then y e T(S1), so thereis an a1 e S1 with11 Y- T(a1)I < ro/4.Sup- pose thatwe continuein thisfashion to choose a1,.. ., ansuch thatai e Si, 1 < i < n, and IIY- Z 1 T(ai)iI < ro/2n+. Then y - Z%1 T(aj) e T(Sn+1) and hence there is an an+1 e Sn+1 with n - Y T(ai) - T(an+ 1) < ro/2n+2. Applying Z1 induction we then obtain a sequence Zi a - a1 = a1 < I1aj+111+ + IHaj+n11 < 2 (j 1) + + 2-(j+n) < 2-j Thus x E A. We also have n jjxjj = lim ||Eai| < lim(2-1 + * + 2 -n) = 1. n-oo i=n1 o Thus lxii< 1. Since I =-I 1 T(aj)ii < r0/2'n+ 1for all n E N, it followsthat T(x) = y, as desired. Thus, withinRCA+, the image of an open unitball in A containsa ball about zero in B. Note thatRCAo was needed only to show that T(S1) containsa basic open set; the remainderof the proof requiresonly RCAo. We are now able to provethe Open Mapping theoremfor separable Banach spaces. THEOREM 5.2 (RCA ). If T is a boundedlinear operator from A ontoB and U is an opensubset of A, thenthere exists an openset V in B suchthat y E V if y = T(x) for somex e U. PROOF. WithLemma 5.1 in hand we can carryout the usual proof usingonly theaxioms of RCAo. See [2]. Thus theOpen Mapping theoremfor open setsis provablein RCA&. Two ques- tionsremain unanswered (as of thiswriting): (1) Can theOpen Mapping theoremfor open setsbe provedin a systemweaker thanRCAo? (2) Whataxioms are neededto provethe Open Mappingtheorem for separably open sets? Since ACAo provesthat every separably open set is open,it is immediatefrom Theorem5.2 that ACAo proves the Open Mapping theoremfor separably open sets.Thus we have a partialanswer to (2). From Theorem5.2 we may now obtain the Bounded Inversetheorem for sepa- rable Banach spaces. THEOREM 5.3 (RCA&). Suppose T is a boundedlinear operator mapping A one- one ontoB. Then T' 1 is bounded. PROOF. Here again the usual proof can be carried out once we have Theo- rem5.2 [2]. We concludethis section by consideringthe Closed Graph theorem.Suppose A and B are codes forseparable Banach spaces withnorms 11I. 1and 11ilB respectively. Considerthe set A x B withoperations: (i) (a,,b1) + (a2,b2)= (a, + a2, b, + b2), (ii) q(a1, b1) = (qa1,qb1), wherea1,a2ce A, bl,b2e B, and q e Q. For a e A and be B define 11(a,b)ii = ||a1lA + ljblIB- It is theneasy to see thatthe completion of A x B underthis norm yields the sep- arable Banach space A x B, i.e., the set of pairs (x,y) wherex E A and y E B. If T: A -* B is a linearoperator (not necessarilybounded), we definethe graphof T THE BAIRE CATEGORY THEOREM 571 to be the"set" of pairs(x, y) in A x B such thatT(x) = y. Note thatthe graph of T does not formallyexist within RCAo. THEOREM5.4 (RCA+). SupposeT: A -> B is a linearoperator such that the graph of T is separablyclosed in A - B. Then T is continuous. PROOF.Let GT = <(xn,,y): n E N> code the graphof T. Then GT is a separable Banach space by Theorem 1.5. Define P: GT-ALATAby P((xn,y)) = xn. Then P is linearand IIP((xn,Yn))I|i = iiXnIIA < IIXnIA + IIYniiB1 = 1(XnYn)I A X so P is bounded.Thus by Theorem5.3, P-': A GTis bounded.Therefore there is an M such thatfor all x E A iiP '(x)iIG < MJx IXA ll(x,T(x)llG- < MIIxIIi => l|xiii+ JJT(x)JJs< MIIxIIJ => |IT(x)IIB< (M - 1)11xIIA. Thus T is boundedand thereforecontinuous. In Theorem5.4 we have the Closed Graph theoremfor linear operators with separablyclosed graphs.It is an open questionas to whataxioms are necessaryto provethe Closed Graph theoremfor linear operators with graphs which are closed sets.We obtain a partialanswer by notingthat, since HI-CAO provesthat every closed set is separablyclosed, it is immediatefrom Theorem 5.4 that no system strongerthan HI-CAo is neededto obtainthis result. ?6. Model theoryof WKL'. In thissection we demonstratethe existence of a modelof WKLo and,in thecourse of this,establish that WKLo is logicallya weak subsystemof axiomsin thatit is conservativeover PrimitiveRecursive Arithmetic (PRA) withrespect to HO sentences. Let L2 be thelanguage of second-orderarithmetic. A modelfor L2 is an ordered 7-tuple M = M = Since L'7induction in M impliesL'? collectionin M' [11] it followsthat we have Jo comprehensionin M' (see [23]). Thus we need onlyshow thatM' satisfiesZ? induction.Suppose cp(n)is a L'? formulain M', say cp(n)=-3k(n, k,X), where0 is Z? and X is a completelist of the parametersfrom SM' occurringin 0. SinceX c JO-DefM, thereis a Z? formulaO_ containing only parameters from IMI and SM such thata E X iffO(a) and, over M', O(n,k, X) - (n,k, 04(a)) (see [23]). Thus,over M', cp(n)=30(n, k,0(a)), i.e., cpis equivalentover M' to a Z? formulacontaining only parametersfrom MI u SM. Z? inductionin M' thenfollows easily from Z? inductionin M, for: M' 1= [qp(O) A Vn(qp(n)-* p(n + 1))] O(d)) + 1,k, O(d))] = M 1= [3kH(O,k, O1(d)) A Vn(3kO(n,k, ak(n = M l= VnikO(n,k, H_(a)) = M' I= Vn3kO(n,k, X) = M' I= Vn~p(n). Thus M' satisfiesZ1? induction and so M' is a model of RCAo. Let M be any countablemodel (i.e.,I MI is countable)of RCAO. Let 2"N be the setof M-finitesequences of zerosand ones. A setD C 2MN is denseif for all a c2e thereis a z e D such thatz D a. A set D C 2MN is definableif D is definableover M withparameters from IMI u SM. A set X c IMI is M-genericif forall definable densesets D C 2MN thereis a a e D such thata e X. As usual,if X is M-genericwe can considerX to be an infinitesequence of zeros and ones,so X e 2' and we will writea( c X fora e X. LEMMA 6.2. Let M be a countablemodel of RCAo. Thereis a modelM' of RCAo suchthat: (i) M is a submodelof M'; (ii) AIl= IM'I; (iii) thereis an X e SM' suchthat X intersectsall M-definabledense sets D. PROOF. Let X be M-generic,and let M[X] = ClearlyM is a submodelof M[X] and IMI = IM[X]I. Since theintegers of M[X] are those of M, it followsthat M[X] satisfiesthe basic axioms of RCAO. Claim M[X] satisfiesZ?? induction. Indeed suppose M[X] I= (p(O)A Vn(cp(n)-* (p(n+ 1)), wherecp(n) is Z??in M[X]. Writecp(n) as 3kO(n,X[k]) whereO(n, k), a e 2N, is z0 withparameters from IMI u SM only(see [23]). For each n e IMI, defineE c 2`N by a e E iffVt D uVk < lh(z) - 0(0,[k]) v 3m < n[3k < lh(a)O(m,a[k]) A Vt D aVk < lh(z) - H(m+ 1,[k])]. THE BAIRE CATEGORY THEOREM 573 Let D c 2 Then D is M-definable.Fix a e 2"N. Eitherthere is a z e E c D such thatz Df a or no such - exists,in whichcase a e D. In eitherevent it followsthat D is dense.Let a0 e D n X, and suppose a0 e E. Then either (i) Vt -aoVk < lh(z) - 0(0,[k]) => Vk - 0(O,X[k]), contradictingour assump- tionthat M[x] 1= cp(O),or (ii) 3m < n[3k < lh(a0)O(m,ao[k])A Vt- aoVk < lh(z) OH(m + 1,[k])]. Fix m as in case (ii). Then 3k < 1h(q0)0(m, ao[k]) 3k(m, X[k]) 3kO(m+ 1,X[k]) (inductionhypothesis) -> 3, : ao3k < lh()O(m + 1,[k]), whichcontradicts a0 e E. Thus a0 s E so, by the definitionof D, if z D a0 then z s E and hence (1) Vm[3k< lh(T)O(m, T[k]) -* D z 3k < lh(y)O(m + 1,y[k])]. Since a0 s E, we have 3DO D ao3k < lh(zo)O(0,oz[k]). Now form e MI suppose we have Tm D ao such that 3k < lh(T,) H(M, ,,[k]). Then by(1) thereis a Tm+ 1D Tm suchthat 3k < lh(cm+1) O(i,Tm+ 1[k]). Thus we have 3D ao3k < lh(c)O(O,[k]) A Vn(3z- ao3k < lh(z)O(n,z[k]) - 3 ' ao3k < lh()O(n + 1,[k]). It followsfrom Z1? induction in M that (2) Vn3z- a03k < lh(c)O(n,z[k]), and thisin factholds witha0 replacedby any a D a0. Now, foreach n e IMI, define nc 2' by a e Dn iff (a is incompatiblewith a0) v 3k < lh(a) O(n,a[k]). Then Dnis M-definablefor each n e Fix e If is incompatiblewith a0 IMI. a 2'.N a thena e Dn. Otherwiseeither: (i) a( c a0, in whichcase by (2) 3, D a0 D a such that 3k < lh(T)O(n,-[k]) so thatT e Dn, or (ii) a0 c a, so by thegeneralized version of (2) we getT z a withT e Dn. Thus we see thateach Dn is dense.Let yn e X n Dnfor n e IMI. Since a0 c X, it fol- lows thateach Ynis compatiblewith a0, so we musthave 3k < 1h(YJ H(n, yn [k] ). Therefore M l= Vn3k< lh(Yn) 0(n,Yn[k]) - M[X] I= Vn3kO(n,X[k]) = M[X] # Vnqp(n). 574 DOUGLAS K. BROWN AND STEPHEN G. SIMPSON ThereforeM[X] satisfiesZ? induction,as desired.Finally, apply Lemma 6.1 to obtain a model M' of RCAo such that M[X] is a submodel of M' and IM[X]I = IM'I. ClearlyM' has thedesired properties. LEMMA 6.3. Anycountable model M of RCAo can be expandedto a countable modelM' of WKLo withIMI = IM'I. PROOF. Harrington(see [21]). THEOREM 6.4. Let M be a countablemodel of RCAo. Thereexists a modelM' of WKL' suchthat M is a submodelof M' and IMI = IM'I. PROOF. ApplyLemmas 6.2 and 6.3 repeatedlyto obtaina sequence M2 i+ I 1=p(0) A Vn(cp(n)-* p(n + 1)) and, since M2i+1 1 RCAO, M2i+1 1=Vnsp(n) so M' I= Vnsp(n).Therefore M' satis- fiesZ 1 induction.Similarly, if T is a treein 2', satisfiedby M' to be infinite,then, for some i, M2i satisfiesthat T is an infinitetree in 2'2. and hence,since M2i # WKLO, thatthere is a path throughT. Thus M' satisfiesthat there exists a path throughT and so M' 1=WKLo. Finally,let upbe an arithmeticformula such thatM' satisfies VnVaE 2m7 cte 2m( D a A p(n, z)). Let i be such that all parametersin upappear by stage 2i + 1. Then each set n={a Ip(n, o} is an M2i+ -definabledense set and so M2i+=1 MXe 2i+l Vn(Dnn X #00). ThereforeM' 1=3X E 2 XVn(Dn n X #00), i.e.,M' satisfies 3X E 2 M ,Vn3kp(n, X[k]). Thus M' = WKL+, as desired. COROLLARY 6.5. WKL+ is a conservativeextension of RCAo withrespect to H7 sentences;i.e., any HI sentenceprovable in WKL+ is provablein RCAo. PROOF. Suppose VXcp,Cp arithmetic, is not provablein RCAo. Then 3X - CPis consistentwith RCAo so, by Gddel's completenesstheorem, there is a countable model M of RCAo + 3X - cp.By Theorem6.4 thereis a modelM' of WKL+ with M a submodelof M' and |MI = IM'I. Then M' 1=3X - upso, by soundness,WKLo+ cannotprove VXcp. LEMMA 6.6. RCAo is a conservativeextension of PRA with respect to Ho2 sentences. PROOF. Parsons (see [21]). THE BAIRE CATEGORY THEOREM 575 COROLLARY 6.7. WKL' is a conservativeextension of PRA withrespect to I7? sentences. PROOF. The corollaryis immediatefrom 6.5 and 6.6. From theremarks made in theintroduction we see thatany I7Hsentence prov- able in WKL' is provablein PRA, i.e.,is provablefinitistically. Thus the results above givea partialrealization of Hilbert'sprogram in termsof theOpen Mapping and Closed Graph theorems. COROLLARY 6.8. WKLO 5 WKL+ ; ACAO. PROOF. By Theorem4.4 WKL+ provesB.C.T.II while,by Theorem3.2, WKLo does not. Thus WKLo S WKL+. For the otherhalf, we note that ACAo proves arithmeticinduction and hence,in particular,2: induction.2: inductionis not provablein RCAo [11] so, since 2: inductionis triviallya HI sentence,WKL+ - does not prove 2: inductionby Corollary6.5. Thus WKLo+ ACAo. We can sharpenthe statementof Corollary6.8 by providingw-models which exhibitthe strict inclusions. In fact,the proof of Theorem3.2 yieldsan w-modelof WKLo whichis not a model of WKL+. Thus we need onlyshow thatthere is an w-modelof WKLo whichis not a model of ACAo. We beginwith the following lemma. LEMMA 6.9. Let M = Rec,an w-modelof RCAo. Thenthe model M' of Lemma6.2 is nota modelof WKLo PROOF. Let S1 = {e Ecw| {e}(e) = O} and S2 = {e Ecw| {e}(e) = 1} be disjoint, recursivelyenumerable, recursively inseparable sets (Kleene [12]), and let iJ'2, N N be 1-1 recursivefunctions which enumerate S1 and S2, respectively.Define TC 2 (ii) Mi+1I= RCAo + (3X [D n X # 0]), forall Mi-definable dense sets D. Then,as in theproof of Theorem6.4, M = U.JM1 is an co-modelof RCA+. Since access to the completerecursively enumerable degree O' would allow us to finda path throughthe tree T of Lemma 6.9 and, conversely,access to a path throughT wouldallow us to compute0', we maystate the conclusion of Lemma6.9 as follows:adding a genericset to a modelof RCAo whichdoes notalready compute O' resultsin a modelwhich still does notcompute 0'. A setX c owis said to be almostrecursive (or hyperimmunefree) if, for all functions co-c w, wheneverf is recursivein X (denotedby f Finally,let M = U. M1. As in the proof of Theorem6.4, it followsthat M is an co-modelof WKL' and that0' 0 M. COROLLARY 6.15. Thereis an co-modelof WKL' whichis nota modelof ACAo. PROOF. Anymodel of ACAo contains0'. REFERENCES [1] A. BLASS, J. HIRST, and S. G. SIMPSON,Logical analysis of some theoremsof combinatorics and topological dynamics, Logic and combinatorics (S. G. 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