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The Baire Category Theorem in Weak Subsystems of Second-Order Arithmetic Author(S): Douglas K

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The Baire Category Theorem in Weak Subsystems of Second-Order Arithmetic Author(S): Douglas K 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 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Association for Symbolic Logic is collaborating with JSTOR to digitize, preserve and extend access to The Journal of Symbolic Logic. 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
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