Blass, A.

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Volume     	 pp   Propositional Connectives and the Set Theory of theContinuumAndreas Blass Mathematics Department University of Michigan Ann Arbor MI  USAe	mail ablassumicheduThis talk is a survey of two topics of recent interest in mathematical logic	 namelylinear logic and cardinal characteristics of the continuum I shall try to explainenough about each of them to be able to point out how they are connectedSince the underlying ideas of the two topics are quite dierent	 I regard theexistence of a connection as surprising  Linear LogicWhat does an implication A  B mean According to classical logic A  Bis true if and only if either A is false or B is true or both This is regarded asspecifying the meaning of implication because quite generally classical logicnds the meaning of a statement in the conditions for its being trueAccording to constructive logic as developed by Brouwer and Heyting aproof of A   B is a construction converting any proof of A into a proof ofB This is regarded as specifying the meaning of implication because quitegenerally constructive logic nds the meaning of a statement in what is requiredto prove itTwo close relatives of the BrouwerHeyting interpretation of implication areKolmogorov	s interpretation in terms of problems and the CurryHoward interpretation in terms of types Kolmogorov regarded statements as representingproblems and interpeted A  B as the problem of reducing B to A ie of solving B given a solution of A Curry and Howard pointed out a correspondencebetween logical systems and type theories where propositions correspond totypes which can safely be regarded simply as sets for the purposes of this talk I thank the organizers of the SMC Congress for inviting me to give this talk and I thankthe NSF for partial support under grant DMS	and A   B is the type of functions from type A to type B If we identify aproposition with the type of its proofs and identify constructions with functions then the CurryHoward correspondence amounts to the BrouwerHeytinginterpretationThe CurryHoward corespondence has been of interest recently in theoretical computer science where one deals with data types and where A  B couldbe the type of procedures with a formal variable of type A and a value of typeBThe preceding comments about implication have analogs for other connectives For example the conjunction A  B is dened classically as being truewhenever both A and B are true It is dened constructively by saying thatto prove A  B one must give a proof of A and a proof of B Under theCurryHoward correspondence conjunction becomes the cartesian product oftypesThe rst central idea of linear logic introduced in the mid	s by Girardis to keep track of how often a hypothesis is used in deducing a conclusionequivalently via the CurryHoward correspondence one keeps track of howoften an input is used in computing an output This and related conceptsseem to me more intuitive in the context of ability to perform actionsrather than knowledge of facts for knowledge is normally permanent andreusable while abilities can be limited in the sense that someone who can do Aand can do B may not necessarily be able to do both Although it is unclear inthe context of traditional set theory what it would mean for a function to usean argument a particular number of times the notion is considerably clearerfor algorithmic procedures and is useful for memory managementThe formal development of linear logic is based on a sequent calculus Intraditional logic    where  and  are lists of statements means that theconjunction of the statements in  entails the disjunction of the statements in The rst step toward linear logic is to abolish the rule of contraction AA   A  which formalized the idea that hypotheses can be reused One hypothesis Ais as good as two copies of it The removal of the contraction rule resultsin a system called ane logic In it a sequent    carries the additionalinformation that each hypothesis is to be used at most onceLinear logic is obtained from ane logic by also abolishing the rule of weakening   A  which formalized the idea that a hypothesis can be ignored In linear logic   requires that each hypothesis in  is used exactly onceThere is also a noncommutative version of linear logic abolishing the ruleof exchange  AB   BA Then    requires that the hypotheses be used in the order listedGirard and others have developed linear logic quite extensively especiallyits proof theory but considerably less is known about noncommutative linearlogic From now on I shall talk only about the commutative systemLinear logic	s insistence that hypotheses be used just once raises a questionabout the meaning of conjunction Should one use of A and one use of Bconstitute one or two uses of AB Girard	s answer is that there are two sortsof conjunction for which he introduced the notations A  B and AB Oneuse of A  B consists of a use of A and a use of B One use of AB consistsof one use of A or one use of B whichever the user wants Notice here thebeginning of an interaction between the hypotheses and a user These twoconjunctions are governed by the rules of inference  A   B  AB  A   B  ABThese rules would be equivalent in the presence of contraction and weakeningThe interaction alluded to above between a user requesting information andhypotheses supplying information or in more customary terminology betweenquestions and answers leads to the second central idea of linear logic namelylinear negation the operation that interchanges questions and answers Ananswer of type A is a question of type A and vice versaThere have been several attempts to model semantically this sort of interaction I introduced a game semantics where propositions or types are modeledby games whose rules specify how the questioner and answerer are to interactThis semantics was modied by Abramsky Jagadeesan Hyland and Ong toimprove its correspondence with Girard	s proof theoretic systemI shall not discuss these developments further here but instead concentrateon a simpler semantics a special case of de Paiva	s Dialecticalike semanticsHere a proposition is represented by a triple A  A A A where Aisthe set of questions of type A Ais the set of answers of type A andA  AAis a binary relation holding between any question and its correctanswersBy a morphism from A to B I mean a pair of functions   B Aand  A Bsuch that for all b  Band a  AbAa  bBaIn the presence of such a morphism if you can answer questions of type Athen you can also answer questions of type B given a question in Bconvertit with  into a question in A produce an answer in A and convert it with into an answer in Bfor the original questionLinear negation is modeled by interchanging questions and answers andinterchanging correct and incorrectA  A A	AThe connectives  and   are modeled byAB  AB BA  ABKwhere f gKa b i fbAa and gaBb andA  B  AB   AB AB BA  Cwhere a bCf g if either not fbAa or bBga In using the notation   Ideviate from the standard notation for this linear implication Unfortunatelythe standard notation a dash with a little circle at the right end is not instandard TEX One pleasant consequence of these denitions is that a morphism A  B is an answer f g that is correct for every question a b inthe sense of A   B De Paiva showed that with suitable interpretations forthe remaining connectives Girard	s proof system is sound for this semanticsIn addition as we shall see in the next section parts of this semantics arisenaturally in a quite dierent context Cardinal Characteristics of the ContinuumOne of set theory	s earliest and most useful contributions to the rest of mathematics was the distinction between dierent innite cardinals and especiallythe distinction between countable innity  and the cardinality of the continuum c   This made it possible to do things in some innite situationscountable ones that would be impossible for continuumsized ones Examplesinclude the Baire category theorem and the countable additivity of Lebesguemeasure Whenever as in these examples and c behave dierently one canask where between these the behavior changes Of course if one believes thecontinuum hypothesis CH ie c    then this question is trivial But itis consistent with the usual axioms of set theory ZFC that there are manycardinals between and c and then it is reasonable to consider cardinals likethe following covB is the minimum number of meager sets countable unions of nowheredense sets whose union is R The B stands for Baire addB is the minimum number of meager sets in R whose union is notmeager d is the minimum number of functions N  N needed to eventually dominate every such function b is the minimum number of functions N  N such that no single functioneventually dominates them allThese and many other cardinal numbers of a similar nature are called cardinalcharacteristics of the continuum and many connections mostly inequalitiesare known between them there are also many independence results saying thatdierent values of these characterisitics are consistent with ZFCFor the cardinals dened above the provable inequalities include  addB  bcovB d  The characteristics for Baire category dened above have analogs for Lebesguemeasure just replace meager with measure zero in the denitions and replace B with L in the notations It is a surprising theorem of Bartoszynskithat addL  addB This inequality like each of the inequalities exhibitedabove can consistently be strict and can consistently reduce to equalityThe denitions of many of the cardinal characteristics and the proofs ofmany of the inequalities between them including all those mentioned abovet into the following framework apparently rst used by Miller and Fremlinand explicitly formulated by VojtasFor two sets Aand Aand a relation A  A Aas in de Paiva	ssemantics described in the preceding section denekA A Ak  minfjZj j Z  Aandx  Az  ZxAzgSuch norms include all the characteristics dened aboveLet M be the set of meager sets or codes for meager Fsets ThencovB  kRMkaddB  kMM kLet be the eventual majorization ordering on NN Thend  kNNNNkb  kNNNN kIf there is a morphism A  B then kAk  kBk All the inequalitiesmentioned above can be deduced from this general fact by constructing explicitmorphisms Since the inequalities all become trivial when CH holds I oncehoped that the existence of morphisms might remain nontrivial and captureeven in the presence of CH the essential content of the proofs of the inequalitiesYiparaki showed that this is not the case CH provides morphisms as wellas inequalities But it does not provide Borel morphisms ie morphismswhose components  and  are Borel functions while the usual proofs of theinequalities do produce Borel morphisms So at the moment Borel morphismsseem to capture the essence of the usual proofs of these inequalitiesThere are a few known inequalities involving three cardinal characteristicsA nice example due to Miller isaddB  minfcovB bgThe reverse inequality follows from those displayed earlier The proofs ofthis and similar examples can be formalized in terms of morphisms to objectsconstructed by means of the dual of the following sequential compositionconnectiveAB  ABA  AB Swhere x fSa b i xAa and faBb This is closely related to de Paiva	sinterpretation of the  of linear logic but it is not commutative In fact theorder of sequential composition is crucial in proofs of inequalities like the oneabove It essentially corresponds to the order of arguments and constructionsin proofs of inequalities In many cases it also corresponds to the order inwhich forcing constructions should be iterated in order to produce models withcertain special properties In other cases recently studied by Mildenbergerforcing cannot detect the order of steps or even the need for several steps butsubtler combinatorial arguments can

Propositional connectives and the set theory of the continuum

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