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An Equational Re-Engineering of Set Theories

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An Equational Re-Engineering of Set Theories An equational reengineering of set theories y z Andrea Formisano Eugenio Omo deo AugustNovember Abstract New successes in dealing with set theories by means of stateoftheart theorem provers may ensue from terse and concise axiomatizations such as can b e moulded in the framework of the fully equational TarskiGivant map calculus In this pap er we carry out this task in detail setting the ground for a number of exp eriments Key words Set theory relation algebras rstorder theoremproving algebraic logic Introduction Like other mature elds of mathematics Set Theory deserves sustained eorts that bring to light richer and richer decidable fragments of it general inference rules for reasoning in it eective pro of strategies based on its domainknowledge and so forth Advances in this sp ecialized area of automated reasoning tend in spite of their steadi ness to b e slow compared to the overall progress in the eld Many exp eriments with set theories have hence b een carried out with standard theoremproving systems Still to day such exp eriments p ose considerable stress on stateoftheart theorem provers or demand man to give much guidance to pro of assistants they therefore constitute ideal b enchmarks Moreover in view of the p ervasiveness of Set Theory they are likely when successful in something tough to have a strong echo amidst computer scientists and mathematicians Even for those who are striving to develop something entirely ad hoc in the challenging arena of set theories it is imp ortant to assess what can to day b e achieved by unsp ecialized pro of metho ds and where the contextspecic b ottlenecks of Set Theory precisely reside In its most p opular rstorder version namely the ZermeloFraenkelSkolem axiomatic system ZF set theory very much like Peano arithmetic presents an immediate obstacle it do es not admit a nite axiomatization This is why the von NeumannGodelBernays theory GB of sets and classes is sometimes preferred to it as a basis for exp erimentation Various authors eg have b een able to retain the traits of ZF by resorting to higherorder features of sp ecic theoremprovers such as Isab elle In this pap er we will pursue a minimalist approach to prop ose a purely equational formulation of b oth ZF and nite set theory Our approach heavily relies on but we go into much ner detail with the axioms ending in such a concise formulation as to oer a go o d starting p oint for exp erimentation with Otter say or with a more markedly equational theoremprover Our formulation of the axioms is based on the formalism L of which is equational and devoid of variables but somewhat out of standards Work partially supp orted by the CNR of Italy co ordinated pro ject SETA and by MURST Tecniche sp eciali p er la sp ecica lanalisi la verica la sintesi e la trasformazione di programmi y University La Sapienza of Rome Dept of Computer Science formisandimiuniudit z University of LAquila Dept of Pure and Applied Mathematics omodeounivaqit Luckily L can easily b e emulated through a rstorder system simply by treating the metavariables that o ccur in the schematic formulation of its axioms b oth the logical ones and those endowed with a genuinely settheoretic content as if they were rstorder variables In practice this means treating ZF as if it were an extension of the theory of relation algebras we can express it through a nite number of axioms b ecause variables are not supp osed to range over sets but over the dyadic relations on the universe of sets Taken in its entirety Set Theory oers a panorama of alternatives cf px that is it consists of axiomatic systems not equivalent and sometimes antithetic cf to one another This is why we will not pro duce the axioms of just one theory and will also touch the theme of individuals ultimate entities entering in the formation of sets Future work will expand the material of this pap er into a to olkit for assembling set theories of all kindsafter we have singled out through exp eriments formulations of the axioms that work decidedly b etter than others Syntax and semantics of L L is a ground equational language where one can state prop erties of dyadic relations maps as we will call them over an unsp ecied yet xed domain U of discourse In this pap er the map whose prop erties we intend to sp ecify is the membership relation over the class U of all sets The language L consists of map equalities QR where Q and R are map expressions Denition Map expressions are al l terms of the fol lowing signature 1 symbol l n y degree priority 1 Of these n y will b e used as leftasso ciative inx op erators as a p ostx as a line topping its argument 2 op erator and For an interpretation of L one must x along with a nonempty U a subset of 2 U U U Then each map expression P comes to designate a sp ecic map P and Def accordingly any equality QR b etween map expressions turns out to b e either true or false on the basis of the following evaluation rules 2 l U fa a a in U g Def Def Def QR f a b Q a b R g Def 2 Q R f a b U a b Q if and only if a b R g Def 2 QR f a b U there are cs in U for which a c Q and c b R g Def 1 Q f b a a b Q g Def Of the op erators and constants in the signature of L only a few deserve b eing re garded as primitive constructs indeed we choose to regard as derived constructs the ones for which we gave no evaluation rule as well as others that we will tacitly add to the signature P P l P yQ P Q Def Def P n Q P Q funPart P P n P Def Def P nQ P Q etc Def The interpretation of L obviously extends to the new constructs eg 2 P yQ f a b U for all c in U either a c P or c b Q g Def funPart P fa b P a c P for any c bg Def so that funPart P P will mean P is a partial function very much like Fun P to b e seen b elow Through abbreviating denitions we can also dene shortening notation for map equal ities that follow certain patterns eg 1 Fun P P P n Total P P ll Def Def so that Total P states that for all a in U there is at least one pair a b in P One often strives to sp ecify the class C of interpretations that are of interest in some application through a collection of equalities that must b e true in every of C The task we are undertaking here is of this nature our aim is to capture through simple map equalities the interpretations of that comply with standard ZermeloFraenkel theory on the one hand a theory of nite sets ultimately based on individuals on the other hand In part the game consists in expressing in L common settheoretic notions To start with something obvious 1 Def Def Def where each stands for one of l 0 1 n 0 1 n i Def To see something slightly more sophisticated Examples With resp ect to an interpretation one says that a intersects b if a and b have some element in common ie there is a c for which c a and c b A map 2 expression P such that P f a b U a intersects b g is Likewise one can dene in L the relation a includes b ie no element of b fails to The expression translates the relation a b elong to a by the map expression is strictly included in b and so on Let a splits b mean that every element of a intersects b and that no two elements of a intersect each other These conditions translate into the map expression dened as follows splits y ( )l Def 2 Secondly the reconstruction of set theory within L consists in restating ordinary axioms and subsequently theorems through map equalities Example One of the many ways of stating the muchdebated axiom of choice under adequately strong remaining axioms is by claiming that when a splits some b there is a c which is also split by a and which does not strictly include any other set split by a Formally splitslsplits n splits Ch Total where the second and third o ccurrence of splits could b e replaced by y 1 To relate the original version of this axiom in with ours notice that a set a splits S some b if and only if a consists of pairwise disjoint sets and accordingly a splits a Moreover an inclusionminimal c split by a must have a singleton intersection with each d in a otherwise of two elements in c d either one could b e removed from c conversely S if c is included in a and has a singleton intersection with each d in a then none of its elements e can b e removed else c n feg would no longer intersect the d in a to which e b elongs 2 In the third place we are to prove theorems ab out sets by equational reasoning moving from the equational sp ecication of the set axioms To discuss this p oint we must refer to an inferential apparatus for L we hence delay this discussion to much later cf Sec For alternative versions of this axiom cf p Extensionality subset sumset and p owerset axioms Two derived constructs and r will b e of great help in stating the prop erties of mem b ership simply P r P P nP P Def Def Plainly a Q b and ar R b will hold in an interpretation if and only if resp ectively all those c in U for which aQ c holds are elements of b in the sense that c b the elements of b are precisely those c in U for which aR c holds Our rst axiom extensionality states that sets are the same whose elements are the same E r A useful strenghtening of this axiom is the scheme Fun r P where P ranges over all map expressions Two rather elementary p ostulates the powerset axiom and the sumset axiom state that for any set a there is a set whose elements comprise all sets included in a and there
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