This is done in such a way as to demonstrate that ikp has the existence property for its veri. Firstly, a detailed rendering of the ordinal analysis of intuitionistic kripkeplatek set theory, ikp, is given. The study of axiom systems for set theory should therefore be the central subject of proof theory. The notion of set is the central notion of mathematics. We take kp set theory without foundation called kp. A particularly interesting classical subtheory of zf is kripke platek set theory, kp. The paper investigates the strength of the antifoundation axiom,afa, on the basis of kripke platek set theory without foundation. Combined with the results of 40 this has important implications for constructive set theory. Power kripkeplatek set theory and the axiom of choice. Our method is remarkably simpler than previous work.
In set theory, zermelofraenkel set theory, named after mathematicians ernst zermelo and abraham fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as russells paradox. The kripke platek axioms of set theory kp, pronounced. According to wikipedia, kp is roughly the predicative part of zfc and czf has. A maximal lconsistent set an lmcs for short is an lconsistent set that has no proper lconsistent superset. It is shownthat the addition of afa considerably increases the proof theoretic strength. Instead, have separation and replacement for 0formulas only. It is, however, possible to relativize the techniques of ordinal analysis developed for kripkeplatek set theory to obtain useful information about power kripkeplatek set theory as shown in 27. A particularly interesting classical subtheory of zf is kripkeplatek set theory, kp. A set is a collection of objects without repetitions.
Kp arises from zf by completely omitting the power set axiom and restricting separation and collection to absolute predicates cf. He has also contributed to set theory see admissible ordinal and kripke platek set theory modal logic two of kripke s earlier works, a completeness theorem in modal logic and semantical considerations on modal logic, the former written while he was still a teenager, were on the subject of modal logic. Fragments of kripkeplatek set theory and the metamathematics. It is the prooftheoretic ordinal of several mathematical theories, such as kripkeplatek set theory with the axiom of infinity and the system czf of constructive set theory. A cutfree labelled sequent calculus for dynamic epistemic logic. The strength of such fragments will be characterized in terms of the smallest ordinal ff such that l ff is a model of every \pi 2 sentence which is provable in the theory. In this paper we shall investigate fragments of kripke platek set theory with infinity which arise from the full theory by restricting foundation to \pi n foundation, where n 2. Beyond background on kripke platek set theory and the theory of dilators as specified earlier, we presuppose the results on computing with dilators expounded in pa 871883j. The foundation scheme in set theory asserts that every nonempty class has an.
The set theoretical universe can therefore be regarded as the mathematical universe. A set of formulas is lconsistent if no contradiction can be derived from it using the theorems of l, and modus ponens. Ordinal analysis of intuitionistic power and exponentiation kripke. Following this trend we present a new method of analyzing extensions of kripkeplatek set theory in the spirit of girards tilogic. Saul kripke project gutenberg selfpublishing ebooks. This is used in order to introduce the basic techniques of proof theory, the relationship of these type theories to so called admissible ordinals, and to variants of a weak version of set theory called kripke platek set theory. Kripkeplatek set theory with one recursively mahlo ordinal. With the pioneering work of jager 10 on kripke platek set theory and its extensions to stronger theories by jager and pohlers 11. Ordinal analysis for kripkeplatek set theory with infinity.
Confirmed refutation of kripkeplatek kp constructive. Antifoundation axiom, kripkeplate set theory, subsystems ofsecond order arithmeic1. Hypersequent calculi for some intermediate logics with. Kripke argues that no identity theory can be correct. All mathematical objects can be represented as sets. Strahm iam, uni bern kripke platek set theory over ptca swansea, april, 2007 1 25. Jan 05, 2018 whilst power kripke platek set theory, kpp, shares many properties with ordinary kripke platek set theory, kp, in several ways it behaves quite differently from kp.
The paper relativizes the method of ordinal analysis developed for kripkeplatek set theory to theories which have the power set axiom. Here this technology is applied to intuitionistic kripkeplatek set theories ikpp and ikpe, where the operation of powerset and exponentiation, respectively, is allowed as a primitive in the separation and collection schemata. Mac lane set theory, kripkeplatek set theory, axiom h spectacles, mostowskis principle, constructibility, forcing over nonstandard models, poweradmissible set, forsterkaye set theory, strati able formula, conservative extension, simple theory of. Pdf an ordinal analysis of admissible set theory using.
Today, zermelofraenkel set theory, with the historically controversial axiom of choice ac included, is the standard form of. In zf, one can prove that is a model of kripkeplatek set theory plus the axiom of infinity and thus that kripkeplatek set theory plus the axiom of infinity is consistent. Kripkeplatek set theory plus infinity hereinafter called kp. In this paper, we investigate the logical strength of the foundation principle in basic set theory and. We take kp set theory without foundation called kp \\ as the base theory. Kripkeplatek set theory kp captures a small part of mathematics stronger than 2nd order arithmetic but noticeably weaker than zf. Though considerably weaker than zf, a great deal of set. The latter change is advanced most forcefully by jager, and for background and recent results we refer to his contribution in this volume ja 2j. The foundation scheme in set theory asserts that every nonempty class has an \\in \minimal element.
Truths, inductive definitions, and kripkeplatek systems over set theory. Acknowledgements the research described in this paper was partly carried out at the universite paris vii. Theory rachael alvir november 2016 1 axioms of kp and admissible sets an admissible setis a transitive set asatisfying the axioms of kripke platek set theory kp. Slim models of zermelo set theory welcome to dpmms. Request pdf power kripke platek set theory and the axiom of choice whilst power kripke platek set theory, kpp, shares many properties with ordinary kripke platek set theory, kp, in several. Powerkripkeplateksettheoryandtheaxiomof choice arxiv. Mancini and zambella introduced a weak fragment dubbed kp. Proof theory of martinl of type theory an overview. On some semiconstructive theories related to kripke. Truths, inductive definitions, and kripke platek systems over set theory. Power kripkeplatek set theory, ordinal analysis and global choice. Whilst power kripkeplatek set theory, kpp, shares many properties with ordinary kripkeplatek set theory, kp, in several ways it behaves quite differently from kp. One of the set theories which is amenable to ordinal analysis is kripkeplatek set theory, kp. John harrison amazon web services aitp 2018, aussois 27th march 2018 10.
It was first conceived for modal logics, and later adapted to intuitionistic logic and other nonclassical systems. Mac lane set theory, kripkeplatek set theory, axiom h spectacles, mostowskis principle, constructibility, forcing over nonstandard models, poweradmissible set, forsterkaye set theory, strati able formula, conservative extension, simple theory of types, failure of collection, failure of induction. Kripke platek set theory over polynomial time computable arithmetic i dieter probst and thomas strahm institut fur. Apr 01, 2001 power kripkeplatek set theory and the axiom of choice a cutfree labelled sequent calculus for dynamic epistemic logic on aggregating probabilistic evidence. Power kripke platek set theory a particularly interesting classical subtheory of zf is kripke platek set theory, kp. Though considerably weaker than zf, a great deal of set theory requires only the axioms of kp. Set theory \a set is a many that allows itself to be thought of as a one. Whilst power kripkeplatek set theory, kpp, shares many properties with ordinary kripkeplatek set theory, kp. The axiom of pair, the axiom of union, and the axiom of. In this paper, we investigate the logical strength of the foundation principle in basic set theory and \\alpha \recursion theory. The strength of mac lane set theory welcome to dpmms. Theory rachael alvir november 2016 1 axioms of kp and admissible sets an admissible setis a transitive set asatisfying the axioms of kripkeplatek set theory kp. Kripkeplatek set theory and the antifoundation axiom.
Kp arises from zf by completely omitting the power set axiom. Prooftheoretic reductions let t1, t2 be a pair of theories with languages l1 and l2. An identity theory of some mental phenomenon is a theory which says that that phenomenon state, property is identical to some physical phenomenon. Canonical kripke models play a role similar to the lindenbaumtarski algebra construction in algebraic semantics. Set theory lecture notes, spring 2009 hendrix college. In particular, ikpp proves the powerset axiom whereas ikpe proves the exponentiation axiom. Georg cantor in the previous chapters, we have often encountered sets, for example, prime numbers form a set, domains in predicate logic form sets as well. Kp is considerably weaker than zermelofraenkel set theory zfc, and can be thought of as roughly the predicative part of zfc.
Power kripkeplatek set theory, ordinal analysis and global. In appendix d we will show that type theory introduced in sect. We show that, given a path f through tkp, representing a model m of kp, and another computable illfounded. Kripke platek set theory over polynomial time computable. The paper investigates the strength of the antifoundation axiom,afa, on the basis of kripkeplatek set theory without foundation. One of the reasons that this is an important theory is that a great deal of set theory requires only the axioms of kp. Kripkeplatek set theory kp and constructive zermelofraenkel set theory czf are two set theoretic systems which are also routinely claimed to be predicative. It was introduced by heinz bachmann and william alvin howard. In mathematics, the bachmannhoward ordinal or howard ordinal is a large countable ordinal. Independence of ch finish proof that mg is a ctm, ramseys theorem. We show that it is possible to use this technique to extract information about power kripkeplatek set theory, kp p.
One of the set theories which is amenable to ordinal analysis is kripke platek set theory, kp. Power kripkeplatek set theory a particularly interesting classical subtheory of zf is kripkeplatek set theory, kp. In the following section, we will present the first step into the study of axiom systems for set. Kripkeplatek is a truely interesting subsystem of zermelofraenkel set theory that plays an important role in the interaction between set theory, recursion theory, model theory, and proof theory. This is used in order to introduce the basic techniques of proof theory, the relationship of these type theories to so called admissible ordinals, and to variants of a weak version of set theory called kripkeplatek set theory. This chapter presents a new method of analyzing extensions of kripkeplatek kp set theory in the spirit of girards. Here this technology is applied to intuitionistic kripke platek set theories ikpp and ikpe, where the operation of powerset and exponentiation, respectively, is allowed as a primitive in the separation and collection schemata. Kripkeplatek set theory, kp though considerably weaker than zf, a great deal of set theory requires only the axioms of kp. An even more important reason is that admissible sets have been a major source of. Kripke on the distinctness of the mind from the body. In this article our general framework is a class version of kripkeplatek set theory kp with in. Power kripkeplatek set theory, ordinal analysis and. For full access to this pdf, sign in to an existing account, or purchase an annual subscription. Indeed, zfc does most of the work that mathematicians ask of a foundation of mathematics, namely, it serves as a.
Independence of ch mg is a ctm, digression on kripkeplatek set theory, axiom of collection, and computability 422. The set theory kpu kripke platek set theory with urelements, for which see barwise 1975 is of interest for technical reasons in model theory. One of the reasons that this is an important theory is that a great deal of set theory 1the theories share the same p 1 theorems, but are still distinct since zermelo set. Notes on constructive set theory draft peter aczel and michael rathjen june 18, 2008 contents 1 introduction 5. It is, however, possible to relativize the techniques of ordinal analysis developed for kripke platek set theory to obtain useful information about power kripke platek set theory as shown in 27. Here this notion is generalized to functions on the universe of sets, using notations for well orderings longer than the class of ordinals. Roughly speaking, kp arises from zf by completely omitting the power set axiom and restricting separation and collection to set bounded formulae but adding set induction or class foundation. This is perhaps most strikingly demonstrated by a result, due to mathias, to the effect that adding the axiom of constructibility to kpp gives rise to a much stronger theory, whereas in the case of kp the constructible hierarchy.
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