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exists y forall x in x exists y in y varphi w_ 1 ldots w_ n x y this axiom is the disjunction of a σ 2 formula with a π 2 formula thus it is δ 3 when n 0 or π 3 when n 0 some but not all authors include an axiom of infinity w n w n 0 m w n m m 0 w m w n w n m m displaystyle exists w forall n in w n 0 lor exists m in w n m cup m land 0 in w land forall m in w exists n in w n m cup m this axiom is a σ 1 formula it defines ω kp with infinity is denoted by kpω these axioms lead to close connections between kp computability theory and the theory of admissible ordinals kp can be studied as a constructive set theory by dropping the law of excluded middle without changing any axioms empty set edit if any set c displaystyle c is postulated to exist such as in the axiom of infinity then the axiom of empty set is redundant because it is equal to the subset x c x x displaystyle x in c mid x neq x furthermore the existence of a member in the universe of discourse i e x x x is implied in certain formulations 1 of first order logic in which case the axiom of empty set follows from the axiom of δ 0 separation and is thus redundant comparison with zermelo fraenkel set theory edit as noted the above axioms are together weaker than zfc as they exclude the power set axiom choice and sometimes infinity also the axioms of separation and collection here are weaker than the corresponding axioms in zfc because the formulas φ used in these are limited to bounded quantifiers only the axiom of induction in the context of kp is stronger than the usual axiom of regularity which amounts to applying induction to the complement of a set the class of all sets not in the given set opposite of infinity edit one could add an axiom of finiteness denying the existence of infinite sets this would amount to saying that the universe is v ω this could be done by saying that every set has an injection to a natural number finite ordinal in formal language s n f k n n k k m n m 0 k n m k k displaystyle forall s exists n exists f big exists k in n n k cup k land forall m in n m 0 lor exists k in n m k cup k land t s k n t k f t s u s k n t k f u k f t u displaystyle forall t in s exists k in n langle t k rangle in f land forall t in s forall u in s exists k in n langle t k rangle in f land langle u k rangle in f implies t u big a key point here is to identify natural numbers without referring to ω which is one of the infinite sets that we are trying to deny a natural number is a set that is either empty or a successor and thus not a limit and all of whose elements are also natural numbers the elements of f are ordered pairs see also the axiom of limitation of size this axiom is π 2 it implies the axiom of choice given a set of nonempty sets apply this to their union then map each set to its element with the smallest value of f related definitions edit a set a displaystyle a is called admissible if it is transitive and a displaystyle langle a in rangle is a model of kripke platek set theory an ordinal number α displaystyle alpha is called an admissible ordinal if l α displaystyle l_ alpha is an admissible set l α displaystyle l_ alpha is called an amenable set if it is a standard model of kp set theory without the axiom of δ 0 collection theorems edit admissible sets edit the ordinal α is an admissible ordinal if and only if α is a limit ordinal and there does not exist a γ α for which there is a σ 1 l α mapping from γ onto α if m is a standard model of kp then the set of ordinals in m is an admissible ordinal cartesian products exist edit theorem if a and b are sets then there is a set a b that consists of all ordered pairs a b of elements a of a and b of b proof the singleton set with member a written a is the same as the unordered pair a a by the axiom of extensionality the singleton the set a b and then also the ordered pair a b a a b displaystyle a b a a b all exist by pairing a possible δ 0 formula ψ a b p displaystyle psi a b p expressing that p stands for the pair a b is given by the lengthy r p a r x r x a displaystyle exists r in p big a in r land forall x in r x a big s p a s b s x s x a x b displaystyle land exists s in p big a in s land b in s land forall x in s x a lor x b big t p a t x t x a a t b t x t x a x b displaystyle land forall t in p big big a in t land forall x in t x a big lor big a in t land b in t land forall x in t x a lor x b big big what follows are two steps of collection of sets followed by a restriction through separation all results are also expressed using set builder notation firstly given b displaystyle b and collecting with respect to a displaystyle a some superset of a b a b a a displaystyle a times b a b mid a in a exists by collection the δ 0 formula a a ψ a b p displaystyle exists a in a psi a b p grants that just a b displaystyle a times b itself exists by separation if p displaystyle p ought to stand for this collection of pairs a b displaystyle a times b then a δ 0 formula characterizing it is a a p p ψ a b p p p a a ψ a b p displaystyle forall a in a exists p in p psi a b p land forall p in p exists a in a psi a b p given a displaystyle a and collecting with respect to b displaystyle b some superset of a b b b displaystyle a times b mid b in b exists by collection putting b b displaystyle exists b in b in front of that last formula and one finds the set a b b b displaystyle a times b mid b in b itself exists by separation finally the desired a b a b b b displaystyle a times b bigcup a times b mid b in b exists by union q e d transitive containment edit transitive containment is the principle that every set is contained in some transitive set it does not hold in certain set theories such as zermelo set theory though its inclusion as an axiom does not add consistency strength 2 theorem if a is a set then there exists a transitive set b such that a is a member of b proof we proceed by induction on the formula ϕ a b a b b b displaystyle phi a exists b a in b land bigcup b subseteq b note that b b displaystyle bigcup b subseteq b is another way of expressing that b is transitive the inductive hypothesis then informs us that a a b a b b b displaystyle forall a in a exists b a in b land bigcup b subseteq b if a a displaystyle a notin a then we set b displaystyle b which is transitive by δ 0 collection we have c a a b c a b b b displaystyle exists c forall a in a exists b in c a in b land bigcup b subseteq b by δ 0 separation the set c c c c displaystyle c in c mid bigcup c subseteq c exists whose union we call d now d is a union of transitive sets and therefore itself transitive and since a d displaystyle a subseteq d we know d a displaystyle d cup a is also transitive and further contains a as required q e d even natural numbers edit in kpω we can prove the existence of a set of all even natural numbers an even number is either zero or the successor of the successor of an even number or a limit ordinal the set of even numbers up to some finite stage can be built up using null set pairing and union from the axiom of infinity we get the existence of ω and its element 2 0 1 we apply the axiom of foundation to get a function that gives k modulo 2 for k n for each n in ω then we use δ 0 collection to send n 0 to n and everything else to 0 we use δ 0 separation and the mod 2 function to remove any extraneous elements this leaves us with the set of even natural numbers φ n f n ω 0 0 f n ω v 2 n v f k v f k 0 v 0 j k u 2 u v k j j j u f displaystyle varphi n equiv exists f n notin omega implies langle 0 0 rangle in f land n in omega implies exists v in 2 langle n v rangle in f land forall langle k v rangle in f k 0 land v 0 lor exists j in k exists u in 2 u neq v land k j cup j land langle j u rangle in f consider any set x suppose φ y holds for all elements y in x if x is not in ω let f be 0 0 which exists by the axioms of null set and pairing likewise if x 0 if x is a nonzero natural number then it is the successor of some natural number m that is an element of x so φ m holds and we can take g to be the function whose existence is ensured by that then there is some u in 2 such that m u is in g let v be the element of 2 different from u let f g x v then f will be a function that satisfies φ x thus by the axiom of induction we know that φ x holds for all sets x initial segments of the modulo 2 function are identified by segmod2 f p f k ω v 2 p k v k 0 v 0 j k u 2 u v k j j j u f displaystyle operatorname segmod2 f equiv forall p in f exists k in omega exists v in 2 p langle k v rangle land k 0 land v 0 lor exists j in k exists u in 2 u neq v land k j cup j land langle j u rangle in f metalogic edit the proof theoretic ordinal of kpω is the bachmann howard ordinal kp fails to prove some common theorems in set theory such as the mostowski collapse lemma 3 see also edit constructible universe admissible ordinal hereditarily countable set kripke platek set theory with urelements references edit poizat bruno 2000 a course in model theory an introduction to contemporary mathematical logic springer isbn 0 387 98655 3 note at end of 2 3 on page 27 those who do not allow relations on an empty universe consider x x x and its consequences as theses we however do not share this abhorrence with so little logical ground of a vacuum mathias a r d 2001 the strength of mac lane set theory annals of pure and applied logic 110 1 3 107 234 doi 10 1016 s0168 0072 00 00031 2 p odifreddi classical recursion theory 1989 p 421 north holland 0 444 87295 7 bibliography edit devlin keith j 1984 constructibility berlin springer verlag isbn 0 387 13258 9 gostanian richard 1980 constructible models of subsystems of zf journal of symbolic logic 45 2 association for symbolic logic 237 doi 10 2307 2273185 jstor 2273185 kripke s 1964 transfinite recursion on admissible ordinals journal of symbolic logic 29 161 162 doi 10 2307 2271646 jstor 2271646 platek richard alan 1966 foundations of recursion theory thesis ph d stanford university mr 2615453 v t e set theory overview set mathematics axioms adjunction choice countable dependent global constructibility v l determinacy projective extensionality infinity limitation of size pairing power set regularity union martin s axiom axiom schema replacement specification operations cartesian product complement i e set difference de morgan s laws disjoint union identities intersection power set symmetric difference union concepts methods almost cardinality cardinal number large class constructible universe continuum hypothesis diagonal argument element ordered pair tuple family forcing one to one correspondence ordinal number set builder notation transfinite induction venn diagram set types amorphous countable empty finite hereditarily filter base subbase ultrafilter fuzzy infinite dedekind infinite recursive singleton subset superset transitive uncountable universal theories alternative axiomatic constructive naive cantor s theorem zermelo general principia mathematica new foundations zermelo fraenkel von neumann bernays gödel morse kelley kripke platek tarski grothendieck paradoxes problems russell s paradox suslin s problem burali forti paradox set theorists paul bernays georg cantor paul cohen richard dedekind abraham fraenkel kurt gödel thomas jech john von neumann willard quine bertrand russell thoralf skolem ernst zermelo v t e mathematical logic general axiom list cardinality first order logic formal proof formal semantics foundations of mathematics information theory lemma logical consequence model theorem theory type theory theorems list paradoxes gödel s completeness incompleteness theorems tarski s undefinability banach tarski paradox cantor s theorem paradox diagonal argument compactness halting problem lindström s löwenheim skolem russell s paradox logics traditional classical logic logical truth tautology proposition inference logical equivalence consistency equiconsistency argument soundness validity syllogism square of opposition venn diagram propositional boolean algebra boolean functions logical connectives propositional calculus propositional formula truth tables many valued logic 3 finite predicate first order list second order monadic higher order fixed point free quantifiers predicate monadic predicate calculus set theory set hereditary class ur element ordinal number extensionality forcing relation equivalence partition set operations intersection union complement cartesian product power set identities types of sets countable uncountable empty inhabited singleton finite infinite transitive ultrafilter recursive fuzzy universal universe constructible grothendieck von neumann maps cardinality function map domain codomain image in sur bi jection schröder bernstein theorem isomorphism gödel numbering enumeration large cardinal inaccessible aleph number operation binary theories zermelo fraenkel axiom of choice continuum hypothesis general kripke platek morse kelley naive new foundations tarski grothendieck von neumann bernays gödel ackermann constructive formal systems list language syntax alphabet arity automata axiom schema expression ground extension by definition conservative relation formation rule grammar formula atomic closed ground open free bound variable language metalanguage logical connective predicate functional variable propositional variable proof quantifier rank sentence atomic spectrum signature string substitution symbol function logical constant non logical variable term theory list example axiomatic systems list of true arithmetic peano second order elementary function primitive recursive robinson skolem of the real numbers tarski s axiomatization of boolean algebras canonical minimal axioms of geometry euclidean elements hilbert s tarski s non euclidean principia mathematica proof theory formal proof natural deduction logical consequence rule of inference sequent calculus theorem systems axiomatic deductive hilbert list complete theory independence from zfc proof of impossibility ordinal analysis reverse mathematics self verifying theories model theory interpretation function of models model atomic equivalence finite prime saturated spectrum submodel non standard model of non standard arithmetic diagram elementary categorical theory model complete theory satisfiability semantics of logic strength theories of truth semantic tarski s kripke s t schema transfer principle truth predicate truth value type ultraproduct validity computability theory church encoding church turing thesis computably enumerable computable function computable set decision problem decidable undecidable p np p versus np problem kolmogorov complexity lambda calculus primitive recursive function recursion recursive set turing machine type theory related abstract logic algebraic logic automated theorem proving category theory concrete abstract category category of sets history of logic history of mathematical logic timeline logicism mathematical object philosophy of mathematics 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