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Kripke–Platek set theory
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== Axioms == In its formulation, a Δ<sub>0</sub> formula is one all of whose quantifiers are [[bounded quantifier|bounded]]. This means any quantification is the form <math>\forall u \in v</math> or <math>\exist u \in v.</math> (See the [[Lévy hierarchy]].) * [[Axiom of extensionality]]: Two sets are the same if and only if they have the same elements. * [[Epsilon-induction|Axiom of induction]]: φ(''a'') being a [[well-formed formula#Predicate logic|formula]], if for all sets ''x'' the assumption that φ(''y'') holds for all elements ''y'' of ''x'' entails that φ(''x'') holds, then φ(''x'') holds for all sets ''x''. * [[Axiom of empty set]]: There exists a set with no members, called the [[empty set]] and denoted {}. * [[Axiom of pairing]]: If ''x'', ''y'' are sets, then so is {''x'', ''y''}, a set containing ''x'' and ''y'' as its only elements. * [[Axiom of union]]: For any set ''x'', there is a set ''y'' such that the elements of ''y'' are precisely the elements of the elements of ''x''. * [[Axiom schema of predicative separation|Axiom of Δ<sub>0</sub>-separation]]: Given any set and any Δ<sub>0</sub> formula φ(''x''), there is a [[subset]] of the original set containing precisely those elements ''x'' for which φ(''x'') holds. (This is an [[axiom schema]].) * [[Axiom of collection|Axiom of Δ<sub>0</sub>-collection]]: Given any Δ<sub>0</sub> formula φ(''x'', ''y''), if for every set ''x'' there exists a set ''y'' such that φ(''x'', ''y'') holds, then for all sets ''X'' there exists a set ''Y'' such that for every ''x'' in ''X'' there is a ''y'' in ''Y'' such that φ(''x'', ''y'') holds. Some but not all authors include an * [[Axiom of infinity]] KP with infinity is denoted by KPω. These axioms lead to close connections between KP, [[Mathematical logic#Recursion theory|generalized recursion theory]], and the theory of [[admissible ordinal]]s. KP can be studied as a [[constructive set theory]] by dropping the [[law of excluded middle]], without changing any axioms. === Empty set === If any set <math>c</math> 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 <math>\{x\in c\mid x\neq x\}</math>. Furthermore, the existence of a member in the universe of discourse, i.e., ∃x(x=x), is implied in certain formulations<ref>{{cite book |title=A course in model theory: an introduction to contemporary mathematical logic |url=https://archive.org/details/courseinmodelthe0000poiz |url-access=registration |last=Poizat |first=Bruno |year=2000 |publisher=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."</ref> of [[first-order logic]], in which case the axiom of empty set follows from the axiom of Δ<sub>0</sub>-separation, and is thus redundant. === Comparison with Zermelo-Fraenkel set theory === As noted, the above are 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).
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