TheKripke–Platek set theory (KP), pronounced/ˈkrɪpkiˈplɑːtɛk/, is anaxiomatic set theory developed bySaul Kripke and Richard Platek.The theory can be thought of as roughly thepredicative part ofZermelo–Fraenkel set theory (ZFC) and is considerably weaker than it.

In its formulation, a Δ₀ formula is one all of whose quantifiers arebounded. This means any quantification is the form${\displaystyle \mathrm{\forall }u\in v}$ or${\displaystyle \mathrm{\exists }u\in v.}$ (See theLévy hierarchy.)

• Axiom of extensionality: Two sets are the same if and only if they have the same elements.
• Axiom of induction: φ(a) being aformula, if for all setsx the assumption that φ(y) holds for all elementsy ofx entails that φ(x) holds, then φ(x) holds for all setsx.
• Axiom of empty set: There exists a set with no members, called theempty set and denoted {}.
• Axiom of pairing: Ifx,y are sets, then so is {x,y}, a set containingx andy as its only elements.
• Axiom of union: For any setx, there is a sety such that the elements ofy are precisely the elements of the elements ofx.
• Axiom of Δ₀-separation: Given any set and any Δ₀ formula φ(x), there is asubset of the original set containing precisely those elementsx for which φ(x) holds. (This is anaxiom schema.)
• Axiom of Δ₀-collection: Given any Δ₀ formula φ(x,y), if for every setx there exists a sety such that φ(x,y) holds, then for all setsX there exists a setY such that for everyx inX there is ay inY 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,generalized recursion theory, and the theory ofadmissible ordinals.KP can be studied as aconstructive set theory by dropping thelaw of excluded middle, without changing any axioms.

If any set${\displaystyle c}$ 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${\displaystyle \{x\in c\mid x\ne x\}}$. Furthermore, the existence of a member in the universe of discourse, i.e., ∃x(x=x), is implied in certain formulations^[1] offirst-order logic, in which case the axiom of empty set follows from the axiom of Δ₀-separation, and is thus redundant.

As noted, the above are weaker than ZFC as they exclude thepower 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 usualaxiom of regularity, which amounts to applying induction to the complement of a set (the class of all sets not in the given set).

• A set${\displaystyle A}$ is calledadmissible if it istransitive and${\displaystyle ⟨A,\in ⟩}$ is amodel of Kripke–Platek set theory.
• Anordinal number${\displaystyle \alpha }$ is called anadmissible ordinal if${\displaystyle L_{\alpha }}$ is an admissible set.
• ${\displaystyle L_{\alpha }}$ is called anamenable set if it is a standard model of KP set theory without the axiom of Δ₀-collection.

The ordinalα is an admissible ordinal if and only ifα is alimit ordinal and there does not exist aγ < α for which there is a Σ₁(L_α) mapping fromγ ontoα. IfM is a standard model of KP, then the set of ordinals inM is an admissible ordinal.

Theorem:IfA andB are sets, then there is a setA×B which consists of allordered pairs (a,b) of elementsa ofA andb ofB.

Proof:

The singleton set with membera, written {a}, is the same as the unordered pair {a,a}, by the axiom ofextensionality.

The singleton, the set {a,b}, and then also the ordered pair

${\displaystyle (a,b):=\{\{a\},\{a,b\}\}}$

all exist bypairing.A possible Δ₀-formula${\displaystyle \psi (a,b,p)}$ expressing thatp stands for the pair (a,b) is given by the lengthy

${\displaystyle \mathrm{\exists }r\in p(a\in r\wedge \mathrm{\forall }x\in r(x=a))}$

${\displaystyle \wedge \mathrm{\exists }s\in p(a\in s\wedge b\in s\wedge \mathrm{\forall }x\in s(x=a\vee x=b))}$

${\displaystyle \wedge \mathrm{\forall }t\in p((a\in t\wedge \mathrm{\forall }x\in t(x=a))\vee (a\in t\wedge b\in t\wedge \mathrm{\forall }x\in t(x=a\vee x=b))).}$

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${\displaystyle b}$ and collecting with respect to${\displaystyle A}$, some superset of${\displaystyle A\times \{b\}=\{(a,b)\mid a\in A\}}$ exists bycollection.

The Δ₀-formula

${\displaystyle \mathrm{\exists }a\in A\psi (a,b,p)}$

grants that just${\displaystyle A\times \{b\}}$ itself exists byseparation.

If${\displaystyle P}$ ought to stand for this collection of pairs${\displaystyle A\times \{b\}}$, then a Δ₀-formula characterizing it is

${\displaystyle \mathrm{\forall }a\in A\mathrm{\exists }p\in P\psi (a,b,p)\wedge \mathrm{\forall }p\in P\mathrm{\exists }a\in A\psi (a,b,p).}$

Given${\displaystyle A}$ and collecting with respect to${\displaystyle B}$, some superset of${\displaystyle \{A\times \{b\}\mid b\in B\}}$ exists bycollection.

Putting${\displaystyle \mathrm{\exists }b\in B}$ in front of that last formula and one finds the set${\displaystyle \{A\times \{b\}\mid b\in B\}}$ itself exists byseparation.

Finally, the desired

${\displaystyle A\times B:=\bigcup \{A\times \{b\}\mid b\in B\}}$

exists byunion.Q.E.D.

Transitive containment is the principle that every set is contained in sometransitive set. It does not hold in certain set theories, such asZermelo set theory (though its inclusion as an axiom does not add consistency strength^[2]).

Theorem:IfA is a set, then there exists a transitive setB such thatA is a member ofB.

Proof:

We proceed byinduction on the formula:

${\displaystyle \varphi (A):=\mathrm{\exists }B(A\in B\wedge \bigcup B\subseteq B)}$

Note that${\displaystyle \bigcup B\subseteq B}$ is another way of expressing thatB is transitive.

The inductive hypothesis then informs us that

${\displaystyle \mathrm{\forall }a\in A\mathrm{\exists }b(a\in b\wedge \bigcup b\subseteq b)}$.

ByΔ₀-collection, we have:

${\displaystyle \mathrm{\exists }C\mathrm{\forall }a\in A\mathrm{\exists }b\in C(a\in b\wedge \bigcup b\subseteq b)}$

ByΔ₀-separation, the set${\displaystyle \{c\in C\mid \bigcup c\subseteq c\}}$ exists, whoseunion we callD.

NowD is a union of transitive sets, and therefore itself transitive. And since${\displaystyle A\subseteq D}$, we know${\displaystyle D\cup \{A\}}$ is also transitive, and further containsA, as required. Q.E.D.

Theproof-theoretic ordinal of KPω is theBachmann–Howard ordinal. KP fails to prove some common theorems in set theory, such as theMostowski collapse lemma.^[3]

• Constructible universe
• Admissible ordinal
• Hereditarily countable set
• Kripke–Platek set theory with urelements

• 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

• Systems of set theory

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