# Kripke–Platek set theory

**Kripke–Platek set theory** (KP) is a weak axiomatic set theory, obtained from [Zermelo–Fraenkel set theory](https://www.edgepedia.org/zermelo-fraenkel-set-theory) by omitting the power set axiom and restricting the Separation and Collection schemes to bounded (Δ₀) formulas. Its standard models are called admissible sets, and the theory serves as the formal setting for generalized recursion theory and for admissible proof theory.<sup>[1](https://plato.stanford.edu/entries/proof-theory/appendix-d.html)</sup><sup> • </sup><sup>[2](https://math.berkeley.edu/~marks/notes/edst_notes3.pdf)</sup>

| Key fact | Detail |
|---|---|
| Origin | Introduced by Saul Kripke (1964 abstracts in the *Journal of Symbolic Logic*) and Richard Platek (1966 Stanford dissertation), independently, from the motivation of a set theory whose models suit generalized recursion theory<sup>[3](https://www.gc.cuny.edu/saul-kripke-center/resources)</sup><sup> • </sup><sup>[4](https://lingenic.com/en/Knowledge_Base/Logic/Formal_Logical_Systems/Theories/Kripke-Platek_Set_Theory)</sup> |
| Axioms | Extensionality, Foundation, Pairing, Union, Δ₀-Separation, Δ₀-Collection; the axiom of Infinity is optional (KP vs. KP + INF)<sup>[2](https://math.berkeley.edu/~marks/notes/edst_notes3.pdf)</sup> |
| Standard models | Transitive models of KP are the admissible sets<sup>[1](https://plato.stanford.edu/entries/proof-theory/appendix-d.html)</sup> |
| Proof-theoretic ordinal | \( \lvert \mathrm{KP} \rvert = \lvert \mathrm{ID}_1 \rvert = \psi_{\Omega_1}(\varepsilon_{\Omega_1+1}) \), the Bachmann–Howard ordinal<sup>[1](https://plato.stanford.edu/entries/proof-theory/appendix-d.html)</sup> |
| Canonical model | \( L_{\omega_1^{CK}} \) is a model of KP, and \( \omega^{\omega} \cap L_{\omega_1^{CK}} = \mathrm{HYP} \), the hyperarithmetic reals<sup>[2](https://math.berkeley.edu/~marks/notes/edst_notes3.pdf)</sup> |
| Admissible ordinals | The first admissible ordinal is \( \omega \); the second is \( \omega_1^{CK} \), whose recursion theory agrees with metarecursion theory<sup>[5](https://api.pageplace.de/preview/DT0400.9781316731697_A29773274/preview-9781316731697_A29773274.pdf)</sup> |

## Axioms and language

The theory is formulated in the usual language of set theory, \( \mathcal{L}_{\in} \), with \( \in \) as the only non-logical symbol besides \( = \). Quantifiers of the forms \( \forall x \in a \) and \( \exists x \in a \) are called bounded; a Δ₀ (bounded) formula is one in which every quantifier is bounded.<sup>[1](https://plato.stanford.edu/entries/proof-theory/appendix-d.html)</sup>

The axioms of KP are Extensionality, Foundation, Pairing, Union, Δ₀-Separation, and Δ₀-Collection. Δ₀-Separation asserts, for each Δ₀ formula \( \varphi \), the existence of \( \{ z \in x : \varphi(z) \} \); Δ₀-Collection asserts that if every element of a set \( u \) has some \( y \) satisfying a Δ₀ relation, then a single set \( v \) contains witnesses for all elements of \( u \) simultaneously.<sup>[2](https://math.berkeley.edu/~marks/notes/edst_notes3.pdf)</sup> The axiom of Infinity, \( (\exists x)[\emptyset \in x \land (\forall y \in x)\, y \cup \{y\} \in x] \), is frequently added, giving the system KP + INF.<sup>[2](https://math.berkeley.edu/~marks/notes/edst_notes3.pdf)</sup>

KP is strong enough to prove a substantial amount of standard set theory, for example that the ordinals are linearly ordered and that the rank function on sets is well defined.<sup>[2](https://math.berkeley.edu/~marks/notes/edst_notes3.pdf)</sup> It remains too weak for other basic facts: an ill-founded computable tree \( T \) belongs to \( L_{\omega_1^{CK}} \), yet \( L_{\omega_1^{CK}} \) satisfies that \( T \) is well founded, because the model contains only hyperarithmetic reals and hence no infinite descending sequence through \( T \).<sup>[2](https://math.berkeley.edu/~marks/notes/edst_notes3.pdf)</sup>

## Admissible sets and higher recursion theory

The point of the restrictions is that the resulting transitive models, the admissible sets, carry a workable recursion theory. An admissible structure is, in effect, an inner model of computation: by taking an appropriate initial segment of the constructible universe \( L \) that forms an admissible structure, a \( \Pi^1_1 \)-set assumes the role that an r.e. set plays in ordinary recursion theory.<sup>[6](https://www.oreilly.com/library/view/recursion-theory/9783110381290/html/chapter3.html)</sup>

Kripke introduced admissible ordinals by means of an equation calculus, and Platek gave an independent equivalent definition using schemata and another using machines: an ordinal \( \alpha \) is admissible if every \( \alpha \)-recursive function, when defined at some \( \beta < \alpha \), takes a value below \( \alpha \), so the initial segment determined by \( \alpha \) is closed under the function.<sup>[5](https://api.pageplace.de/preview/DT0400.9781316731697_A29773274/preview-9781316731697_A29773274.pdf)</sup> The first admissible ordinal is \( \omega \); the second is \( \omega_1^{CK} \), and the \( \omega_1^{CK} \)-recursion theory of Kripke and Platek agrees with metarecursion theory.<sup>[5](https://api.pageplace.de/preview/DT0400.9781316731697_A29773274/preview-9781316731697_A29773274.pdf)</sup> An ordinal \( \alpha \) is admissible precisely when \( L_{\alpha} \) is an admissible set, and the resulting \( \alpha \)-recursion theory can be developed directly.<sup>[7](https://www.sciencedirect.com/science/chapter/bookseries/pii/S0049237X08708814)</sup>

Admissible sets have been a major source of interaction between model theory, recursion theory, and set theory, consolidated in Jon Barwise's 1975 monograph *Admissible Sets and Structures*.<sup>[1](https://plato.stanford.edu/entries/proof-theory/appendix-d.html)</sup> A salient feature of admissible sets is that they satisfy Bounded Collection, a principle that, on the basis of the other axioms of KP, is equivalent to Σ-Reflection.<sup>[1](https://plato.stanford.edu/entries/proof-theory/appendix-d.html)</sup> Admissible sets of the form \( L_{\kappa} \) also satisfy \( \Pi_2 \)-reflection: if \( L_{\kappa} \models \forall x \exists y\, C(x,y,\vec{a}) \) with \( C \) bounded, then some earlier level \( L_{\rho} \), \( \rho < \kappa \), already satisfies the same statement with the same parameters.<sup>[1](https://plato.stanford.edu/entries/proof-theory/appendix-d.html)</sup>

## Variants

**KPU.** Kripke–Platek set theory with urelements, KPU, weakens Extensionality to allow non-sets (urelements) and keeps Pairing, Union, Δ₀-Separation, Δ₀-Collection, and \( \in \)-induction for arbitrary formulas. It is of technical interest in model theory; its constructions are absolute in a suitable sense.<sup>[8](https://plato.stanford.edu/entries/settheory-alternative/)</sup>

**Power and exponentiation KP.** Power Kripke–Platek set theory, KP(P), adds the power set operation \( P \) as a primitive symbol to the language with its own axiom; KP(P) is not the same theory as KP + Pow.<sup>[9](https://www.researchgate.net/publication/259119137_Relativized_ordinal_analysis_The_case_of_Power_Kripke-Platek_set_theory)</sup> Cook and Rathjen carried out ordinal analyses of the intuitionistic versions IKP(P) and IKP(E), where powerset and, respectively, exponentiation (the assertion that for any sets \( A \) and \( B \) the collection of all functions from \( A \) to \( B \) is a set) are allowed as primitives in the separation and collection schemata. The treatment of IKP(E) posed considerable obstacles: in the infinitary system, the levels of terms become a moving target, since they cannot be assigned a fixed level in the formal cumulative hierarchy from their syntactic structure alone.<sup>[10](https://arxiv.org/abs/1603.03358)</sup>

**Weakenings.** The theory ReR, studied by Zachiri McKenzie, replaces KP's bounded collection scheme with bounded replacement; ReR proves that every set is contained in a transitive set, and the theories obtained by adding the negation of Infinity to ReR and to KP have the same consequences.<sup>[11](https://arxiv.org/abs/2608.23398)</sup>

## Ordinal analysis and strength

With the work of Gerhard Jäger and Wolfram Pohlers in 1982, the forum of ordinal analysis switched from [second-order arithmetic](https://www.edgepedia.org/second-order-arithmetic) to set theory, founding what is now called admissible proof theory, named after the models of KP.<sup>[1](https://plato.stanford.edu/entries/proof-theory/appendix-d.html)</sup> KP is the weakest theory in the line analyzed by the Munich school in the late 1970s and 1980s, and its proof-theoretic ordinal is the Bachmann–Howard ordinal, \( \psi_{\Omega_1}(\varepsilon_{\Omega_1+1}) \), the same ordinal as that of \( \mathrm{ID}_1 \), the theory of one positive inductive definition.<sup>[1](https://plato.stanford.edu/entries/proof-theory/appendix-d.html)</sup><sup> • </sup><sup>[12](https://www.sciencedirect.com/science/article/pii/S0168007224000228)</sup>

The theory scales upward through extensions. KPl, KPi without Bounded Collection, proves the same \( \mathcal{L}_2 \)-formulas as \( \Pi^1_1 \)-CA, while KPi, which adds an axiom that every set is contained in an admissible set, corresponds to \( (\Delta^1_2\text{-}\mathrm{CA}) + \mathrm{BI} \).<sup>[1](https://plato.stanford.edu/entries/proof-theory/appendix-d.html)</sup><sup> • </sup><sup>[16](https://plato.stanford.edu/archives/win2021/entries/proof-theory/appendix-d.html)</sup> The ordinal system for KPl assumes infinitely many large ordinals \( \Omega_1, \Omega_2, \Omega_3, \ldots \), each with its own collapsing function \( \alpha \mapsto \psi_{\Omega_n}(\alpha) \); the system for KPi is built using the assumption of an inaccessible cardinal.<sup>[1](https://plato.stanford.edu/entries/proof-theory/appendix-d.html)</sup>

Cook and Rathjen also classified the provably total set-functions of KP: KP proves the totality of a set function precisely when it falls within a hierarchy of set functions based on a relativized constructible hierarchy stretching up to any ordinal below the Bachmann–Howard ordinal. For KP(P), the corresponding hierarchy is based on a relativized von Neumann hierarchy of the same length.<sup>[13](https://arxiv.org/abs/1610.02194)</sup>

A Kreisel–Lévy-type result connects KP's strength to its own soundness: over KP with infinity, transfinite induction along the ordinal \( \varepsilon_{\Omega+1} \), where \( \Omega \) denotes the supremum of all ordinals in the universe, is equivalent to the schema asserting the soundness of KP, mirroring the equivalence over [Peano arithmetic](https://www.edgepedia.org/peano-arithmetic) between induction along \( \varepsilon_0 \) and the soundness of PA.<sup>[14](https://arxiv.org/abs/2212.02843)</sup>

## Intuitionistic KP and the existence property

Intuitionistic Kripke–Platek set theory, IKP, consists of extensionality, empty set, union, pairing, infinity, bounded separation, \( \in \)-induction, and bounded collection, formulated in intuitionistic logic.<sup>[15](https://pmc.ncbi.nlm.nih.gov/articles/PMC10083100/)</sup> The ordinal analyses of IKP(P) and IKP(E) are an important tool in showing that several intuitionistic set theories with the collection axiom possess the existence property: if such a theory proves an existential theorem, then a witness can be provably described in the theory. One example is intuitionistic Zermelo–Fraenkel set theory with bounded separation.<sup>[10](https://arxiv.org/abs/1603.03358)</sup>

## References

1. [Proof Theory, Appendix D: Proof Theory of Set Theories, Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/entries/proof-theory/appendix-d.html)
2. [Admissible sets, admissible computability, KP (lecture notes), University of California, Berkeley](https://math.berkeley.edu/~marks/notes/edst_notes3.pdf)
3. [Resources, Saul Kripke Center, CUNY Graduate Center](https://www.gc.cuny.edu/saul-kripke-center/resources)
4. [Kripke-Platek Set Theory, Knowledge Base](https://lingenic.com/en/Knowledge_Base/Logic/Formal_Logical_Systems/Theories/Kripke-Platek_Set_Theory)
5. [Admissible Sets and Structures (preview), Jon Barwise, Cambridge University Press](https://api.pageplace.de/preview/DT0400.9781316731697_A29773274/preview-9781316731697_A29773274.pdf)
6. [Recursion Theory, Chapter 3: Admissibility and constructibility, O'Reilly](https://www.oreilly.com/library/view/recursion-theory/9783110381290/html/chapter3.html)
7. [Admissible Set Recursion Theories, Studies in Logic and the Foundations of Mathematics](https://www.sciencedirect.com/science/chapter/bookseries/pii/S0049237X08708814)
8. [Alternative Axiomatic Set Theories, Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/entries/settheory-alternative/)
9. [Relativized ordinal analysis: The case of Power Kripke–Platek set theory, ResearchGate](https://www.researchgate.net/publication/259119137_Relativized_ordinal_analysis_The_case_of_Power_Kripke-Platek_set_theory)
10. [Jacob Cook and Michael Rathjen (2016). Ordinal analysis of intuitionistic power and exponentiation Kripke Platek set theory. arXiv.](https://arxiv.org/abs/1603.03358)
11. [Zachiri McKenzie. On a slight weakening of Kripke-Platek Set Theory. arXiv.](https://arxiv.org/abs/2608.23398)
12. [Admissible extensions of subtheories of second order arithmetic, Annals of Pure and Applied Logic](https://www.sciencedirect.com/science/article/pii/S0168007224000228)
13. [Jacob Cook and Michael Rathjen (2016). Classifying the provably total set functions of KP and KP(P). arXiv.](https://arxiv.org/abs/1610.02194)
14. [Kreisel-Lévy-type theorems for Kripke-Platek and other set theories. arXiv.](https://arxiv.org/abs/2212.02843)
15. [Logics and admissible rules of constructive set theories, PMC](https://pmc.ncbi.nlm.nih.gov/articles/PMC10083100/)
16. [Proof Theory > D. Proof Theory of Set Theories (Stanford Encyclopedia of Philosophy/Winter 2021 Edition)](https://plato.stanford.edu/archives/win2021/entries/proof-theory/appendix-d.html)

---
*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Logic and discrete mathematics › Formal logic and foundations › Set theory › Axiomatic set theories › Alternative set theories*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*

License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license