# Importance of Kripke–Platek set theory

What would be importance of Kripke-Platek set theory? I know that Saul Kripke is one of the most important philosophers in the world, but curious in what place he is in set theory.

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Importance for what? –  Qiaochu Yuan Oct 5 '12 at 1:33
General development of set theory. –  W12 Oct 5 '12 at 2:13
I do not understand what you mean by importance. If what you are asking is whether it is a central part of the subject, or it is isolated, or a curiosity, the theory of admissible sets, which is what Kripke-Platek is, is used all over the place, in recursion theory and set theory. Both in classical results, and in fairly recent ones. (I do not know how to measure "people's place in set theory", nor do I think this is the right venue to address that.) –  Andres Caicedo Oct 5 '12 at 3:01
@Mathemagician1234, that is an pretty extreme example of name dropping... –  Mariano Suárez-Alvarez Oct 5 '12 at 6:39

Kripke-Platek set theory is important foundationally as a natural intermediate between weaker theories such as second-order arithmetic and stronger theories such as ZF.

The transitive models of KP are known as "admissible sets". The ordinals $\alpha$ for which $L_\alpha$ is admissible are called "admissible ordinals". These are key subjects of study in a generalized computability theory known as $\alpha$ recursion theory. In turn, admissible sets are closely related to descriptive set theory. The simplest nontrivial admissible ordinal is $\omega^{CK}_1$ which is tightly connected to hyperarithmetical theory, which $\alpha$ recursion theory generalizes.

From a different point of view, a key feature of KP is that separation and collection have been restricted to "predicative" formulas. The $\Sigma_0$ formulas that are allowed in these schemes in KP have only bounded quantifiers, which means that they do not quantify over the entire universe of sets, only over a fixed subset of the universe.

There are also connections between admissible sets and the infinitary logic $L_{{\omega_1},\omega}$. One example of this is the Barwise compactness theorem.

Two detailed references on this are Generalized Recursion Theory by Sacks and Admissible Sets and Structures by Barwise. I believe both of these are now freely available in Project Euclid. A shorter survey by Makkai is in the Handbook of Mathematical Logic.

• Higher Recursion Theory by Sacks at projecteuclid
• Admissible Sets and Structures by Barwise at projecteulid
• Makkai: Admissible Sets and Infinitary Logic (from Handbook of Mathematical Logic By Jon Barwise) at Google Books
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I've added links to the references you've mentioned - I hope you don't mind. Of course, if you don't like the format, you can edit the post. (I've added the links at the end, since I did not want to make changes in your post.) I think having links here might be useful for the people reading your post. –  Martin Sleziak Oct 5 '12 at 6:44
I am also not sure, whether the link which I gave to Sacks' book is the one you had in mind. –  Martin Sleziak Oct 5 '12 at 6:54
@Martin: Thank you for the links. That was exactly the book I had in mind. I typed the wrong title originally. –  Carl Mummert Oct 5 '12 at 11:15