Importance of Kripke–Platek set theory

Question

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.

Answer 1

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