On the relative strengths of fragments of collection

TL;DR

This paper analyzes the relative strength of various fragments of collection in set theory, establishing consistency and conservativity results for theories with restricted collection schemes.

Contribution

It provides new results on the consistency and conservativity of set theories with fragments of collection, especially relating to $ ext{Pi}_n$-collection and strong $ ext{Pi}_n$-collection.

Findings

Proves that certain collection schemes imply the consistency of Zermelo Set Theory.

Shows $ ext{Pi}_{n+3}$-conservativity of $ ext{M}+ ext{Pi}_{n+1} ext{-collection}$ over strong $ ext{Pi}_n$-collection.

Extends results to theories without the powerset axiom, including Kripke-Platek Set Theory.

Abstract

Let $\mathbf{M}$ be the basic set theory that consists of the axioms of extensionality, emptyset, pair, union, powerset, infinity, transitive containment, $\Delta_0$-separation and set foundation. This paper studies the relative strength of set theories obtained by adding fragments of the set-theoretic collection scheme to $\mathbf{M}$. We focus on two common parameterisations of collection: $\Pi_n$-collection, which is the usual collection scheme restricted to $\Pi_n$-formulae, and strong $\Pi_n$-collection, which is equivalent to $\Pi_n$-collection plus $\Sigma_{n+1}$-separation. The main result of this paper shows that for all $n \geq 1$, (1) $\mathbf{M}+\Pi_{n+1}\textrm{-collection}+\Sigma_{n+2}\textrm{-induction on } \omega$ proves the consistency of Zermelo Set Theory plus $\Pi_{n}$-collection, (2) the theory $\mathbf{M}+\Pi_{n+1}\textrm{-collection}$ is $\Pi_{n+3}$-conservative over the theory $\mathbf{M}+\textrm{strong }\Pi_n \textrm{-collection}$. It is also shown that (2) holds for $n=0$ when the Axiom of Choice is included in the base theory. The final section indicates how the proofs of (1) and (2) can be modified to obtain analogues of these results for theories obtained by adding fragments of collection to a base theory (Kripke-Platek Set Theory with Infinity and $V=L$) that does not include the powerset axiom.