On the relative strengths of fragments of collection

Abstract

Let M be the basic set theory that consists of the axioms of extensionality, emptyset, pair, union, powerset, infinity, transitive containment, 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 M. We focus on two common parameterisations of the collection: n -collection, which is the usual collection scheme restricted to n -formulae, and strong n -collection, which is equivalent to n -collection plus n+1 -separation. The main result of this paper shows that for all n ≥ 1,(1) M + n+1 -collection + n+2 -induction on ω proves that there exists a transitive model of Zermelo Set Theory plus n -collection, (2) the theory M + n+1 -collection is n+3 -conservative over the theory M + strong n -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 plus V=L) that does not include the powerset axiom.Both Collection and Strong Collection yield ZF when added to M. In § 2, we note that, over M, the restriction of the Strong Collection scheme to n -formulae (strong n -collection) is equivalent to the restriction of the Collection scheme to n -formulae ( n -collection) plus separation for all n+1 -formulae. This means that M plus n+1 -collection proves all instances of strong n -collection. One of the many achievements of [7] is showing that if M is consistent, then so is M plus the Axiom of Choice and strong 0 -collection. In § 3, we investigate the strength of adding ℘ 0 -collection to four of the weak set theories studied in [7]. We show that if T is one of the theories M, Mac, M + H or MOST, then T plus *

End extending models of set theory via power admissible covers

“…The results of [Mat01] and [M] (see [M,Corollary 3.5]) show that Mac and MOST + Π 1 -Collection have the same consistency strength. Therefore, Theorem 3.7 yields: Theorem 3.9 The consistency of MOST+Π 1 -Collection is provable in MOST+Π 1 -Collection+ Σ P 1 -Foundation.…”

Weak set theories in foundational debates

“…So “I am a moderate generic” is Σ11 as meeting every HYP dense open set of conditions is a Σ11 property. By a corollary to the Gandy basis theorem (Sacks Higher Recursion Theory, [61] page 54, Corollary 1.5), there is a generic pair falsefalse⟨f,gfalsefalse⟩ with ω1f=ω1g=ω1normalCK: so by results in [48], both f and g are Σ11-generic, and Mfalse[ffalse],Mfalse[gfalse] are both admissible, and by standard facts about product forcing, f∉Mfalse[gfalse] and g∉Mfalse[ffalse].…”

“…1 : so by results in [49], both f and g are Σ 1 1 -generic, and M[f ], M[g] are both admissible, and by standard facts about product forcing, f ∈ M[g] and g ∈ M[f ].…”