The Bulletin of Symbolic Logic | 2021
LEVEL THEORY, PART 1: AXIOMATIZING THE BARE IDEA OF A CUMULATIVE HIERARCHY OF SETS
Abstract
This story says nothing at all about the height of any hierarchy, and apparently says almost nothing about the order-type of the stages. It lays down nothing more than the bare idea of a pure cumulative hierarchy. Surprisingly, though, this bare idea already guarantees that the sets are arranged in well-ordered levels. Indeed, this bare idea is quasi-categorical. Otherwise put: the Basic Story pins down any cumulative hierarchy completely, modulo that hierarchy’s height, on which the Story takes no stance. The aim of this paper is to show all of this. I begin by axiomatizing the Basic Story in the most obvious way possible, obtaining Stage Theory, ST. It is clear that any pure cumulative hierarchy satisfies ST. Unfortunately, ST has multiple primitives. To overcome this, I develop Level Theory, LT. Its only primitive is ∈, but LT and ST say exactly the same things about sets (see §§1–4). As such, any cumulative hierarchy satisfies LT. Moreover, LT proves that the levels are well-ordered, and LT is quasi-categorical (see §§5–6). My theory LT builds on work by Dana Scott, Richard Montague, John Derrick, and Michael Potter. I discuss their theories in §8, but I wish to be very clear at the outset: LT is significantly technically simpler than its predecessors, but it owes everything to them.