Characterizations of pretameness and the Ord-cc

TL;DR

The paper explores various characterizations of pretameness in class forcing, demonstrating its role as a key dividing line for well-behaved notions and linking it to the Ord-chain condition.

Contribution

It provides multiple new characterizations of pretameness, clarifying its significance in class forcing and its relation to the Ord-chain condition.

Findings

Pretameness implies the forcing theorem.

Pretameness preserves ZF^- axioms and replacement.

Pretameness is characterized by several equivalent conditions.

Abstract

It is well known that pretameness implies the forcing theorem, and that pretameness is characterized by the preservation of the axioms of $\mathsf{ZF}^-$, that is $\mathsf{ZF}$ without the power set axiom, or equivalently, by the preservation of the axiom scheme of replacement, for class forcing over models of $\mathsf{ZF}$. We show that pretameness in fact has various other characterizations, for instance in terms of the forcing theorem, the preservation of the axiom scheme of separation, the forcing equivalence of partial orders and their dense suborders, and the existence of nice names for sets of ordinals. These results show that pretameness is a strong dividing line between well and badly behaved notions of class forcing, and that it is exactly the right notion to consider in applications of class forcing. Furthermore, for most properties under consideration, we also present a corresponding characterization of the $\mathrm{Ord}$-chain condition.