Peter Holy

CLASS FORCING, THE FORCING THEOREM AND BOOLEAN COMPLETIONS

The forcing theorem is the most fundamental result about set forcing, stating that the forcing relation for any set forcing is definable and that the truth lemma holds, that is everything that holds in a generic extension is forced by a condition in the relevant generic filter. We show that both the definability (and, in fact, even the amenability) of the forcing relation and the truth lemma can fail for class forcing. In addition to these negative results, we show that the forcing theorem is equivalent to the existence of a (certain kind of) Boolean completion, and we introduce a weak combinatorial property (approachability by projections) that implies the forcing theorem to hold. Finally, we show that unlike for set forcing, Boolean completions need not be unique for class forcing.

A quasi-lower bound on the consistency strength of PFA

A long-standing open question is whether supercompactness provides a lower bound on the consistency strength of the Proper Forcing Axiom (PFA). In this article we establish a quasi lower bound by showing that there is a model with a proper class of subcompact cardinals such that PFA (indeed the weaker statement that PFA holds for (2א0)+-linked forcings) fails in all of its proper forcing extensions. Neeman obtained such a result assuming the existence of “fine structural” models containing very large cardinals, however the existence of such models remains open. We show that Neeman’s arguments go through for a similar notion of “L-like” model and establish the existence of Llike models containing very large cardinals. The main technical result needed is the compatibility of Local Club Condensation with Acceptability in the presence of very large cardinals, a result which constitutes further progress in the outer model programme. The core model programme (initiated by Jensen, see Steel’s [16] for a survey) has had considerable success in establishing lower bounds on the consistency strength of set-theoretic statements, up to the level of Woodin cardinals. But the consistency strength of the Proper Forcing Axiom (PFA) is conjectured to be that of a supercompact cardinal, for which no core model theory is currently available. It is therefore worthwhile to consider quasi lower bounds on the consistency strength of PFA and the main result of this paper is that a proper class of subcompact cardinals serves as such a quasi lower bound: Theorem 1. Assuming the consistency of a proper class of subcompact cardinals, it is consistent that there is a proper class of subcompact cardinals, but PFA (even restricted to posets which are (2א0)+-linked) holds in no proper extension of the universe. What exactly is meant by a quasi lower bound? The necessary ingredients are • the desired set-theoretic principle φ for which we want to obtain a quasi-lower bound result 2000 Mathematics Subject Classification. 03E35, 03E55, 03E57.

Characterizations of pretameness and the Ord-cc

It is well known that pretameness implies the forcing theorem, and that pretameness is characterized by the preservation of the axioms of

Forcing lightface definable well-orders without the GCH

\mathsf{ZF}^-

Small embedding characterizations for large cardinals

, that is

Local club condensation and L-likeness

\mathsf{ZF}

LARGE CARDINALS AND LIGHTFACE DEFINABLE WELL-ORDERS, WITHOUT THE GCH

without the power set axiom, or equivalently, by the preservation of the axiom scheme of replacement, for class forcing over models of

Σ1-wellorders without collapsing

\mathsf{ZF}

Condensation and large cardinals

. 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

Locally

\mathrm{Ord}