The class forcing theorem, which asserts that every class forcing notion $\mathbb{P}$ admits a forcing relation $\Vdash_{\mathbb{P}}$, that is, a relation satisfying the forcing relation recursion -- it follows that statements true in the corresponding forcing extensions are forced and forced statements are true -- is equivalent over Godel-Bernays set theory GBC to the principle of elementary transfinite recursion $\text{ETR}_{\text{Ord}}$ for class recursions of length $\text{Ord}$. It is also equivalent to the existence of truth predicates for the infinitary languages $\mathcal{L}_{\text{Ord},\omega}(\in,A)$, allowing any class parameter $A$; to the existence of truth predicates for the language $\mathcal{L}_{\text{Ord},\text{Ord}}(\in,A)$; to the existence of $\text{Ord}$-iterated truth predicates for first-order set theory $\mathcal{L}_{\omega,\omega}(\in,A)$; to the assertion that every separative class partial order $\mathbb{P}$ has a set-complete class Boolean completion; to a class-join separation principle; and to the principle of determinacy for clopen class games of rank at most $\text{Ord}+1$. Unlike set forcing, if every class forcing notion $\mathbb{P}$ has a forcing relation merely for atomic formulas, then every such $\mathbb{P}$ has a uniform forcing relation applicable simultaneously to all formulas. Our results situate the class forcing theorem in the rich hierarchy of theories between GBC and Kelley-Morse set theory KM.

The exact strength of the class forcing theorem.

Wir definieren lokale Clubmengenkondensation (Local Club Condensation), ein Prinzip, welches Eigenschaften von Godels Kondensationsprinzip isoliert und verallgemeinert. Wir zeigen, dass wir uber einem beliebigen Modell der Mengenlehre durch die Erzwingungsmethode zu einem Modell der Mengenlehre gelangen konnen, welches lokale Clubmengenkondensation erfullt und zugleich verschiedene grosse Kardinalzahlen erhalten werden konnen; insbesondere zeigen wir, dass lokale Clubmengenkondensation mit der Existenz
einer omega-superstarken Kardinalzahl konsistent ist. Wir gehen ahnlich fur Acceptability vor, ein weiteres Prinzip welches Aspekte
von Godels Kondensationsprinzip isoliert und verallgemeinert. Dies setzt das Outer Model Program (zu deutsch Programm der ausseren Modelle) von Sy Friedman fort. Wir fuhren auch eine mogliche Zukunftsanwendung in Bezug auf das Proper Forcing Axiom PFA
oben beschriebener Ergebnisse am Ende der Arbeit auf.

/pdf/condensation-and-large-cardinals-3csixf31bn.pdf

Condensation and large cardinals

This dissertation is a contribution to the project of second-order set theory, which has seen a revival in recent years. The approach is to understand second-order set theory by studying the structure of models of second-order set theories. The main results are the following, organized by chapter. First, I investigate the poset of $T$-realizations of a fixed countable model of $\mathsf{ZFC}$, where $T$ is a reasonable second-order set theory such as $\mathsf{GBC}$ or $\mathsf{KM}$, showing that it has a rich structure. In particular, every countable partial order embeds into this structure. Moreover, we can arrange so that these embedding preserve the existence/nonexistence of upper bounds, at least for finite partial orders. Second I generalize some constructions of Marek and Mostowski from $\mathsf{KM}$ to weaker theories. They showed that every model of $\mathsf{KM}$ plus the Class Collection schema "unrolls" to a model of $\mathsf{ZFC}^-$ with a largest cardinal. I calculate the theories of the unrolling for a variety of second-order set theories, going as weak as $\mathsf{GBC} + \mathsf{ETR}$. I also show that being $T$-realizable goes down to submodels for a broad selection of second-order set theories $T$. Third, I show that there is a hierarchy of transfinite recursion principles ranging in strength from $\mathsf{GBC}$ to $\mathsf{KM}$. This hierarchy is ordered first by the complexity of the properties allowed in the recursions and second by the allowed heights of the recursions. Fourth, I investigate the question of which second-order set theories have least models. I show that strong theories---such as $\mathsf{KM}$ or $\Pi^1_1\text{-}\mathsf{CA}$---do not have least transitive models while weaker theories---from $\mathsf{GBC}$ to $\mathsf{GBC} + \mathsf{ETR}_\mathrm{Ord}$---do have least transitive models.

The Structure of Models of Second-order Set Theories

pdf/characterizations-of-pretameness-and-the-ord-cc-2hbdrco68t.pdf

Characterizations of pretameness and the Ord-cc

The resurrection axioms are forcing axioms introduced recently by Hamkins and Johnstone, developing on ideas of Chalons and Velickovic. We introduce a stronger form of resurrection axioms (the iterated resurrection axioms RAα(Γ) for a class of forcings Γ and a given ordinal α), and show that RAω(Γ) implies generic absoluteness for the first-order theory of Hγ+ with respect to forcings in Γ preserving the axiom, where γ = γΓ is a cardinal which depends on Γ (γΓ = ω1 if Γ is any among the classes of countably closed, proper, semiproper, stationary set preserving forcings). We also prove that the consistency strength of these axioms is below that of a Mahlo cardinal for most forcing classes, and below that of a stationary limit of supercompact cardinals for the class of stationary set preserving posets. Moreover, we outline that simultaneous generic absoluteness for Hγ0+ with respect to Γ0 and for Hγ1+ with respect to Γ1 with γ0 = γΓ0≠γΓ1 = γ1 is in principle possible, and we present several natural models of the Morse–Kelley set theory where this phenomenon occurs (even for all Hγ simultaneously). Finally, we compare the iterated resurrection axioms (and the generic absoluteness results we can draw from them) with a variety of other forcing axioms, and also with the generic absoluteness results by Woodin and the second author.

Absoluteness via resurrection

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.

Class forcing, the forcing theorem, and Boolean completions

Class forcing, the forcing theorem and Boolean completions

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