Ideal Projections and Forcing Projections

It is well known that saturation of ideals is closely related to the “antichain-catching” phenomenon from Foreman-Magidor-Shelah [10]. We consider several antichain-catching properties that are weaker than saturation, and prove: (1) If I is a normal ideal on ω2 which satisfies stationary antichain catching, then there is an inner model with a Woodin cardinal; (2) For any n ∈ ω, it is consistent relative to large cardinals that there is a normal ideal I on ωn which satisfies projective antichain catching, yet I is not saturated (or even strong). This provides a negative answer to Open Question number 13 from Foreman’s chapter in the Handbook of Set Theory ([7])

Martin's maximum and the non-stationary ideal

We analyze the non-stationary ideal and the club filter at aleph_1 under MM

Absolute model companionship, forcibility, and the continuum problem

Absolute model companionship (AMC) is a strict strengthening of model companionship defined as follows: For a theory $T$, $T_{\exists\vee\forall}$ denotes the logical consequences of $T$ which are boolean combinations of universal sentences. $T^*$ is the AMC of $T$ if it is model complete and $T_{\exists\vee\forall}=T^*_{\exists\vee\forall}$. The $\{+,\cdot,0,1\}$-theory $\mathsf{ACF}$ of algebraically closed field is the model companion of the theory of $\mathsf{Fields}$ but not its AMC as $\exists x(x^2+1=0)$ is in $\mathsf{ACF}_{\exists\vee\forall}\setminus \mathsf{Fields}_{\exists\vee\forall}$. We use AMC to study the continuum problem and to gauge the expressive power of forcing. We show that (a definable version of) $2^{\aleph_0}=\aleph_2$ is the unique solution to the continuum problem which can be in the AMC of a "partial Morleyization" of the $\in$-theory $\mathsf{ZFC}+$"there are class many supercompact cardinals". We also show that (assuming large cardinals) forcibility overlaps with the apparently weaker notion of consistency for any mathematical problem $\psi$ expressible as a $\Pi_2$-sentence of a (very large fragment of) third order arithmetic ($\mathsf{CH}$, the Suslin hypothesis, the Whitehead conjecture for free groups are a small sample of such problems $\psi$). Partial Morleyizations can be described as follows: let $\mathsf{Form}_{\tau}$ be the set of first order $\tau$-formulae; for $A\subseteq \mathsf{Form}_\tau$, $\tau_A$ is the expansion of $\tau$ adding atomic relation symbols $R_\phi$ for all formulae $\phi$ in $A$ and $T_{\tau,A}$ is the $\tau_A$-theory asserting that each $\tau$-formula $\phi(\vec{x})\in A$ is logically equivalent to the corresponding atomic formula $R_\phi(\vec{x})$. For a $\tau$-theory $T$ $T+T_{\tau,A}$ is the partial Morleyization of $T$ induced by $A\subseteq \mathsf{Form}_\tau$.Comment: This paper systematizes and improves the results appearing in arxiv submissions arXiv:2101.07573, arXiv:2003.07114, arXiv:2003.0712

A hierarchy of Ramsey-like cardinals

We introduce a hierarchy of large cardinals between weakly compact and measurable cardinals, that is closely related to the Ramsey-like cardinals introduced by Victoria Gitman, and is based on certain infinite filter games, however also has a range of equivalent characterizations in terms of elementary embeddings. The aim of this paper is to locate the Ramsey-like cardinals studied by Gitman, and other well-known large cardinal notions, in this hierarchy

Stationary set preserving L-forcings and the extender algebra

Wir konstruieren das Jensensche L-Forcing und nutzen dieses um die Pi_2 Konsequenzen der Theorie ZFC+BMM+"das nichtstationäre Ideal auf omega_1 ist abschüssig" zu studieren. Viele natürliche Konsequenzen der Theorie ZFC+MM folgen schon aus dieser schwächeren Theorie. Wir geben eine neue Charakterisierung des Axioms Dagger ("Alle Forcings welche stationäre Teilmengen von omega_1 bewahren sind semiproper") in dem wir eine Klasse von L-Forcings isolieren deren Semiproperness äquivalent zu Dagger ist. Wir verallgemeinern ein Resultat von Todorcevic: wir zeigen, dass Rado's Conjecture Dagger impliziert. Des weiteren studieren wir Generizitätsiterationen im Kontext einer messbaren Woodinzahl. Mit diesem Werkzeug erhalten wir eine Verallgemeinerung des Woodinschen Sigma^2_1 Absolutheitstheorems. We review the construction of Jensen's L-forcing which we apply to study the Pi_2 consequences of the theory ZFC + BMM + "the nonstationary ideal on omega_1 is precipitous". Many natural consequences ZFC + MM follow from this weaker theory. We give a new characterization of the axiom dagger ("All stationary set preserving forcings are semiproper") by isolating a class of stationary set preserving L-forcings whose semiproperness is equivalent to dagger. This characterization is used to generalize work of Todorcevic: we show that Rado's Conjecture implies dagger. Furthermore we study genericity iterations beginning with a measurable Woodin cardinal. We obtain a generalization of Woodin's Sigma^2_1 absoluteness theorem