We investigate a variety of cut and choose games, their relationship with (generic) large cardinals, and show that they can be used to characterize a number of properties of ideals and of partial orders: certain notions of distributivity, strategic closure, and precipitousness.

We show that assuming modest large cardinals, there is a definable class of ordinals, closed and unbounded beneath every uncountable cardinal, so that for any closed and unbounded subclasses $P, Q, {\langle L[P],\in ,P \rangle }$ and ${\langle L[Q],\in ,Q \rangle }$ possess the same reals, satisfy the Generalised Continuum Hypothesis, and moreover are elementarily equivalent. Examples of such P are Card, the class of uncountable cardinals, I the uniform indiscernibles, or for any n the class $C^{n}{=_{{\operatorname {df}}}}\{ \lambda \, | \, V_{\lambda } \prec _{{\Sigma }_{n}}V\}$; moreover the theory of such models is invariant under ZFC-preserving extensions. They also all have a rich structure satisfying many of the usual combinatorial principles and a definable wellorder of the reals. The inner model constructed using definability in the language augmented by the Härtig quantifier is thus also characterized.

We define a weak iterability notion that is sufficient for a number of arguments concerning $\Sigma _{1}$-definability at uncountable regular cardinals. In particular we give its exact consistency strength first in terms of the second uniform indiscernible for bounded subsets of $\kappa $: $u_2(\kappa )$, and secondly to give the consistency strength of a property of Lücke’s.

Theorem The following are equiconsistent:

1. (i) There exists $\kappa $ which is stably measurable;

2. (ii) for some cardinal$\kappa $, $u_2(\kappa )=\sigma (\kappa )$;

3. (iii) The $\boldsymbol {\Sigma }_{1}$-club property holds at a cardinal$\kappa $.

Here $\sigma (\kappa )$ is the height of the smallest $M \prec _{\Sigma _{1}} H ( \kappa ^{+} )$ containing $\kappa +1$ and all of $H ( \kappa )$. Let $\Phi (\kappa )$ be the assertion:

Theorem Assume $\kappa $ is stably measurable. Then $\Phi (\kappa )$.

And a form of converse:

Theorem Suppose there is no sharp for an inner model with a strong cardinal. Then in the core model K we have: $\mbox {``}\exists \kappa \Phi (\kappa ) \mbox {''}$ is (set)-generically absolute${\,\longleftrightarrow \,}$ There are arbitrarily large stably measurable cardinals.

When $u_2(\kappa ) < \sigma (\kappa )$ we give some results on inner model reflection.

The Clinical and Translational Science Award (CTSA) Consortium and the National Center for Advancing Translational Science (NCATS) undertook a Common Metrics Initiative to improve research processes across the national CTSA Consortium. This was implemented by Tufts Clinical and Translational Science Institute at the 64 CTSA academic medical centers. Three metrics were collaboratively developed by NCATS staff, CTSA Consortium teams, and outside consultants for Institutional Review Board Review Duration, Careers in Clinical and Translational Research, and Pilot Award Publications and Subsequent Funding. The implementation program included training on the metric operational guidelines, data collection, data reporting system, and performance improvement framework. The implementation team provided small-group coaching and technical assistance. Collaborative learning sessions, driver diagrams, and change packages were used to disseminate best and promising practices. After 14 weeks, 84% of hubs had produced a value for one metric and about half had produced an initial improvement plan. Overall, hubs reported that the implementation activities facilitated their Common Metrics performance improvement process. Experiences implementing the first three metrics can inform future directions of the Common Metrics Initiative and other research groups implementing standardized metrics and performance improvement processes, potentially including other National Institutes of Health institutes and centers.

A correction is needed to our paper: to the definition contained within the statement of Lemma 1.5 and thus arguments around it in §3.

We generalise the α-Ramsey cardinals introduced in Holy and Schlicht (2018) for cardinals α to arbitrary ordinals α, and answer several questions posed in that paper. In particular, we show that α-Ramseys are downwards absolute to the core model K for all α of uncountable cofinality, that strategic ω-Ramsey cardinals are equiconsistent with remarkable cardinals and that strategic α-Ramsey cardinals are equiconsistent with measurable cardinals for all α > ω. We also show that the n-Ramseys satisfy indescribability properties and use them to provide a game-theoretic characterisation of completely ineffable cardinals, as well as establishing further connections between the α-Ramsey cardinals and the Ramsey-like cardinals introduced in Gitman (2011), Feng (1990), and Sharpe and Welch (2011).

The use of Extended Logics to replace ordinary second order definability in Kleene’s Ramified Analytical Hierarchy is investigated. This mirrors a similar investigation of Kennedy, Magidor and Väänänen [11] where Gödel’s universe L of constructible sets is subjected to similar variance. Enhancing second order definability allows models to be defined which may or may not coincide with the original Kleene hierarchy in domain. Extending the logic with game quantifiers, and assuming strong axioms of infinity, we obtain minimal correct models of analysis. A wide spectrum of models can be so generated from abstract definability notions: one may take an abstract Spector Class and extract an extended logic for it. The resultant structure is then a minimal model of the given kind of definability.

We present a forcing to obtain a localized version of Local Club Condensation, a generalized Condensation principle introduced by Sy Friedman and the first author in [3] and [5]. This forcing will have properties nicer than the forcings to obtain this localized version that could be derived from the forcings presented in either [3] or [5]. We also strongly simplify the related proofs provided in [3] and [5]. Moreover our forcing will be capable of introducing this localized principle at κ while simultaneously performing collapses to make κ become the successor of any given smaller regular cardinal. This will be particularly useful when κ has large cardinal properties in the ground model. We will apply this to measure how much L-likeness is implied by Local Club Condensation and related principles. We show that Local Club Condensation at κ+ is consistent with ¬☐κ whenever κ is regular and uncountable, generalizing and improving a result of the third author in [14], and that if κ ≥ ω2 is regular, CC(κ+) - Chang’s Conjecture at κ+ - is consistent with Local Club Condensation at κ+, both under suitable large cardinal consistency assumptions.

This article explores ways in which the Revision Theory of Truth can be expressed in the object language. In particular, we investigate the extent to which semantic deficiency, stable truth, and nearly stable truth can be so expressed, and we study different axiomatic systems for the Revision Theory of Truth.

The Inner Model Hypothesis (IMH) and the Strong Inner Model Hypothesis (SIMH) were introduced in [4]. In this article we establish some upper and lower bounds for their consistency strength.

We repeat the statement of the IMH, as presented in [4]. A sentence in the language of set theory is internally consistent iff it holds in some (not necessarily proper) inner model. The meaning of internal consistency depends on what inner models exist: If we enlarge the universe, it is possible that more statements become internally consistent. The Inner Model Hypothesis asserts that the universe has been maximised with respect to internal consistency:

The Inner Model Hypothesis (IMH): If a statement φ without parameters holds in an inner model of some outer model of V (i.e., in some model compatible with V), then it already holds in some inner model of V.

Equivalently: If φ is internally consistent in some outer model of V then it is already internally consistent in V. This is formalised as follows. Regard V as a countable model of Gödel-Bernays class theory, endowed with countably many sets and classes. Suppose that V* is another such model, with the same ordinals as V. Then V* is an outer model of V (V is an inner model of V*) iff the sets of V* include the sets of V and the classes of V* include the classes of V. V* is compatible with V iff V and V* have a common outer model.

We prove that a form of the Erdӧs property (consistent with V = L[Hω2] and strictly weaker than the Weak Chang's Conjecture at ω1), together with Bounded Martin's Maximum implies that Woodin's principle ψAC holds, and therefore . We also prove that ψAC implies that every function f: ω1 → ω1 is bounded by some canonical function on a club and use this to produce a model of the Bounded Semiproper Forcing Axiom in which Bounded Martin's Maximum fails.