The concepts of closed unbounded (club) and stationary sets are generalised to γ-club and γ-stationary sets, which are closely related to stationary reflection principles. We use these notions to define generalisations of Jensen's combinatorial principles □ as □γ and □<γ sequences. We define Πγ1-indescribability and show first that in L if γ<κ is an ordinal and κ is Σγ1-indescribable but not Πγ1-indescribable, and A⊆κ is γ-stationary, then there is EA⊆A and a □<γ sequence S on κ such that EA is γ-stationary in κ and S avoids EA. This generalises a result of Jensen for γ=1. As a corollary we also extend the result of Jensen that in L a regular cardinal is stationary reflecting if and only if it is Π11-indescribable by showing that such a κ as above is not γ-reflecting, yielding a different proof of a result appearing in [3]. Thus in L a cardinal is Πγ1-indescribable iff it reflects γ-stationary sets. We define □γ(κ), as stating that there is an unthreadable □γ-sequence at κ; we show this implies that κ is not γ+1-reflecting. Certain assumptions on the γ-club filter allow us to prove that γ-stationarity is downwards absolute to L, and allows for splitting of γ-stationary sets. We define γ-ineffability, and look into the relation between γ-ineffability and various ⋄γ principles.

Abstract 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.

Abstract It is shown that the determinacy of $G_{\delta \sigma }$ games of length $\omega ^2$ is equivalent to the existence of a transitive model of ${\mathsf {KP}} + {\mathsf {AD}} + \Pi _1\textrm {-MI}_{\mathbb {R}}$ containing $\mathbb {R}$ . Here, $\Pi _1\textrm {-MI}_{\mathbb {R}}$ is the axiom asserting that every monotone $\Pi _1$ operator on the real numbers has an inductive fixpoint.

Abstract 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. TheoremThe following are equiconsistent: (i) There exists $\kappa $ which is stably measurable; (ii) for some cardinal $\kappa $ , $u_2(\kappa )=\sigma (\kappa )$ ; (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: TheoremAssume $\kappa $ is stably measurable. Then $\Phi (\kappa )$ . And a form of converse: TheoremSuppose 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.

Abstract 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).

Abstract 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.

Abstract The Infinite Time Turing Machine model [8] of Hamkins and Kidder is, in an essential sense, a “Σ2-machine” in that it uses a Σ2Liminf Rule to determine cell values at limit stages of time. We give a generalisation of these machines with an appropriate Σn rule. Such machines either halt or enter an infinite loop by stage , again generalising precisely the ITTM case. The collection of such machines taken together computes precisely those reals of the least model of analysis.

Abstract This paper continues the study of the Ramsey-like large cardinals introduced in [5] and [14]. Ramsey-like cardinals are defined by generalizing the characterization of Ramsey cardinals via the existence of elementary embeddings. Ultrafilters derived from such embeddings are fully iterable and so it is natural to ask about large cardinal notions asserting the existence of ultrafilters allowing only α-many iterations for some countable ordinal α. Here we study such α-iterable cardinals. We show that the α-iterable cardinals form a strict hierarchy for α ≤ ω1, that they are downward absolute to L for , and that the consistency strength of Schindler's remarkable cardinals is strictly between 1-iterable and 2-iterable cardinals. We show that the strongly Ramsey and super Ramsey cardinals from [5] are downward absolute to the core model K. Finally, we use a forcing argument from a strongly Ramsey cardinal to separate the notions of Ramsey and virtually Ramsey cardinals. These were introduced in [14] as an upper bound on the consistency strength of the Intermediate Chang's Conjecture.

Abstract We locate winning strategies for various -games in the L-hierarchy in order to prove the following: Theorem 1. KP + Σ2-Comprehension -Determinacy.” Alternatively: “there is a β-model of -Determinacy.” The implication is not reversible. (The antecedent here may be replaced with instances of Comprehension with only -lightface definable parameters—or even weaker theories.) Theorem 2. KP + Δ2-Comprehension + Σ2-Replacement + -Determinacy. (Here AQI is the assertion that every arithmetical quasi-inductive definition converges.) Alternatively: -Determinacy. Hence the theories: , and are in strictly descending order of strength.

Abstract We give limits defined in terms of abstract pointclasses of the amount of determinacy available in certain canonical inner models involving strong cardinals. We show for example: Theorem A. Det( -IND) ⇒ there exists an inner model with a strong cardinal. Theorem B. Det(AQI) ⇒ there exist type-l mice and hence inner models with proper classes of strong cardinals. where -IND(AQI) is the pointclass of boldface -inductive (respectively arithmetically quasi-inductive) sets of reals.

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.

Abstract We look at various notions of a class of definability operations that generalise inductive operations, and are characterised as “revision operations”. More particularly we: (i) characterise therevision theoretically definablesubsets of a countable acceptable structure; (ii) show that the categorical truth set of Belnap and Gupta's theory of truth over arithmetic usingfullyvaried revision sequences yields a complete Σ31set of integers; (iii) the set ofstably categoricalsentences using their revision operator Ψ is similarly Σ31and which is complete in GÖdel's universe of constructive setsL; (iv) give an alternative account of a theory of truth—realistic variancethat simplifies full variance, whilst at the same time arriving at Kripkean fixed points.

Abstract 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.

Abstract We consider the following question of Kunen: Does Con(ZFC + ∃M a transitive inner model and a non-trivial elementary embedding j: M → V) imply Con(ZFC + ∃ a measurable cardinal)? We use core model theory to investigate consequences of the existence of such a j: M → V. We prove, amongst other things, the existence of such an embedding implies that the core model K is a model of “there exists a proper class of almost Ramsey cardinals”. Conversely, if On is Ramsey, then such a j. M are definable. We construe this as a negative answer to the question above. We consider further the consequences of strengthening the closure assumption on j to having various classes of fixed points.

Abstract We characterise explicitly the decidable predicates on integers of Infinite Time Turing machines, in terms of admissibility theory and the constructible hierarchy. We do this by pinning down ζ, the least ordinal not the length of any eventual output of an Infinite Time Turing machine (halting or otherwise); using this the Infinite Time Turing Degrees are considered, and it is shown how the jump operator coincides with the production of mastercodes for the constructible hierarchy; further that the natural ordinals associated with the jump operator satisfy a Spector criterion, and correspond to the Lζ-stables. It also implies that the machines devised are “Σ2 Complete” amongst all such other possible machines. It is shown that least upper bounds of an “eventual jump” hierarchy exist on an initial segment.

Abstract We follow [8] in asking when a set of ordinals X ⊆ α is a countable union of sets in K, the core model. We show that, analogously to L, an X closed under the canonical Σ1 Skolem function for Kα can be so decomposed provided K is such that no ω-closed filters are put on its measure sequence, but not otherwise. This proviso holds if there is no inner model of a weak Erdős-type property.

Abstract A small large cardinal upper bound in V for proving when certain subsets of ω1 (including the universally Baire subsets) are precisely those constructible from a real is given. In the core model we find an exact equivalence in terms of the length of the mouse order; we show that ∀B ⊆ ω1 [B is universally Baire ⇔ B ϵ L[r] for some real r] is preserved under set-sized forcing extensions if and only if there are arbitrarily large “admissibly measurable” cardinals.

Abstract We set = ‹ , ≤L, # ›, where is the set of degrees of nonconstructibility for countable sets of countable ordinals. We show how to define inductively over this structure the degrees of such sets of ordinals in Κ, the core model, and the next few core models thereafter, i.e. without reference to mice, premice or measurable cardinals

Abstract We show in ZFC, assuming all reals have sharps, that a countable collection of Δ⅓-degrees without a minimal upper bound implies the existence of inner models with measurable cardinals.

Abstract. We investigate the set S2 of “quickly sharped” reals: in the manner of [K] defining a natural hierarchy and quasi-hierarchy of constructibility degrees and identifying their termination points.

In the December 1982 issue of this Journal Weitkamp [W] posed some questions concerning the incomparability of certain “r.e.” sets for the notion of Kleene reducibility. He asked whether the incomparability of, for example, the Friedman set F (defined below) and the set WI0 (the set of reals coding wellfounded trees of admissible height) was equivalent to the existence of 0#, since forcing over L with a set of conditions could not achieve this. We answer this by showing that in a certain class generic extension of L they are comparable, but 0# does not exist. This is an application of Jensen's coding theorem (cf. [BJW]), using a modified construction due to René David [D]. Indeed the result here is a simple application of his result. Define F as follows: Harrington showed, in effect, that one could not add a cone of Turing degrees to this set by forcing with sets of conditions over L. The method used here does add a cone of Turing degrees to a much simpler set RI1 (defined below)—and indeed the whole process could be viewed as forcing over L to obtain the determinacy of certain rather simple sets. It is the determinacy of the game with payoff set RI1 that ensures the comparability of F and WI0 (amongst many others). We shall refrain from repeating all the basic definitions and lemmas since the reader can readily refer to [W]; we shall give the basic necessities.