We analyze the models $L[T_{2n} ]$, where $T_{2n}$ is a tree on $\omega \times \kappa _{2n + 1}^1 $ projecting to a universal ${\rm{\Pi }}_{2n}^1 $ set of reals, for $n > 1$. Following Hjorth’s work on $L[T_2 ]$, we show that under ${\rm{Det}}\left( {{\rm{}}_{2n}^1 } \right)$, the models $L[T_{2n} ]$ are unique, that is they do not depend of the choice of the tree $T_{2n}$. This requires a generalization of the Kechris–Martin theorem to all pointclasses${\rm{\Pi }}_{2n + 1}^1$. We then characterize these models as constructible models relative to the direct limit of all countable nondropping iterates of${\cal M}_{2n + 1}^\# $. We then show that the GCH holds in $L[T_{2n} ]$, for every $n < \omega $, even though they are not extender models. This analysis localizes the HOD analysis of Steel and Woodin at the even levels of the projective hierarchy.

We introduce a mouse whose derived model satisfies $AD_ + {\rm{\Theta }} \ge \theta _{\aleph _2 } $. More generally, we will introduce a class of large cardinal properties yielding mice whose derived models can satisfy properties as strong as $AD_ + {\rm{\Theta }} = \theta _{\rm{\Theta }} $.

Motivated by the search for a logic for polynomial time, we study rank logic (FPR) which extends fixed-point logic with counting (FPC) by operators that determine the rank of matrices over finite fields. While FPR can express most of the known queries that separate FPC from Ptime, almost nothing was known about the limitations of its expressive power.

In our first main result we show that the extensions of FPC by rank operators over different prime fields are incomparable. This solves an open question posed by Dawar and Holm and also implies that rank logic, in its original definition with a distinct rank operator for every field, fails to capture polynomial time. In particular we show that the variant of rank logic ${\text{FPR}}^{\text{*}}$ with an operator that uniformly expresses the matrix rank over finite fields is more expressive than FPR.

One important step in our proof is to consider solvability logic FPS which is the analogous extension of FPC by quantifiers which express the solvability problem for linear equation systems over finite fields. Solvability logic can easily be embedded into rank logic, but it is open whether it is a strict fragment. In our second main result we give a partial answer to this question: in the absence of counting, rank operators are strictly more expressive than solvability quantifiers.

A category has the amalgamation property (AP) if every pushout diagram has a cocone, and the joint embedding property (JEP) if every finite coproduct diagram has a cocone. We show that for a finitely generated category I, the following are equivalent: (i) every I-shaped diagram in a category with the AP and the JEP has a cocone; (ii) every I-shaped diagram in the category of sets and injections has a cocone; (iii) a certain canonically defined category ${\cal L}\left( {\bf{I}} \right)$ of “paths” in I has only idempotent endomorphisms. When I is a finite poset, these are further equivalent to: (iv) every upward-closed subset of I is simply-connected; (v) I can be built inductively via some simple rules. Our proof also shows that these conditions are decidable for finite I.

We give examples of calculi that extend Gentzen’s sequent calculus LK by unsound quantifier inferences in such a way that (i) derivations lead only to true sequents, and (ii) proofs therein are nonelementarily shorter than LK-proofs.

The notion of a symmetric extension extends the usual notion of forcing by identifying a particular class of names which forms an intermediate model of $ZF$ between the ground model and the generic extension, and often the axiom of choice fails in these models. Symmetric extensions are generally used to prove choiceless consistency results. We develop a framework for iterating symmetric extensions in order to construct new models of $ZF$. We show how to obtain some well-known and lesser-known results using this framework. Specifically, we discuss Kinna–Wagner principles and obtain some results related to their failure.

We will describe the Ziegler spectrum over the ring of entire complex valued functions.

We study the number of normal measures on a tall cardinal. Our main results are that:

• • The least tall cardinal may coincide with the least measurable cardinal and carry as many normal measures as desired.

• • The least measurable limit of tall cardinals may carry as many normal measures as desired.

We examine recursive monotonic functions on the Lindenbaum algebra of $EA$. We prove that no such function sends every consistent φ to a sentence with deductive strength strictly between φ and $\left( {\varphi \wedge Con\left( \varphi \right)} \right)$. We generalize this result to iterates of consistency into the effective transfinite. We then prove that for any recursive monotonic function f, if there is an iterate of $Con$ that bounds f everywhere, then f must be somewhere equal to an iterate of $Con$.

We determine the proof-theoretic ordinals (i) of ${\cal C} - {\bf{TI}}[\alpha ]$, the transfinite induction along α, for any hyperarithmetical level ${\cal C}$, in the first order setting and (ii) of any combination of iterated arithmetical comprehension and ${\cal C} - {\bf{TI}}[\alpha ]$ for ${\cal C}\, \equiv \,{\rm{\Pi }}_k^i ,{\rm{\Sigma }}_k^i$ ($i\, = \,0,1$) in the second order setting.

Using a nonLaver modification of Uri Abraham’s minimal $\Delta _3^1$ collapse function, we define a generic extension $L[a]$ by a real a, in which, for a given $n \ge 3$, $\left\{ a \right\}$ is a lightface $\Pi _n^1 $ singleton, a effectively codes a cofinal map $\omega \to \omega _1^L $ minimal over L, while every $\Sigma _n^1 $ set $X \subseteq \omega $ is still constructible.

It is proved that there is a formula $\pi \left( {h,x} \right)$ in the first-order language of group theory such that each component and each non-abelian minimal normal subgroup of a finite group G is definable by $\pi \left( {h,x} \right)$ for a suitable element h of G; in other words, each such subgroup has the form $\left\{ {x|x\pi \left( {h,x} \right)} \right\}$ for some h. A number of consequences for infinite models of the theory of finite groups are described.

Woodin and Vopěnka cardinals are established notions in the large cardinal hierarchy and it is known that Vopěnka cardinals are the Woodin analogue for supercompactness. Here we give the definition of Woodin for strong compactness cardinals, the Woodinised version of strong compactness, and we prove an analogue of Magidor’s identity crisis theorem for the first strongly compact cardinal.

We study the notion of tightly stationary sets which was introduced by Foreman and Magidor in [8]. We obtain two consistency results showing that certain sequences of regular cardinals ${\langle {\kappa _n}\rangle _{n < \omega }}$ can have the property that in some generic extension, every ground-model sequence of fixed-cofinality stationary sets ${S_n} \subseteq {\kappa _n}$ is tightly stationary. The results are obtained using variations of the short-extenders forcing method.

In this paper, we study finitely axiomatizable conservative extensions of a theory U in the case where U is recursively enumerable and not finitely axiomatizable. Stanisław Krajewski posed the question whether there are minimal conservative extensions of this sort. We answer this question negatively.

Consider a finite expansion of the signature of U that contains at least one predicate symbol of arity ≥ 2. We show that, for any finite extension α of U in the expanded language that is conservative over U, there is a conservative extension β of U in the expanded language, such that $\alpha \vdash \beta$ and $\beta \not \vdash \alpha$. The result is preserved when we consider either extensions or model-conservative extensions of U instead of conservative extensions. Moreover, the result is preserved when we replace $\dashv$ as ordering on the finitely axiomatized extensions in the expanded language by a relevant kind of interpretability, to wit interpretability that identically translates the symbols of the U-language.

We show that the result fails when we consider an expansion with only unary predicate symbols for conservative extensions of U ordered by interpretability that preserves the symbols of U.

Given a family ${\cal C}$ of infinite subsets of ${\Bbb N}$, we study when there is a Borel function $S:2^{\Bbb N} \to 2^{\Bbb N} $ such that for every infinite $x \in 2^{\Bbb N} $, $S\left( x \right) \in {\Cal C}$ and $S\left( x \right) \subseteq x$. We show that the family of homogeneous sets (with respect to a partition of a front) as given by the Nash-Williams’ theorem admits such a Borel selector. However, we also show that the analogous result for Galvin’s lemma is not true by proving that there is an $F_\sigma $ tall ideal on ${\Bbb N}$ without a Borel selector. The proof is not constructive since it is based on complexity considerations. We construct a ${\bf{\Pi }}_2^1 $ tall ideal on ${\Bbb N}$ without a tall closed subset.

We prove the following two basis theorems for ${\rm{\Sigma }}_2^1$-sets of reals:

1. (1) Every nonthin ${\rm{\Sigma }}_2^1$-set has a perfect ${\rm{\Delta }}_2^1$-subset if and only if it has a nonthin ${\rm{\Delta }}_2^1$-subset, and this is equivalent to the statement that there is a nonconstructible real.

2. (2) Every uncountable ${\rm{\Sigma }}_2^1$-set has an uncountable ${\rm{\Delta }}_2^1$-subset if and only if either every real is constructible or $\omega _1^L$ is countable.

We also apply the method that proves (2) to show that if there is a nonconstructible real, then there is a perfect ${\rm{\Pi }}_2^1$-set with no nonempty ${\rm{\Pi }}_2^1$-thin subset, strengthening a result of Harrington [4].

We prove that, in order to establish that a theory is NSOP1, it suffices to show that no formula in a single free variable has SOP1.

In this paper, we study the power and limitations of computing effectively generic sequences using effectively random oracles. Previously, it was known that every 2-random sequence computes a 1-generic sequence (as shown by Kautz) and every 2-random sequence forms a minimal pair in the Turing degrees with every 2-generic sequence (as shown by Nies, Stephan, and Terwijn). We strengthen these results by showing that every Demuth random sequence computes a 1-generic sequence and that every Demuth random sequence forms a minimal pair with every pb-generic sequence (where pb-genericity is an effective notion of genericity that is strictly between 1-genericity and 2-genericity). Moreover, we prove that for every comeager ${\cal G} \subseteq {2^\omega }$, there is some weakly 2-random sequence X that computes some $Y \in {\cal G}$, a result that allows us to provide a fairly complete classification as to how various notions of effective randomness interact in the Turing degrees with various notions of effective genericity.

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).