The hierarchies of C(n)-cardinals were introduced by Bagaria in [1] and were further studied and extended by the author in [18] and in [20]. The case of C(n)-extendible cardinals, and of their C(n)+-extendibility variant, is of particular interest since such cardinals have found applications in the areas of category theory, of homotopy theory, and of model theory (see [2], [3], and [4], respectively). However, the exact relation between these two notions had been left unclarified. Moreover, the question of whether the Generalized Continuum Hypothesis (GCH) can be forced while preserving C(n)-extendible cardinals (for n1) also remained open. In this note, we first establish results in the direction of exactly controlling the targets of C(n)-extendibility embeddings. As a corollary, we show that every C(n)-extendible cardinal is in fact C(n)+-extendible; this, in turn, clarifies the assumption needed in some applications obtained in [3]. At the same time, we underline the applicability of our arguments in the context of C(n)-ultrahuge cardinals as well, as these were introduced in [20]. Subsequently, we show that C(n)-extendible cardinals carry their own Laver functions, making them the first known example of C(n)-cardinals that have this desirable feature. Finally, we obtain an alternative characterization of C(n)-extendibility, which we use to answer the question regarding forcing the GCH affirmatively.

With the help of various square principles, we obtain results concerning the consistency strength of several statements about trees containing ascent paths, special trees, and strong chain conditions. Building on a result that shows that Todorčević’s principle $\square \left( {\kappa ,\lambda } \right)$ implies an indexed version of $\square \left( {\kappa ,\lambda } \right)$, we show that for all infinite, regular cardinals $\lambda < \kappa$, the principle $\square \left( \kappa \right)$ implies the existence of a κ-Aronszajn tree containing a λ-ascent path. We then provide a complete picture of the consistency strengths of statements relating the interactions of trees with ascent paths and special trees. As a part of this analysis, we construct a model of set theory in which ${\aleph _2}$-Aronszajn trees exist and all such trees contain ${\aleph _0}$-ascent paths. Finally, we use our techniques to show that the assumption that the κ-Knaster property is countably productive and the assumption that every κ-Knaster partial order is κ-stationarily layered both imply the failure of $\square \left( \kappa \right)$.

In [16], the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway’s ordered field ${\bf{No}}$ of surreal numbers was brought to the fore and employed to provide necessary and sufficient conditions for an ordered field to be isomorphic to an initial subfield of ${\bf{No}}$, i.e., a subfield of ${\bf{No}}$ that is an initial subtree of ${\bf{No}}$. In this sequel to [16], analogous results for ordered abelian groups and ordered domains are established which in turn are employed to characterize the convex subgroups and convex subdomains of initial subfields of ${\bf{No}}$ that are themselves initial. It is further shown that an initial subdomain of ${\bf{No}}$ is discrete if and only if it is a subdomain of ${\bf{No}}$’s canonical integer part ${\bf{Oz}}$ of omnific integers. Finally, making use of class models the results of [16] are extended by showing that the theories of nontrivial divisible ordered abelian groups and real-closed ordered fields are the sole theories of nontrivial densely ordered abelian groups and ordered fields all of whose models are isomorphic to initial subgroups and initial subfields of ${\bf{No}}$.

For each infinite cardinal κ, the set of algebraic hypergraphs having chromatic number no larger than κ is decidable.

In ZFC, the class Ord of ordinals is easily seen to satisfy the definable version of strong inaccessibility. Here we explore deeper ZFC-verifiable combinatorial properties of Ord, as indicated in Theorems A & B below. Note that Theorem A shows the unexpected result that Ord is never definably weakly compact in any model of ZFC.

Theorem A. Let ${\cal M}$ be any model of ZFC.

1. (1) The definable tree property fails in ${\cal M}$: There is an ${\cal M}$-definable Ord-tree with no ${\cal M}$-definable cofinal branch.

2. (2) The definable partition property fails in ${\cal M}$: There is an ${\cal M}$-definable 2-coloring $f:{[X]^2} \to 2$ for some ${\cal M}$-definable proper class X such that no ${\cal M}$-definable proper classs is monochromatic for f.

3. (3) The definable compactness property for ${{\cal L}_{\infty ,\omega }}$ fails in ${\cal M}$: There is a definable theory ${\rm{\Gamma }}$ in the logic ${{\cal L}_{\infty ,\omega }}$ (in the sense of ${\cal M}$) of size Ord such that every set-sized subtheory of ${\rm{\Gamma }}$ is satisfiable in ${\cal M}$, but there is no ${\cal M}$-definable model of ${\rm{\Gamma }}$.

Theorem B. The definable ⋄Ord principle holds in a model ${\cal M}$ of ZFC iff ${\cal M}$ carries an ${\cal M}$-definable global well-ordering.

Theorems A and B above can be recast as theorem schemes in ZFC, or as asserting that a single statement in the language of class theory holds in all ‘spartan’ models of GB (Gödel-Bernays class theory); where a spartan model of GB is any structure of the form $\left( {{\cal M},{D_{\cal M}}} \right)$, where ${\cal M} \models {\rm{ZF}}$ and ${D_{\cal M}}$ is the family of${\cal M}$-definable classes. Theorem C gauges the complexity of the collection GBspa of (Gödel-numbers of) sentences that hold in all spartan models of GB.

Theorem C. GBspa is ${\rm{\Pi }}_1^1$-complete.

We prove a variation of Easton’s lemma for strongly proper forcings, and use it to prove that, unlike the stronger principle IGMP, GMP together with 2ω ≤ ω2 is consistent with the existence of an ω1-distributive nowhere c.c.c. forcing poset of size ω1. We introduce the idea of a weakly guessing model, and prove that many of the strong consequences of the principle GMP follow from the existence of stationarily many weakly guessing models. Using Namba forcing, we construct a model in which there are stationarily many indestructibly weakly guessing models which have a bounded countable subset not covered by any countable set in the model.

We show that if there is a supercompact cardinal, then Keisler’s order is not linear. More specifically, let Tn,k be the theory of the generic n-clique free k-ary graph for any n > k ≥ 3, and let TCas be the simple nonlow theory described by Casanovas in [2]. Then we show that TCas$$Tn,k always, and if there is a supercompact cardinal then Tn,k$$TCas.

We give a model of set theory based on multisets in homotopy type theory. The equality of the model is the identity type. The underlying type of iterative sets can be formulated in Martin-Löf type theory, without Higher Inductive Types (HITs), and is a sub-type of the underlying type of Aczel’s 1978 model of set theory in type theory. The Voevodsky Univalence Axiom and mere set quotients (a mild kind of HITs) are used to prove the axioms of constructive set theory for the model. We give an equivalence to the model provided in Chapter 10 of “Homotopy Type Theory” by the Univalent Foundations Program.

We introduce and study the framework of compact metric structures and their associated notions of isomorphisms such as homeomorphic and bi-Lipschitz isomorphism. This is subsequently applied to model various classification problems in analysis such as isomorphism of C*-algebras and affine homeomorphism of Choquet simplices, where among other things we provide a simple proof of the completeness of the isomorphism relation of separable, simple, nuclear C*-algebras recently established by M. Sabok.

Starting points of this article are fixed point axioms for set-bounded monotone Σ1 definable operators in the context of Kripke–Platek set theory $KP$. We analyze their relationship to other principles such as maximal iterations, bounded proper injections, and Σ1 subset-bounded separation. One of our main results states that in $KP + (V\, = \,L)$ all these principles are equivalent to Σ1 separation.

Profinite algebras are exactly those that are isomorphic to inverse limits of finite algebras. Such algebras are naturally equipped with Boolean topologies. A variety ${\cal V}$ is standard if every Boolean topological algebra with the algebraic reduct in ${\cal V}$ is profinite.

We show that there is no algorithm which takes as input a finite algebra A of a finite type and decide whether the variety $V\left( {\bf{A}} \right)$ generated by A is standard. We also show the undecidability of some related properties. In particular, we solve a problem posed by Clark, Davey, Freese, and Jackson.

We accomplish this by combining two results. The first one is Moore’s theorem saying that there is no algorithm which takes as input a finite algebra A of a finite type and decides whether $V\left( {\bf{A}} \right)$ has definable principal subcongruences. The second is our result saying that possessing definable principal subcongruences yields possessing finitely determined syntactic congruences for varieties. The latter property is known to yield standardness.

We develop an algebraic notion of recognizability for languages of words indexed by countable linear orderings. We prove that this notion is effectively equivalent to definability in monadic second-order (MSO) logic. We also provide three logical applications. First, we establish the first known collapse result for the quantifier alternation of MSO logic over countable linear orderings. Second, we solve an open problem posed by Gurevich and Rabinovich, concerning the MSO-definability of sets of rational numbers using the reals in the background. Third, we establish the MSO-definability of the set of yields induced by an MSO-definable set of trees, confirming a conjecture posed by Bruyère, Carton, and Sénizergues.

We show that from large cardinals it is consistent to have the tree property simultaneously at ${\aleph _{{\omega ^2} + 1}}$ and ${\aleph _{{\omega ^2} + 2}}$ with ${\aleph _{{\omega ^2}}}$ strong limit.

We show that if a first-order structure ${\cal M}$, with universe ℤ, is an expansion of (ℤ,+,0) and a reduct of (ℤ,+,<,0), then ${\cal M}$ must be interdefinable with (ℤ ,+,0) or (ℤ ,+,<,0).

I construct, in ZFC, a forcing notion that collapses $\aleph _3 $ and preserves all other cardinals. The existence of such a forcing answers a question of Uri Abraham from 1983.

We study the complexity of the topological isomorphism relation for various classes of closed subgroups of the group of permutations of the natural numbers. We use the setting of Borel reducibility between equivalence relations on Borel spaces. For profinite, locally compact, and Roelcke precompact groups, we show that the complexity is the same as the one of countable graph isomorphism. For oligomorphic groups, we merely establish this as an upper bound.

We introduce a forcing technique to construct three-dimensional arrays of generic extensions through FS (finite support) iterations of ccc posets, which we refer to as 3D-coherent systems. We use them to produce models of new constellations in Cichoń’s diagram, in particular, a model where the diagram can be separated into 7 different values. Furthermore, we show that this constellation of 7 values is consistent with the existence of a ${\rm{\Delta }}_3^1$ well-order of the reals.

Let R be a commutative integral unital domain and L a free noncommutative Lie algebra over R. In this article we show that the ring R and its action on L are 0-interpretable in L, viewed as a ring with the standard ring language $+ , \cdot ,0$. Furthermore, if R has characteristic zero then we prove that the elementary theory $Th\left( L \right)$ of L in the standard ring language is undecidable. To do so we show that the arithmetic ${\Bbb N} = \langle {\Bbb N}, + , \cdot ,0\rangle $ is 0-interpretable in L. This implies that the theory of $Th\left( L \right)$ has the independence property. These results answer some old questions on model theory of free Lie algebras.

Givant [6] generalized the notion of an atomic pair-dense relation algebra from Maddux [13] by defining the notion of a measurable relation algebra, that is to say, a relation algebra in which the identity element is a sum of atoms that can be measured in the sense that the “size” of each such atom can be defined in an intuitive and reasonable way (within the framework of the first-order theory of relation algebras). In Andréka--Givant [2], a large class of examples of such algebras is constructed from systems of groups, coordinated systems of isomorphisms between quotients of the groups, and systems of cosets that are used to “shift” the operation of relative multiplication. In Givant--Andréka [8], it is shown that the class of these full coset relation algebras is adequate to the task of describing all measurable relation algebras in the sense that every atomic and complete measurable relation algebra is isomorphic to a full coset relation algebra.

Call an algebra $\mathfrak{A}$ a coset relation algebra if $\mathfrak{A}$ is embeddable into some full coset relation algebra. In the present article, it is shown that the class of coset relation algebras is equationally axiomatizable (that is to say, it is a variety), but that no finite set of sentences suffices to axiomatize the class (that is to say, the class is not finitely axiomatizable).

We extend some recent results about bounded invariant equivalence relations and invariant subgroups of definable groups: we show that type-definability and smoothness are equivalent conditions in a wider class of relations than heretofore considered, which includes all the cases for which the equivalence was proved before.

As a by-product, we show some analogous results in purely topological context (without direct use of model theory).