Fisher [10] and Baur [6] showed independently in the seventies that if T is a complete first-order theory extending the theory of modules, then the class of models of T with pure embeddings is stable. In [25, 2.12], it is asked if the same is true for any abstract elementary class $(K, \leq _p)$ such that K is a class of modules and $\leq _p$ is the pure submodule relation. In this paper we give some instances where this is true:Theorem.

Assume R is an associative ring with unity. Let $(K, \leq _p)$ be an AEC such that $K \subseteq R\text {-Mod}$

and K is closed under finite direct sums, then:

• • If K is closed under pure-injective envelopes, then $\mathbf {K}$ is $\lambda $-stable for every $\lambda \geq \operatorname {LS}(\mathbf {K})$ such that $\lambda ^{|R| + \aleph _0}= \lambda $.

• • If K is closed under pure submodules and pure epimorphic images, then $\mathbf {K}$ is $\lambda $-stable for every $\lambda $ such that $\lambda ^{|R| + \aleph _0}= \lambda $.

• • Assume R is Von Neumann regular. If $\mathbf {K}$ is closed under submodules and has arbitrarily large models, then $\mathbf {K}$ is $\lambda $-stable for every $\lambda $ such that $\lambda ^{|R| + \aleph _0}= \lambda $.

As an application of these results we give new characterizations of noetherian rings, pure-semisimple rings, Dedekind domains, and fields via superstability. Moreover, we show how these results can be used to show a link between being good in the stability hierarchy and being good in the axiomatizability hierarchy.

Another application is the existence of universal models with respect to pure embeddings in several classes of modules. Among them, the class of flat modules and the class of $\mathfrak {s}$-torsion modules.

Standard Type Theory, ${\textrm {STT}}$, tells us that $b^n(a^m)$ is well-formed iff $n=m+1$. However, Linnebo and Rayo [23] have advocated the use of Cumulative Type Theory, $\textrm {CTT}$, which has more relaxed type-restrictions: according to $\textrm {CTT}$, $b^\beta (a^\alpha )$ is well-formed iff $\beta>\alpha $. In this paper, we set ourselves against $\textrm {CTT}$. We begin our case by arguing against Linnebo and Rayo’s claim that $\textrm {CTT}$ sheds new philosophical light on set theory. We then argue that, while $\textrm {CTT}$’s type-restrictions are unjustifiable, the type-restrictions imposed by ${\textrm {STT}}$ are justified by a Fregean semantics. What is more, this Fregean semantics provides us with a principled way to resist Linnebo and Rayo’s Semantic Argument for $\textrm {CTT}$. We end by examining an alternative approach to cumulative types due to Florio and Jones [10]; we argue that their theory is best seen as a misleadingly formulated version of ${\textrm {STT}}$.

We solve a longstanding question of Soare by showing that if ${\mathbf d}$ is a non-low$_2$ computably enumerable degree then ${\mathbf d}$ contains a c.e. set with no r-maximal c.e. superset.

In this paper we examine various requirements on the formalisation choices under which self-reference can be adequately formalised in arithmetic. In particular, we study self-referential numberings, which immediately provide a strong notion of self-reference even for expressively weak languages. The results of this paper suggest that the question whether truly self-referential reasoning can be formalised in arithmetic is more sensitive to the underlying coding apparatus than usually believed. As a case study, we show how this sensitivity affects the formal study of certain principles of self-referential truth.

A bipartite graph $H = \left (V_1, V_2; E \right )$ with $\lvert V_1\rvert + \lvert V_2\rvert = n$ is semilinear if $V_i \subseteq \mathbb {R}^{d_i}$ for some $d_i$ and the edge relation E consists of the pairs of points $(x_1, x_2) \in V_1 \times V_2$ satisfying a fixed Boolean combination of s linear equalities and inequalities in $d_1 + d_2$ variables for some s. We show that for a fixed k, the number of edges in a $K_{k,k}$-free semilinear H is almost linear in n, namely $\lvert E\rvert = O_{s,k,\varepsilon }\left (n^{1+\varepsilon }\right )$ for any $\varepsilon> 0$; and more generally, $\lvert E\rvert = O_{s,k,r,\varepsilon }\left (n^{r-1 + \varepsilon }\right )$ for a $K_{k, \dotsc ,k}$-free semilinear r-partite r-uniform hypergraph.

As an application, we obtain the following incidence bound: given $n_1$ points and $n_2$ open boxes with axis-parallel sides in $\mathbb {R}^d$ such that their incidence graph is $K_{k,k}$-free, there can be at most $O_{k,\varepsilon }\left (n^{1+\varepsilon }\right )$ incidences. The same bound holds if instead of boxes, one takes polytopes cut out by the translates of an arbitrary fixed finite set of half-spaces.

We also obtain matching upper and (superlinear) lower bounds in the case of dyadic boxes on the plane, and point out some connections to the model-theoretic trichotomy in o-minimal structures (showing that the failure of an almost-linear bound for some definable graph allows one to recover the field operations from that graph in a definable manner).

We present a natural standard translation of inquisitive modal logic $\mathrm{InqML}$ into first-order logic over the natural two-sorted relational representations of the intended models, which captures the built-in higher-order features of $\mathrm{InqML}$. This translation is based on a graded notion of flatness that ties the inherent second-order, team-semantic features of $\mathrm{InqML}$ over information states to subsets or tuples of bounded size. A natural notion of pseudo-models, which relaxes the non-elementary constraints on the intended models, gives rise to an elementary, purely model-theoretic proof of the compactness property for $\mathrm{InqML}$. Moreover, we prove a Hennessy-Milner theorem for $\mathrm{InqML}$, which crucially uses $\omega $-saturated pseudo-models and the new standard translation. As corollaries we also obtain van Benthem style characterisation theorems.

We address Steel’s Programme to identify a ‘preferred’ universe of set theory and the best axioms extending $\mathsf {ZFC}$ by using his multiverse axioms $\mathsf {MV}$ and the ‘core hypothesis’. In the first part, we examine the evidential framework for $\mathsf {MV}$, in particular the use of large cardinals and of ‘worlds’ obtained through forcing to ‘represent’ alternative extensions of $\mathsf {ZFC}$. In the second part, we address the existence and the possible features of the core of $\mathsf {MV}_T$ (where T is $\mathsf {ZFC}$+Large Cardinals). In the last part, we discuss the hypothesis that the core is Ultimate-L, and examine whether and how, based on this fact, the Core Universist can justify V=Ultimate-L as the best (and ultimate) extension of $\mathsf {ZFC}$. To this end, we take into account several strategies, and assess their prospects in the light of $\mathsf {MV}$’s evidential framework.

This paper is dedicated to extending and adapting to modal logic the approach of fractional semantics to classical logic. This is a multi-valued semantics governed by pure proof-theoretic considerations, whose truth-values are the rational numbers in the closed interval $[0,1]$. Focusing on the modal logic K, the proposed methodology relies on three key components: bilateral sequent calculus, invertibility of the logical rules, and stability (proof-invariance). We show that our semantic analysis of K affords an informational refinement with respect to the standard Kripkean semantics (a new proof of Dugundji’s theorem is a case in point) and it raises the prospect of a proof-theoretic semantics for modal logic.

The general theory developed by Ben Yaacov for metric structures provides Fraïssé limits which are approximately ultrahomogeneous. We show here that this result can be strengthened in the case of relational metric structures. We give an extra condition that guarantees exact ultrahomogenous limits. The condition is quite general. We apply it to stochastic processes, the class of diversities, and its subclass of $L_1$ diversities.

We analyze the hereditarily ordinal definable sets $\operatorname {HOD} $ in $M_n(x)[g]$ for a Turing cone of reals x, where $M_n(x)$ is the canonical inner model with n Woodin cardinals build over x and g is generic over $M_n(x)$ for the Lévy collapse up to its bottom inaccessible cardinal. We prove that assuming $\boldsymbol \Pi ^1_{n+2}$-determinacy, for a Turing cone of reals x, $\operatorname {HOD} ^{M_n(x)[g]} = M_n(\mathcal {M}_{\infty } | \kappa _{\infty }, \Lambda ),$ where $\mathcal {M}_{\infty }$ is a direct limit of iterates of $M_{n+1}$, $\delta _{\infty }$ is the least Woodin cardinal in $\mathcal {M}_{\infty }$, $\kappa _{\infty }$ is the least inaccessible cardinal in $\mathcal {M}_{\infty }$ above $\delta _{\infty }$, and $\Lambda $ is a partial iteration strategy for $\mathcal {M}_{\infty }$. It will also be shown that under the same hypothesis $\operatorname {HOD}^{M_n(x)[g]} $ satisfies $\operatorname {GCH} $.

By a classical theorem of Harvey Friedman (1973), every countable nonstandard model $\mathcal {M}$ of a sufficiently strong fragment of ZF has a proper rank-initial self-embedding j, i.e., j is a self-embedding of $\mathcal {M}$ such that $j[\mathcal {M}]\subsetneq \mathcal {M}$, and the ordinal rank of each member of $j[\mathcal {M}]$ is less than the ordinal rank of each element of $\mathcal {M}\setminus j[\mathcal {M}]$. Here, we investigate the larger family of proper initial-embeddings j of models $\mathcal {M}$ of fragments of set theory, where the image of j is a transitive submodel of $\mathcal {M}$. Our results include the following three theorems. In what follows, $\mathrm {ZF}^-$ is $\mathrm {ZF}$ without the power set axiom; $\mathrm {WO}$ is the axiom stating that every set can be well-ordered; $\mathrm {WF}(\mathcal {M})$ is the well-founded part of $\mathcal {M}$; and $\Pi ^1_\infty \text{-}\mathrm {DC}_\alpha $ is the full scheme of dependent choice of length $\alpha $.

Theorem A.

There is an $\omega $-standard countable nonstandard model $\mathcal {M}$ of $\mathrm {ZF}^-+\mathrm {WO}$ that carries no initial self-embedding $j:\mathcal {M} \longrightarrow \mathcal {M}$ other than the identity embedding.

Theorem B.

Every countable $\omega $-nonstandard model $\mathcal {M}$ of $\ \mathrm {ZF}$ is isomorphic to a transitive submodel of the hereditarily countable sets of its own constructible universe $L^{\mathcal {M}}$.

Theorem C.

The following three conditions are equivalent for a countable nonstandard model $\mathcal {M}$ of $\mathrm {ZF}^{-}+\mathrm {WO}+\forall \alpha \ \Pi ^1_\infty \text{-}\mathrm {DC}_\alpha $.

1. (I) There is a cardinal in $\mathcal {M}$ that is a strict upper bound for the cardinality of each member of $\mathrm {WF}(\mathcal {M})$.

2. (II) $\mathrm {WF}(\mathcal {M})$ satisfies the powerset axiom.

3. (III) For all $n \in \omega $ and for all $b \in M$, there exists a proper initial self-embedding $j: \mathcal {M} \longrightarrow \mathcal {M}$ such that $b \in \mathrm {rng}(j)$ and $j[\mathcal {M}] \prec _n \mathcal {M}$.

In 2001, the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway’s ordered field ${\mathbf {No}}$ of surreal numbers was brought to the fore by the first author and employed to provide necessary and sufficient conditions for an ordered field (ordered $K$-vector space) to be isomorphic to an initial subfield ($K$-subspace) of ${\mathbf {No}}$, i.e. a subfield ($K$-subspace) of ${\mathbf {No}}$ that is an initial subtree of ${\mathbf {No}}$. In this sequel, analogous results are established for ordered exponential fields, making use of a slight generalization of Schmeling’s conception of a transseries field. It is further shown that a wide range of ordered exponential fields are isomorphic to initial exponential subfields of $({\mathbf {No}}, \exp )$. These include all models of $T({\mathbb R}_W, e^x)$, where ${\mathbb R}_W$ is the reals expanded by a convergent Weierstrass system W. Of these, those we call trigonometric-exponential fields are given particular attention. It is shown that the exponential functions on the initial trigonometric-exponential subfields of ${\mathbf {No}}$, which includes ${\mathbf {No}}$ itself, extend to canonical exponential functions on their surcomplex counterparts. The image of the canonical map of the ordered exponential field ${\mathbb T}^{LE}$ of logarithmic-exponential transseries into ${\mathbf {No}}$ is shown to be initial, as are the ordered exponential fields ${\mathbb R}((\omega ))^{EL}$ and ${\mathbb R}\langle \langle \omega \rangle \rangle $.

M. E. Rudin (1971) proved, under CH, that for each P-point p there exists a P-point q strictly RK-greater than p. This result was proved under ${\mathfrak {p}= \mathfrak {c}}$ by A. Blass (1973), who also showed that each RK-increasing $ \omega $-sequence of P-points is upper bounded by a P-point, and that there is an order embedding of the real line into the class of P-points with respect to the RK-ordering. In this paper, the results cited above are proved under the (weaker) assumption that $\mathfrak { b}=\mathfrak {c}$. A. Blass asked in 1973 which ordinals can be embedded in the set of P-points, and pointed out that such an ordinal cannot be greater than $ \mathfrak {c}^{+}$. In this paper it is proved, under $\mathfrak {b}=\mathfrak {c}$, that for each ordinal $\alpha < \mathfrak {c}^{+}$, there is an order embedding of $ \alpha $ into P-points. It is also proved, under $\mathfrak {b}=\mathfrak {c}$, that there is an embedding of the long line into P-points.

A structure ${\mathbb Y}$ of a relational language L is called almost chainable iff there are a finite set $F \subset Y$ and a linear order $\,<$ on the set $Y\setminus F$ such that for each partial automorphism $\varphi $ (i.e., local automorphism, in Fraïssé’s terminology) of the linear order $\langle Y\setminus F, <\rangle $ the mapping $\mathop {\mathrm {id}}\nolimits _F \cup \varphi $ is a partial automorphism of ${\mathbb Y}$. By theorems of Fraïssé and Pouzet, an infinite structure ${\mathbb Y}$ is almost chainable iff the profile of ${\mathbb Y}$ is bounded; namely, iff there is a positive integer m such that ${\mathbb Y}$ has $\leq m$ non-isomorphic substructures of size n, for each positive integer n. A complete first order L-theory ${\mathcal T}$ having infinite models is called almost chainable iff all models of ${\mathcal T}$ are almost chainable and it is shown that the last condition is equivalent to the existence of one countable almost chainable model of ${\mathcal T}$. In addition, it is proved that an almost chainable theory has either one or continuum many non-isomorphic countable models and, thus, the Vaught conjecture is confirmed for almost chainable theories.

A permutation group G on a set A is ${\kappa }$-homogeneous iff for all $X,Y\in \bigl [ {A} \bigr ]^ {\kappa } $ with $|A\setminus X|=|A\setminus Y|=|A|$ there is a $g\in G$ with $g[X]=Y$. G is ${\kappa }$-transitive iff for any injective function f with $\operatorname {dom}(f)\cup \operatorname {ran}(f)\in \bigl [ {A} \bigr ]^ {\le {\kappa }} $ and $|A\setminus \operatorname {dom}(f)|=|A\setminus \operatorname {ran}(f)|=|A|$ there is a $g\in G$ with $f\subset g$.

Giving a partial answer to a question of P. M. Neumann [6] we show that there is an ${\omega }$-homogeneous but not ${\omega }$-transitive permutation group on a cardinal ${\lambda }$ provided

1. (i) ${\lambda }<{\omega }_{\omega }$, or

2. (ii) $2^{\omega }<{\lambda }$, and ${\mu }^{\omega }={\mu }^+$ and $\Box _{\mu }$ hold for each ${\mu }\le {\lambda }$ with ${\omega }=\operatorname {cf}({\mu })<{{\mu }}$, or

3. (iii) our model was obtained by adding $(2^{\omega })^+$ many Cohen generic reals to some ground model.

For ${\kappa }>{\omega }$ we give a method to construct large ${\kappa }$-homogeneous, but not ${\kappa }$-transitive permutation groups. Using this method we show that there exist ${\kappa }^+$-homogeneous, but not ${\kappa }^+$-transitive permutation groups on ${\kappa }^{+n}$ for each infinite cardinal ${\kappa }$ and natural number $n\ge 1$ provided $V=L$.

We make comments on some problems Erdős and Hajnal posed in their famous problem list. Let X be a graph on $\omega _1$ with the property that every uncountable set A of vertices contains a finite set s such that each element of $A-s$ is joined to one of the elements of s. Does then X contain an uncountable clique? (Problem 69) We prove that both the statement and its negation are consistent. Do there exist circuitfree graphs $\{X_n:n<\omega \}$ on $\omega _1$ such that if $A\in [\omega _1]^{\aleph _1}$, then $\{n<\omega :X_n\cap [A]^2=\emptyset \}$ is finite? (Problem 61) We show that the answer is yes under CH, and no under Martin’s axiom. Does there exist $F:[\omega _1]^2\to 3$ with all three colors appearing in every uncountable set, and with no triangle of three colors. (Problem 68) We give a different proof of Todorcevic’ theorem that the existence of a $\kappa $-Suslin tree gives $F:[\kappa ]^2\to \kappa $ establishing $\kappa \not \to [\kappa ]^2_{\kappa }$ with no three-colored triangles. This statement in turn implies the existence of a $\kappa $-Aronszajn tree.

In light of a gap found by Krupiński, we give a new proof of associativity for the Morley (or “nonforking”) product of invariant measures in NIP theories.

The $\Omega $ numbers—the halting probabilities of universal prefix-free machines—are known to be exactly the Martin-Löf random left-c.e. reals. We show that one cannot uniformly produce, from a Martin-Löf random left-c.e. real $\alpha $, a universal prefix-free machine U whose halting probability is $\alpha $. We also answer a question of Barmpalias and Lewis-Pye by showing that given a left-c.e. real $\alpha $, one cannot uniformly produce a left-c.e. real $\beta $ such that $\alpha - \beta $ is neither left-c.e. nor right-c.e.

Motivated by recent results and questions of Raghavan and Shelah, we present ZFC theorems on the bounding and various almost disjointness numbers, as well as on reaping and dominating families on uncountable, regular cardinals. We show that if $\kappa =\lambda ^+$ for some $\lambda \geq \omega $ and $\mathfrak {b}(\kappa )=\kappa ^+$ then $\mathfrak {a}_e(\kappa )=\mathfrak {a}_p(\kappa )=\kappa ^+$. If, additionally, $2^{<\lambda }=\lambda $ then $\mathfrak {a}_g(\kappa )=\kappa ^+$ as well. Furthermore, we prove a variety of new bounds for $\mathfrak {d}(\kappa )$ in terms of $\mathfrak {r}(\kappa )$, including $\mathfrak {d}(\kappa )\leq \mathfrak {r}_\sigma (\kappa )\leq \operatorname {\mathrm {cf}}([\mathfrak {r}(\kappa )]^\omega )$, and $\mathfrak {d}(\kappa )\leq \mathfrak {r}(\kappa )$ whenever $\mathfrak {r}(\kappa )<\mathfrak {b}(\kappa )^{+\kappa }$ or $\operatorname {\mathrm {cf}}(\mathfrak {r}(\kappa ))\leq \kappa $ holds.

This paper presents hitherto unpublished writings of Kurt Gödel concerning logical, epistemological, theological, and physical antinomies, which he generally considered as “the most interesting facts in modern logic,” and which he used as a basis for his famous metamathematical results. After investigating different perspectives on the notion of the logical structure of the antinomies and presenting two “antinomies of the intensional,” a new kind of paradox closely related to Gödel’s ontological proof for the existence of God is introduced and completed by a compilation of further theological antinomies. Finally, after a presentation of unpublished general philosophical remarks concerning the antinomies, Gödel’s type-theoretic variant of Leibniz’ Monadology, discovered in his notes on the foundations of quantum mechanics, is examined. Most of the material presented here has been transcribed from the Gabelsberger shorthand system for the first time.