This paper studies which truth-values are most likely to be taken on finite models by arbitrary sentences of a many-valued predicate logic. The classical zero-one law (independently proved by Fagin and Glebskiĭ et al.) states that every sentence in a purely relational language is almost surely false or almost surely true, meaning that the probability that the formula is true in a randomly chosen finite structures of cardinal n is asymptotically $0$ or $1$ as n grows to infinity. We obtain generalizations of this result for any logic with values in a finite lattice-ordered algebra, and for some infinitely valued logics, including Łukasiewicz logic. The finitely valued case is reduced to the classical one through a uniform translation and Oberschelp’s generalization of the zero-one law. Moreover, it is shown that the complexity of determining the almost sure value of a given sentence is PSPACE-complete (generalizing Grandjean’s result for the classical case), and for some logics we describe completely the set of truth-values that can be taken by sentences almost surely.

We characterize the epimorphisms in homotopy type theory (HoTT) as the fiberwise acyclic maps and develop a type-theoretic treatment of acyclic maps and types in the context of synthetic homotopy theory as developed in univalent foundations. We present examples and applications in group theory, such as the acyclicity of the Higman group, through the identification of groups with 0-connected, pointed 1-types. Many of our results are formalized as part of the agda-unimath library.

In $\mathsf {ZF}$ (i.e., Zermelo–Fraenkel set theory minus the axiom of choice ($\mathsf {AC}$)), we investigate the open problem of the deductive strength of the principle

which was introduced by Herzberg, Kanovei, Katz, and Lyubetsky [(2018), Journal of Symbolic Logic, 83(1), 385–391]. Typical results are:

1. (1) “$\aleph _{1}\leq 2^{\aleph _{0}}$” is strictly weaker than $\mathsf {UF_{wob}}(\omega )$ in $\mathsf {ZF}$.

2. (2) “There exists a free ultrafilter on $\omega $” does not imply “$\aleph _{1}\leq 2^{\aleph _{0}}$” in $\mathsf {ZF}$, and thus (by (1)) neither does it imply $\mathsf {UF_{wob}}(\omega )$ in $\mathsf {ZF}$. This fills the gap in information in Howard and Rubin [Mathematical Surveys and Monographs, American Mathematical Society, 1998], as well as in Herzberg et al. (2018).

3. (3) Martin’s Axiom ($\mathsf {MA}$) implies “no free ultrafilter on $\omega $ has a well-orderable base of cardinality $<2^{\aleph _{0}}$”, and the latter principle is not implied by $\aleph _{0}$-Martin’s Axiom ($\mathsf {MA}(\aleph _{0})$) in $\mathsf {ZF}$.

4. (4) $\mathsf {MA} + \mathsf {UF_{wob}}(\omega )$ implies $\mathsf {AC}(\mathbb {R})$ (the axiom of choice for non-empty sets of reals), which in turn implies $\mathsf {UF_{wob}}(\omega )$. Furthermore, $\mathsf {MA}$ and $\mathsf {UF_{wob}}(\omega )$ are mutually independent in $\mathsf {ZF}$.

5. (5) For any infinite linearly orderable set X, each of “every filter base on X can be well ordered” and “every filter on X has a well-orderable base” is equivalent to “$\wp (X)$ can be well ordered”. This yields novel characterizations of the principle “every linearly ordered set can be well ordered” in $\mathsf {ZFA}$ (i.e., Zermelo–Fraenkel set theory with atoms), and of $\mathsf {AC}$ in $\mathsf {ZF}$.

6. (6) “Every filter on $\mathbb {R}$ has a well-orderable base” implies “every filter on $\omega $ has a well-orderable base”, which in turn implies $\mathsf {UF_{wob}}(\omega )$, and none of these implications are reversible in $\mathsf {ZF}$.

7. (7) “Every filter on $\omega $ can be extended to an ultrafilter with a well-orderable base” is equivalent to $\mathsf {AC}(\mathbb {R}),$ and thus is strictly stronger than $\mathsf {UF_{wob}}(\omega )$ in $\mathsf {ZF}$.

8. (8) “Every filter on $\omega $ can be extended to an ultrafilter” implies “there exists a free ultrafilter on $\omega $ which has no well-orderable base of cardinality ${<2^{\aleph _{0}}}$”. The former principle does not imply “there exists a free ultrafilter on $\omega $ which has no well-orderable base” in $\mathsf {ZF}$, and the latter principle is true in the Basic Cohen Model.

We explore the interplay between $\omega $-categoricity and pseudofiniteness for groups, and we conjecture that $\omega $-categorical pseudofinite groups are finite-by-abelian-by-finite. We show that the conjecture reduces to nilpotent p-groups of class 2, and give a proof that several of the known examples of $\omega $-categorical p-groups satisfy the conjecture. In particular, we show by a direct counting argument that for any odd prime p the ($\omega $-categorical) model companion of the theory of nilpotent class 2 exponent p groups, constructed by Saracino and Wood, is not pseudofinite, and that an $\omega $-categorical group constructed by Baudisch with supersimple rank 1 theory is not pseudofinite. We also survey some scattered literature on $\omega $-categorical groups over 50 years.

We investigate the tower spectrum in the generalized Baire space, i.e., the set of lengths of towers in $\kappa ^\kappa $. We show that both small and large tower spectra at all regular cardinals simultaneously are consistent. Furthermore, based on previous work by Bağ, the first author and Friedman, we prove that globally, a small tower spectrum is consistent with an arbitrarily large spectrum of maximal almost disjoint families. Finally, we show that any non-trivial upper bound on the tower spectrum in $\kappa ^\kappa $ is consistent.

Condensed mathematics, developed by Clausen and Scholze over the last few years, proposes a generalization of topology with better categorical properties. It replaces the concept of a topological space by that of a condensed set, which can be defined as a sheaf for the coherent topology on a certain category of compact Hausdorff spaces. In this case, the sheaf condition has a fairly simple explicit description, which arises from studying the relationship between the coherent, regular, and extensive topologies. In this paper, we establish this relationship under minimal assumptions on the category, going beyond the case of compact Hausdorff spaces. Along the way, we also provide a characterization of sheaves and covering sieves for these categories. All results in this paper have been fully formalized in the Lean proof assistant.

We study the question of $\mathcal {L}_{\mathrm {ring}}$-definability of non-trivial henselian valuation rings. Building on previous work of Jahnke and Koenigsmann, we provide a characterization of henselian fields that admit a non-trivial definable henselian valuation. In particular, we treat the cases where the canonical henselian valuation has positive residue characteristic, using techniques from the model theory and algebra of tame fields.

We start by showing how to approximate unitary and bounded self-adjoint operators by operators in finite dimensional spaces. Using ultraproducts we give a precise meaning for the approximation. In this process we see how the spectral measure is obtained as an ultralimit of counting measures that arise naturally from the finite dimensional approximations. Then we see how generalized distributions can be interpreted in the ultraproduct. Finally we study how one can calculate kernels of operators K by calculating them in the finite dimensional approximations and how one needs to interpret Dirac deltas in the ultraproduct in order to get the kernels as propagators $\langle x_{1}|K|x_{0}\rangle $.

Reverse Mathematics (RM) is a program in the foundations of mathematics where the aim is to find the minimal axioms needed to prove a given theorem of ordinary mathematics. Generally, the minimal axioms are equivalent to the theorem at hand, assuming a weak logical system called the base theory. Moreover, many theorems are either provable in the base theory or equivalent to one of four logical systems, together called the Big Five. For instance, the Weierstrass approximation theorem, i.e., that a continuous function can be approximated uniformly by a sequence of polynomials, has been classified in RM as being equivalent to weak König’s lemma, the second Big Five system. In this paper, we study approximation theorems for discontinuous functions via Bernstein polynomials from the literature. We obtain many equivalences between the latter and weak König’s lemma. We also show that slight variations of these approximation theorems fall far outside of the Big Five but fit in the recently developed RM of new ‘big’ systems, namely the uncountability of ${\mathbb R}$, the enumeration principle for countable sets, the pigeon-hole principle for measure, and the Baire category theorem.

For any $n<\omega $ we construct an infinite $(n+1)$-generated Heyting algebra whose n-generated subalgebras are of cardinality $\leq m_n$ for some positive integer $m_n$. From this we conclude that for every $n<\omega $ there exists a variety of Heyting algebras which contains an infinite $(n+1)$-generated algebra, but which contains only finite n-generated algebras. For the case $n=2$ this provides a negative answer to a question posed by G. Bezhanishvili and R. Grigolia in [4].

All spaces are assumed to be Tychonoff. Given a realcompact space X, we denote by $\mathsf {Exp}(X)$ the smallest infinite cardinal $\kappa $ such that X is homeomorphic to a closed subspace of $\mathbb {R}^\kappa $. Our main result shows that, given a cardinal $\kappa $, the following conditions are equivalent:

• • There exists a countable crowded space X such that $\mathsf {Exp}(X)=\kappa $.

• • $\mathfrak {p}\leq \kappa \leq \mathfrak {c}$.

$\mathfrak {d}\leq \kappa \leq \mathfrak {c}$

$2^\kappa $

$\kappa $

$\kappa $

$\mathsf {Exp}(X)$

It is shown that every model of NF admits a permutation model containing an internal automorphism.

Let K denote prefix-free Kolmogorov complexity, and let $K^A$ denote it relative to an oracle A. We show that for any n, $K^{\emptyset ^{(n)}}$ is definable purely in terms of the unrelativized notion K. It was already known that 2-randomness is definable in terms of K (and plain complexity C) as those reals which infinitely often have maximal complexity. We can use our characterization to show that n-randomness is definable purely in terms of K. To do this we extend a certain “limsup” formula from the literature, and apply Symmetry of Information. This extension entails a novel use of semilow sets, and a more precise analysis of the complexity of $\Delta _2^0$ sets of minimal descriptions.

The property of countable metacompactness of a topological space gets its importance from Dowker’s 1951 theorem that the product of a normal space X with the unit interval $[0,1]$ is again normal iff X is countably metacompact. In a recent paper, Leiderman and Szeptycki studied $\Delta $-spaces, which is a superclass of the class of countably metacompact spaces. They proved that a single Cohen real introduces a ladder system $ L$ over the first uncountable cardinal for which the corresponding space $X_{ L}$ is not a $\Delta $-space, and asked whether there is a ZFC example of a ladder system $ L$ over some cardinal $\kappa $ for which $X_{ L}$ is not countably metacompact, in particular, not a $\Delta $-space. We prove that an affirmative answer holds for the cardinal $\kappa =\operatorname {\mathrm {cf}}(\beth _{\omega +1})$. Assuming $\beth _\omega =\aleph _\omega $, we get an example at a much lower cardinal, namely $\kappa =2^{2^{2^{\aleph _0}}}$, and our ladder system L is moreover $\omega $-bounded.

A recent article of Chernikov, Hrushovski, Kruckman, Krupinski, Moconja, Pillay, and Ramsey finds the first examples of simple structures with formulas which do not fork over the empty set but are universally measure zero. In this article we give the first known simple $\omega $-categorical counterexamples. These happen to be various $\omega $-categorical Hrushovski constructions. Using a probabilistic independence theorem from Jahel and Tsankov, we show how simple $\omega $-categorical structures where a formula forks over $\emptyset $ if and only if it is universally measure zero must satisfy a stronger version of the independence theorem.

We investigate the statement “all automorphisms of ${\mathcal {P}}(\lambda )/[\lambda ]^{<\lambda }$ are trivial.” We show that MA implies the statement for regular uncountable $\lambda <2^{\aleph _0}$, that the statement is false for measurable $\lambda $ if $2^\lambda =\lambda ^+$, and that for “densely trivial” it can be forced (together with $2^\lambda =\lambda ^{++}$) for inaccessible $\lambda $.

There are many results in the literature where superstablity-like independence notions, without any categoricity assumptions, have been used to show the existence of larger models. In this paper we show that stability is enough to construct larger models for small cardinals assuming a mild locality condition for Galois types.Theorem 0.1.

Suppose $\lambda <2^{\aleph _0}$. Let ${\mathbf {K}}$ be an abstract elementary class with $\lambda \geq {\operatorname {LS}}({\mathbf {K}})$. Assume ${\mathbf {K}}$ has amalgamation in $\lambda $, no maximal model in $\lambda $, and is stable in $\lambda $. If ${\mathbf {K}}$ is $(<\lambda ^+, \lambda )$-local, then ${\mathbf {K}}$ has a model of cardinality $\lambda ^{++}$.

The set theoretic assumption that $\lambda <2^{\aleph _0}$ and model theoretic assumption of stability in $\lambda $ can be weakened to the model theoretic assumptions that $|{\mathbf {S}}^{na}(M)|< 2^{\aleph _0}$ for every $M \in {\mathbf {K}}_\lambda $ and stability for $\lambda $-algebraic types in $\lambda $. This is a significant improvement of Theorem 0.1, as the result holds on some unstable abstract elementary classes.

We show that every definable subset of an uncountably categorical pseudofinite structure has pseudofinite cardinality which is polynomial (over the rationals) in the size of any strongly minimal subset, with the degree of the polynomial equal to the Morley rank of the subset. From this fact, we show that classes of finite structures whose ultraproducts all satisfy the same uncountably categorical theory are polynomial R-mecs as well as N-dimensional asymptotic classes, where N is the Morley rank of the theory.

Let M be a short extender mouse. We prove that if $E\in M$ and $M\models $“E is a countably complete short extender whose support is a cardinal $\theta $ and $\mathcal {H}_\theta \subseteq \mathrm {Ult}(V,E)$”, then E is in the extender sequence $\mathbb {E}^M$ of M. We also prove other related facts, and use them to establish that if $\kappa $ is an uncountable cardinal of M and $\kappa ^{+M}$ exists in M then $(\mathcal {H}_{\kappa ^+})^M$ satisfies the Axiom of Global Choice. We prove that if M satisfies the Power Set Axiom then $\mathbb {E}^M$ is definable over the universe of M from the parameter $X=\mathbb {E}^M\!\upharpoonright \!\aleph _1^M$, and M satisfies “Every set is $\mathrm {OD}_{\{X\}}$”. We also prove various local versions of this fact in which M has a largest cardinal, and a version for generic extensions of M. As a consequence, for example, the minimal proper class mouse with a Woodin limit of Woodin cardinals models “$V=\mathrm {HOD}$”. This adapts to many other similar examples. We also describe a simplified approach to Mitchell–Steel fine structure, which does away with the parameters $u_n$.

In this paper we consider the classes of all continuous $\mathcal {L}$-(pre-)structures for a continuous first-order signature $\mathcal {L}$. We characterize the moduli of continuity for which the classes of finite, countable, or all continuous $\mathcal {L}$-(pre-)structures have the amalgamation property. We also characterize when Urysohn continuous $\mathcal {L}$-(pre)-structures exist, establish that certain classes of finite continuous $\mathcal {L}$-structures are countable Fraïssé classes, prove the coherent EPPA for these classes of finite continuous $\mathcal {L}$-structures, and show that actions by automorphisms on finite $\mathcal {L}$-structures also form a Fraïssé class. As consequences, we have that the automorphism group of the Urysohn continuous $\mathcal {L}$-structure is a universal Polish group and that Hall’s universal locally finite group is contained in the automorphism group of the Urysohn continuous $\mathcal {L}$-structure as a dense subgroup.