Jump inversions of algebraic structures and Σ‐definability

It is proved that for every countable structure and a computable successor ordinal α there is a countable structure which is ‐least among all countable structures such that is Σ‐definable in the αth jump . We also show that this result does not hold for the limit ordinal . Moreover, we prove that there is no countable structure with the degree spectrum for .     

Factorials and the finite sequences of sets

We write for the cardinality of the set of finite sequences of a set which is of cardinality . With the Axiom of Choice (), for every infinite cardinal where is the cardinality of the permutations on a set which is of cardinality . In this paper, we show that “ for every cardinal ”  is provable in and this is the best possible result in the absence of . Similar results are also obtained for : the cardinality of the set of finite sequences without repetition of a set which is of cardinality .     

The cofinality of the least Berkeley cardinal and the extent of dependent choice

This paper is concerned with the possible values of the cofinality of the least Berkeley cardinal. Berkeley cardinals are very large cardinal axioms incompatible with the Axiom of Choice, and the interest in the cofinality of the least Berkeley arises from a result in [1], showing it is connected with the failure of . In fact, by a theorem of Bagaria, Koellner and Woodin, if γ is the cofinality of the least Berkeley cardinal then γ‐ fails. We shall prove that this result is optimal for or . In particular, it will follow that the cofinality of the least Berkeley is independent of .     

A wild model of linear arithmetic and discretely ordered modules

Linear arithmetics are extensions of Presburger arithmetic () by one or more unary functions, each intended as multiplication by a fixed element (scalar), and containing the full induction schemes for their respective languages. In this paper, we construct a model of the 2‐linear arithmetic (linear arithmetic with two scalars) in which an infinitely long initial segment of “Peano multiplication” on is ‐definable. This shows, in particular, that is not model complete in contrast to theories and that are known to satisfy quantifier elimination up to disjunctions of primitive positive formulas. As an application, we show that , as a discretely ordered module over the discretely ordered ring generated by the two scalars, does not have the NIP, answering negatively a question of Chernikov and Hils.     

Neutrally expandable models of arithmetic

A subset of a model of is called neutral if it does not change the relation. A model with undefinable neutral classes is called neutrally expandable. We study the existence and non‐existence of neutral sets in various models of . We show that cofinal extensions of prime models are neutrally expandable, and ω₁‐like neutrally expandable models exist, while no recursively saturated model is neutrally expandable. We also show that neutrality is not a first‐order property. In the last section, we study a local version of neutral expandability.     

On the relative strengths of fragments of collection

Let be the basic set theory that consists of the axioms of extensionality, emptyset, pair, union, powerset, infinity, transitive containment, Δ₀‐separation and set foundation. This paper studies the relative strength of set theories obtained by adding fragments of the set‐theoretic collection scheme to . We focus on two common parameterisations of the collection: ‐collection, which is the usual collection scheme restricted to ‐formulae, and strong ‐collection, which is equivalent to ‐collection plus ‐separation. The main result of this paper shows that for all ,

1. proves that there exists a transitive model of Zermelo Set Theory plus ‐collection,
2. the theory is ‐conservative over the theory .

It is also shown that (2) holds for when the Axiom of Choice is included in the base theory. The final section indicates how the proofs of (1) and (2) can be modified to obtain analogues of these results for theories obtained by adding fragments of collection to a base theory (Kripke‐Platek Set Theory with Infinity plus ) that does not include the powerset axiom.     

A variant of Shelah's characterization of Strong Chang's Conjecture

Shelah considered a certain version of Strong Chang's Conjecture which we denote , and proved that it is equivalent to several statements, including the assertion that Namba forcing is semiproper. We introduce an apparently weaker version, denoted , and prove an analogous characterization of it. In particular, is equivalent to the assertion that the the Friedman‐Krueger poset is semiproper. This strengthens and sharpens results by Cox and sheds some light on problems posed by Usuba, Torres‐Perez and Wu.     

Reductions on equivalence relations generated by universal sets

Let X, Y be Polish spaces, , . We say A is universal for Γ provided that each x‐section of A is in Γ and each element of Γ occurs as an x‐section of A. An equivalence relation generated by a set is denoted by , where . The following results are shown:

• (1) If A is a set universal for all nonempty closed subsets of Y, then is a equivalence relation and .
• (2) If A is a set universal for all countable subsets of Y, then is a equivalence relation, and
  • (i) and ;
  • (ii) if , then ;
  • (iii) if every set is Lebesgue measurable or has the Baire property, then .
  • (iv) for , if every set has the Baire property, and E is any equivalence relation, then .

     

Indivisible sets and well‐founded orientations of the Rado graph

Every set can been thought of as a directed graph whose edge relation is ∈ . We show that many natural examples of directed graphs of this kind are indivisible: for every infinite κ, for every indecomposable λ, and every countable model of set theory. All of the countable digraphs we consider are orientations of the countable random graph. In this way we find indivisible well‐founded orientations of the random graph that are distinct up to isomorphism, and ?₁ that are distinct up to siblinghood.     

Algebraic numbers with elements of small height

In this paper, we study the field of algebraic numbers with a set of elements of small height treated as a predicate. We prove that such structures are not simple and have the independence property. A real algebraic integer is called a Salem number if α and are Galois conjugate and all other Galois conjugates of α lie on the unit circle. It is not known whether 1 is a limit point of Salem numbers. We relate the simplicity of a certain pair with Lehmer's conjecture and obtain a model‐theoretic characterization of Lehmer's conjecture for Salem numbers.     

The first omitting cardinal for Magidority

An infinite cardinal λ is Magidor if and only if . It is known that if λ is Magidor then for some , and the first such α is denoted by . In this paper we try to understand some of the properties of . We prove that can be the successor of a supercompact cardinal, when λ is a Magidor cardinal. From this result we obtain the consistency of being a successor of a singular cardinal with uncountable cofinality.     

Distal and non‐distal behavior in pairs

The aim of this work is an analysis of distal and non‐distal behavior in dense pairs of o‐minimal structures. A characterization of distal types is given through orthogonality to a generic type in , non‐distality is geometrically analyzed through Keisler measures, and a distal expansion for the case of pairs of ordered vector spaces is computed.     

Fermat's last theorem and Catalan's conjecture in weak exponential arithmetics

We study Fermat's last theorem and Catalan's conjecture in the context of weak arithmetics with exponentiation. We deal with expansions of models of arithmetical theories (in the language ) by a binary (partial or total) function e intended as an exponential. We provide a general construction of such expansions and prove that it is universal for the class of all exponentials e which satisfy a certain natural set of axioms . We construct a model and a substructure with e total and (Presburger arithmetic) such that in both and Fermat's last theorem for e is violated by cofinally many exponents n and (in all coordinates) cofinally many pairwise linearly independent triples . On the other hand, under the assumption of ABC conjecture (in the standard model), we show that Catalan's conjecture for e is provable in (even in a weaker theory) and thus holds in and . Finally, we also show that Fermat's last theorem for e is provable (again, under the assumption of ABC in ) in “coprimality for e ”.     

Parametric Presburger arithmetic: complexity of counting and quantifier elimination

We consider an expansion of Presburger arithmetic which allows multiplication by k parameters $t_1,\dots ,t_k$. A formula in this language defines a parametric set $S_{\mathbf{t}}?{\mathbb{Z}}^d$ as $\mathbf{t}$ varies in ${\mathbb{Z}}^k$, and we examine the counting function $|S_{\mathbf{t}}|$ as a function of t . For a single parameter, it is known that $|S_t|$ can be expressed as an eventual quasi‐polynomial (there is a period m such that, for sufficiently large t, the function is polynomial on each of the residue classes mod m). We show that such a nice expression is impossible with 2 or more parameters. Indeed (assuming $\mathbf{P}\ne \mathbf{NP}$) we construct a parametric set $S_{t_1,t_2}$ such that $|S_{t_1,t_2}|$ is not even polynomial‐time computable on input $(t_1,t_2)$. In contrast, for parametric sets $S_{\mathbf{t}}?{\mathbb{Z}}^d$ with arbitrarily many parameters, defined in a similar language without the ordering relation, we show that $|S_{\mathbf{t}}|$ is always polynomial‐time computable in the size of t , and in fact can be represented using the gcd and similar functions.     

A note on prime models in weakly o‐minimal structures

Let be a weakly o‐minimal structure with the strong cell decomposition property. In this note, we show that the canonical o‐minimal extension of is the unique prime model of the full first order theory of over any set . We also show that if two weakly o‐minimal structures with the strong cell decomposition property are isomorphic then, their canonical o‐minimal extensions are isomorphic too. Finally, we show the uniqueness of the prime models in a complete weakly o‐minimal theory with prime models.     

Filtrations of generalized Veltman models

The filtration method is often used to prove the finite model property of modal logics. We adapt this technique to the generalized Veltman semantics for interpretability logics. In order to preserve the defining properties of generalized Veltman models, we use bisimulations to define adequate filtrations. We give an alternative proof of the finite model property of interpretability logic with respect to Veltman models, and we prove the finite model property of the systems and with respect to generalized Veltman models.     

The first measurable cardinal can be the first uncountable regular cardinal at any successor height

We use techniques due to Moti Gitik to construct models in which for an arbitrary ordinal ?, is both the least measurable and least regular uncountable cardinal.     

Classical provability of uniform versions and intuitionistic provability

Along the line of Hirst‐Mummert 9 and Dorais 4 , we analyze the relationship between the classical provability of uniform versions Uni(S) of Π₂‐statements S with respect to higher order reverse mathematics and the intuitionistic provability of S. Our main theorem states that (in particular) for every Π₂‐statement S of some syntactical form, if its uniform version derives the uniform variant of over a classical system of arithmetic in all finite types with weak extensionality, then S is not provable in strong semi‐intuitionistic systems including bar induction in all finite types but also nonconstructive principles such as K?nig's lemma and uniform weak K?nig's lemma . Our result is applicable to many mathematical principles whose sequential versions imply .     

Incomparable ω1‐like models of set theory

We show that the analogues of the embedding theorems of [3], proved for the countable models of set theory, do not hold when extended to the uncountable realm of ω₁‐like models of set theory. Specifically, under the ⋄ hypothesis and suitable consistency assumptions, we show that there is a family of many ω₁‐like models of , all with the same ordinals, that are pairwise incomparable under embeddability; there can be a transitive ω₁‐like model of that does not embed into its own constructible universe; and there can be an ω₁‐like model of whose structure of hereditarily finite sets is not universal for the ω₁‐like models of set theory.