We answer the question: which levels of Selivanov's fine hierarchy contain new Turing degrees?
Given an algebraically closed field K of characteristic zero, we study the incidence relation between points and irreducible projective curves, or more precisely the poset of irreducible proper subvarieties of ℙ^2(K). Answering a question of Marcus Tressl, we prove that the poset interprets the field, and it is in fact bi-interpretable with the two-sorted structure consisting of the field K and a sort for its finite subsets. In this structure one can define the integers, so the theory is undecidable. When K is the field of complex numbers we can nevertheless obtain a recursive axiomatization modulo the theory of the integers. We also show that the integers are stably embedded and that the poset of irreducible varieties over the complex numbers is not elementarily equivalent to the one over the algebraic numbers.
In this paper, we prove that definable ring topologies on NIP fields are closely connected to NIP integral domains. More precisely, we show that up to elementary equivalence, any NIP topological field arises from an NIP integral domain. As an application, we prove several results about definable ring topologies on NIP fields, including the following. Let K be an NIP field or expansion of a field. Let tau be a definable ring topology on K. Then tau is a field topology, and tau is locally bounded. If K has characteristic p or finite dp-rank, then tau is "generalized t-henselian" in the sense of Dittman, Walsberg and Ye, meaning that the implicit function theorem holds for polynomials. If K has finite dp-rank, then tau must be a topology of "finite breadth" (a Wn-topology). Using these techniques, we give some reformulations of the conjecture that NIP local rings are henselian.
Kruskal's theorem famously states that finite trees (ordered using an infima-preserving embeddability relation) form a well partial order. Freund, Rathjen and Weiermann extended this result to general recursive data types with their uniform Kruskal theorem. They do not only show that this principle is true but also, in the context of reverse mathematics, that their theorem is equivalent to Pi(1)(1)-comprehension, the characterizing axiom of Pi(1)(1) -CA(0). However, their proof is not carried out directly over RCA(0), the usual base system of reverse mathematics. Instead, it additionally requires a weak consequence of Ramsey's theorem for pairs and two colors: the chain antichain principle. In this paper, we show that this additional assumption is not necessary and the considered equivalence between the uniform Kruskal theorem and Pi(1)(1)-comprehension already holds over RCA(0). For this, we improve Girard's characterization of arithmetical comprehension using ordinal exponentiation by showing that his result even remains correct if only a certain subclass of well orders is considered.
It is well known that ordered exponential fields with a compatible non-trivial valuation cannot be spherically complete, but there are some that are “complete enough”. This paper gives analogues of Kaplansky's theorem on maximally valued fields that hold for a suitable class of elementary extensions of some ordered exponential fields with a compatible valuation. More precisely it does so for models of any theory T_convex given by the expansion of a fixed complete o-minimal theory of ordered fields T, by a predicate 𝒪 for a non-trivial T-convex valuation ring. For λ an uncountable cardinal, say that a unary type p(x) over a model of T_convex is λ-bounded weakly immediate if its cut is defined by an empty intersection of fewer than λ many nested valuation balls. Call an elementary extension λ-bounded wim-constructible if it is obtained as a transfinite composition of extensions each generated by one element whose type is λ-bounded weakly immediate. I show that λ-bounded wim-constructible extensions do not extend the residue-field sort and that any two wim-constructible extensions can be amalgamated in an extension which is again λ-bounded wim-constructible over both. A consequence of this is that given an uncountable cardinal λ, every model of T_convex has a unique-up-to-isomorphism λ-spherically complete λ-bounded wim-constructible extension providing an analogue of Kaplansky's theorem. I call this extension the T-λ-spherical completion. Another consequence is that T_convex is definably spherically complete. When T is power bounded wim-constructible extensions are just the immediate extensions. I discuss the example of power bounded theories expanded by exp (simply exponential theories).
We study reduced products $M=\prod_n M_n/\mathrm{Fin}$ of countable structures in a countable language associated with the Fr\'echet ideal. We prove that such $M$ is $2^{\aleph_0}$-saturated if its theory is stable and not $\aleph_2$-saturated otherwise (regardless of whether the Continuum Hypothesis holds). This implies that $M$ is isomorphic to an ultrapower (associated with an ultrafilter on $\mathbb N$) if its theory is stable, even if the CH fails. We also improve a result of Farah and Shelah and prove that there is a forcing extension in which such reduced product $M$ is isomorphic to an ultrapower if and only if the theory of $M$ is stable. All of these conclusions apply for reduced products associated with $F_\sigma$ ideals or more general layered ideals. We also prove that a reduced product associated with the asymptotic density zero ideal $\mathcal Z_0$, or any other analytic P-ideal that is not $F_\sigma$, is not even $\aleph_1$-saturated if its theory is unstable.
In this paper, we prove that the topological conjugacy relations both for minimal compact systems and pointed minimal compact systems are not Borel-reducible to any Borel S(infinity-)action.
In recent years, much work has been done to measure and compare the complexity of orbit equivalence relations, especially for certain classes of Polish groups. We start by introducing some language to organize this previous work, namely the notion of classification strength of Polish groups. Broadly speaking, a Polish group $G$ has stronger classification strength than $H$ if every orbit equivalence relation induced by a continuous action of $H$ on a Polish space can be "emulated" by such an action of $G$ in the sense of Borel reduction. Among the non-Archimedean Polish groups, the groups with the highest classification strength are those that involve $S_\infty$, the Polish group of permutations of a countably-infinite set. We prove that several properties, including a weakening of the disjoint amalgamation in Fra\"{i}ssé theory, a weakening of the existence of an absolute set of generating indiscernibles, and not having ordinal rank for a particular coanalytic rank function, are all equivalent to a non-Archimedean Polish group involving $S_\infty$. Furthermore, we show the equivalence relation $=^+$, which is a relatively simple benchmark equivalence relation in the theory of Borel reducibility, can only be classified by such groups that involve $S_\infty$.
In this paper, we prove the following dichotomy. Given an analytic equivalence relation [Formula: see text], either [Formula: see text] or else any Borel homomorphism from [Formula: see text] to [Formula: see text] is “very far from a reduction”, specifically, it factors, on a comeager set, through the projection map [Formula: see text] for some [Formula: see text]. As a corollary, we prove that [Formula: see text] is a prime equivalence relation, answering a question on Clemens.
The point-to-set principle [J. H. Lutz and N. Lutz, ACM Trans. Comput. Theory, 10(2), 7 (2018).] characterizes the Hausdorff dimension of a subset E subset of & Ropf;n by the effective (or algorithmic) dimension of its individual points. This characterization has been used to prove several results in classical, i.e. without any computability requirements, analysis. Recent work has shown that algorithmic techniques can be fruitfully applied to Marstrand's projection theorem, a fundamental result in fractal geometry. In this paper, we introduce the notion of optimal oracles for subsets E subset of & Ropf;n. One of the primary motivations of this definition is that, if E has optimal oracles, then the conclusion of Marstrand's projection theorem holds for E. We show that every analytic set has optimal oracles. We also prove that if the Hausdorff and packing dimensions of E agree, then E has optimal oracles. Moreover, we show that the existence of sufficiently nice outer measures on E implies the existence of optimal Hausdorff oracles. In particular, the existence of exact gauge functions for a set E is sufficient for the existence of optimal Hausdorff oracles, and is therefore sufficient for Marstrand's theorem. Thus, the existence of optimal oracles extends the currently known sufficient conditions for Marstrand's theorem to hold. Under certain assumptions, every set has optimal oracles. However, assuming the axiom of choice and the continuum hypothesis, we construct sets which do not have optimal oracles. This construction naturally leads to a generalization of Davies theorem on projections.
In this paper, we use our descriptions of Borel Wadge classes from [A. R. Day, N. Greenberg, M. Harrison-Trainor and D. Turetsky, An effective classification of Borel Wadge classes] to characterize those Borel Wadge classes that have the separation property, and those that have the reduction property. Our analysis shows that both properties are equivalent to their effective versions. To do so, we give a characterization of containment between Borel Wadge classes based on their descriptions, and give a direct proof that all such classes admit admissible descriptions.
In this paper, we introduce a hierarchy dividing the set {sigma is an element of Pi(1)(2) : Pi(1)(1)-CA(0) proves sigma}. Then, we give some characterizations of this set using weaker variants of some principles equivalent to Pi(1)(1)-CA(0): leftmost path principle, Ramsey's theorem for Sigma(0)(n) classes of [& Nopf;](& Nopf;) and determinacy for (Sigma(0)(1)) n classes of & Nopf;(& Nopf;).
Let phi be an L(omega 1,omega )sentence which is a minimal counterexample to Vaught's Conjecture and for alpha < omega(1) let F alpha be the fragment of L-omega 1,L-omega consisting of formulas with quantifier rank less than alpha. For a cub set of beta < omega(1), for all countable models M and N of phi with Scott rank at least beta, for all alpha < beta, M is F-alpha-embeddable into N.
A linear order A is called strongly surjective if for every nonempty suborder B <= A, there is an epimorphism from A onto B (denoted by B (sic) A). We show that under MA(aleph 1) there is a strongly surjective Countryman line, answering some questions of D & aacute;niel T. Soukup. We also study the general structure of the class of Aronszajn lines under (sic), and compare it with the well-known embeddability relation <=. Under PFA, the class of Aronszajn lines and the class of countable linear orders enjoy similar nice properties when viewed under the embeddability relation; both are well-quasi-ordered and have a finite basis. We show that this analogy does not extend perfectly to the (sic) relation; while it is known that the countable linear orders are still well-quasi-ordered under (sic), we show that already in ZFC the class of Aronszajn lines has an infinite antichain, and under MA(aleph 1) an infinite decreasing chain as well. We show that some of the analogy survives by proving that under PFA, for some carefully constructed Countryman line C, C and C* form a (sic)-basis for the class of Aronszajn lines. Finally, we show that this does not extend to all uncountable linear orders by proving that there is never a finite (sic)-basis for the uncountable real orders.
Deciding the amalgamation property for a given class of finite structures is an important subroutine in classifying countable finitely homogeneous structures. We study the computational complexity of the amalgamation decision problem for finitely bounded classes, i.e. classes specified by a finite set of forbidden finite substructures, or equivalently by a finite set of universal axioms. We link the amalgamation decision problem to the problem of testing the containment between the reducts of two given finitely bounded amalgamation classes to a given common subset of their signatures. On the one hand, this link enables polynomial-time reductions from various decision problems that can be represented within the reduct containment problem for finitely bounded amalgamation classes, e.g. the 2-exponential square tiling problem, leading to a new lower bound for the complexity of the amalgamation decision problem: 2NEXPTIME-hardness. On the other hand, the link also allows us to show that the amalgamation decision problem is decidable under the assumption that every finitely bounded strong amalgamation class has a computable finitely bounded Ramsey expansion. The runtime of our conditional decision procedure depends 2-exponentially on the size of a minimal Ramsey expansion. We subsequently prove that the closely related problem of testing homogenizability is already undecidable, by a polynomial-time reduction from the regularity of context-free languages. Our results indicate that the relationship between finitely bounded amalgamation classes and arbitrary finitely bounded classes shares similarities with the relationship between regular grammars and context-free grammars. A key difference is that the regularity of context-free grammars can be tested in linear time, while the problem of testing the amalgamation property for finitely bounded classes is 2NEXPTIME-hard.
In this paper, we investigate iterating the construction of C(aa), the L-like inner model constructed using stationary logic. We show that it is possible to force over generic extensions of L to obtain a model of V = C(aa), and to obtain models in which the sequence of iterated C(aa)s is decreasing of arbitrarily large order types. For this, we prove distributivity and stationary-set preservation properties for countable iterations of club-shooting forcings using mutually stationary sets, and introduce the notion of mutually fat sets which yields better distributivity results even for uncountable iterations.
A predilator is a particularly uniform transformation of linear orders. We have a dilator when the transformation preserves well-foundedness. Over the theory ACA(0) from reverse mathematics, any Pi(1)(2)-formula is equivalent to the statement that some predilator is a dilator. We show how this completeness result breaks down without arithmetical comprehension: over RCA(0) + PA, the statements from a large part of the reverse mathematics zoo are not equivalent to some predilator being a dilator.
For each $n\in\mathbb{N}$, let $[n]\phi$ mean "the sentence $\phi$ is true in all $\Sigma_{n+1}$-correct transitive sets." Assuming G\"odel's axiom $V = L$, we prove the following graded variant of Solovay's completeness theorem: the set of formulas valid under this interpretation is precisely the set of theorems of the linear provability logic GLP.3. We also show that this result is not provable in ZFC, so the hypothesis V = L cannot be removed. As part of the proof, we derive (in ZFC) the following purely modal-logical results which are of independent interest: the logic GLP.3 coincides with the logic of closed substitutions of GLP, and is the maximal non-degenerate, normal extension of GLP.
We prove Zilber's trichotomy for reducts of ACVF expanding (K, +) or (K & lowast;,& sdot;).
