Showing papers in "Notre Dame Journal of Formal Logic in 2020"
TL;DR: It is shown in this paper that there is a plethora of (natural) splittings and disjunctions in Kohlenbach's higher-order RM.
Abstract: Reverse Mathematics (RM hereafter) is a program in the foundations of mathematics founded by Friedman and developed extensively by Simpson and others. The aim of RM is to find the minimal axioms needed to prove a theorem of ordinary, i.e. non-set-theoretic, mathematics. As suggested by the title, this paper deals with two (relatively rare) RM-phenomena, namely splittings and disjunctions. As to splittings, there are some examples in RM of theorems $A, B, C$ such that $A\leftrightarrow (B\wedge C)$, i.e. $A$ can be split into two independent (fairly natural) parts $B$ and $C$. As to disjunctions, there are (very few) examples in RM of theorems $D, E, F$ such that $D\leftrightarrow (E\vee F)$, i.e. $D$ can be written as the disjunction of two independent (fairly natural) parts $E$ and $F$. By contrast, we show in this paper that there is a plethora of (natural) splittings and disjunctions in Kohlenbach's higher-order RM. Finally, we discuss the role of these results in the grand scheme of things.
18 citations
TL;DR: A simplified proof of elimination of imaginaries (in the geometric sorts) in ACVF, based on ideas of Hrushovski is given, which manages to avoid many of the technical issues which arose in the original proof by Haskell, HRushovski, and Macpherson.
Abstract: We give a simplified proof of elimination of imaginaries (in the geometric sorts) in ACVF, based on ideas of Hrushovski. This proof manages to avoid many of the technical issues which arose in the original proof by Haskell, Hrushovski, and Macpherson.
11 citations
TL;DR: In this paper, the authors propose an intuitionistic team semantics, where teams are embedded within intuitionistic Kripke models, and the associated logic is a conservative extension of intuitionistic logic with questions and dependence formulas, including a normal form result, a completeness result, and translations to classical inquisitive logic and modal dependence logic.
Abstract: In recent years, the logic of questions and dependencies has been investigated in the closely related frameworks of inquisitive logic and dependence logic. These investigations have assumed classical logic as the background logic of statements, and added formulas expressing questions and dependencies to this classical core. In this paper, we broaden the scope of these investigations by studying questions and dependency in the context of intuitionistic logic. We propose an intuitionistic team semantics, where teams are embedded within intuitionistic Kripke models. The associated logic is a conservative extension of intuitionistic logic with questions and dependence formulas. We establish a number of results about this logic, including a normal form result, a completeness result, and translations to classical inquisitive logic and modal dependence logic.
9 citations
TL;DR: In this article, the authors studied the dual notions of radicals and socles in module theory from the standpoint of reverse mathematics and showed that the existence of socles of modules over a commutative ring with identity is equivalent to ACA0 over RCA0.
Abstract: This paper studies two dual notions in module theory—namely, radicals and socles—from the standpoint of reverse mathematics. We first consider radicals of Z-modules, where the radical of a Z-module M is defined as the intersection of pM={px:x∈M} with p taken from all primes. It shows that ACA0 is equivalent to the existence of radicals of Z-modules over RCA0. We then study socles of modules over commutative rings with identity. The socle of an R-module M is the largest semisimple submodule of M. We show that the existence of socles of modules over a commutative ring with identity is equivalent to ACA0 over RCA0. Vector spaces are semisimple modules over fields. In general, semisimple modules possess nice properties of vector spaces. Lastly, we study characterizations of semisimple modules over commutative rings using techniques of reverse mathematics.
8 citations
TL;DR: In this article, it was shown that Peano arithmetic (PA) has polynomial proofs of Con(PA+Con∗(PA))↾n, where Con∗ is the slow consistency statement for PA, introduced by S.-D. Friedman, Rathjen and Weiermann.
Abstract: Let Con(T)↾x denote the finite consistency statement “there are no proofs of contradiction in T with ≤x symbols.” For a large class of natural theories T, Pudlak has shown that the lengths of the shortest proofs of Con(T)↾n in the theory T itself are bounded by a polynomial in n. At the same time he conjectures that T does not have polynomial proofs of the finite consistency statements Con(T+Con(T))↾n. In contrast, we show that Peano arithmetic (PA) has polynomial proofs of Con(PA+Con∗(PA))↾n, where Con∗(PA) is the slow consistency statement for Peano arithmetic, introduced by S.-D. Friedman, Rathjen, and Weiermann. We also obtain a new proof of the result that the usual consistency statement Con(PA) is equivalent to e0 iterations of slow consistency. Our argument is proof-theoretic, whereas previous investigations of slow consistency relied on nonstandard models of arithmetic.
5 citations
TL;DR: It is proved that if T is a complete $L$-theory, then T is mutually algebraic if and only if there is some model of T for which every atomic formula has uniformly bounded arrays.
Abstract: We define an easily verifiable notion of an atomic formula having uniformly bounded arrays in a structure M. We prove that if T is a complete L-theory, then T is mutually algebraic if and only if there is some model M of T for which every atomic formula has uniformly bounded arrays. Moreover, an incomplete theory T is mutually algebraic if and only if every atomic formula has uniformly bounded arrays in every model M of T.
4 citations
TL;DR: In this paper, the authors develop a general framework in which various theories of ground decycling can be compared, drawing on graph-theoretic and topological ideas, and show that the notion of asymmetric ground may be indeterminate.
Abstract: Seemingly natural principles about the logic of ground generate cycles of ground; how can this be if ground is asymmetric? The goal of the theory of decycling is to find systematic and principled ways of getting rid of such cycles of ground. In this paper—drawing on graph-theoretic and topological ideas—I develop a general framework in which various theories of decycling can be compared. This allows us to improve on proposals made earlier by Fine and Litland. However, it turns out that there is no unique method of decycling. An important upshot is that the notion of asymmetric ground may be indeterminate.
4 citations
TL;DR: This paper has put forward systematic characterisations of presentism and expansionism which are full blown logics, each logic comprising an axiomatic proof system and an intuitive semantics with respect to which the system is both sound and complete.
Abstract: Temporaryism—the view that not always everything always exists— comes in two main versions: presentism and expansionism (aka the growing block theory of time). Both versions of the view are commonly formulated using the notion of being present, which we, among others, find problematic. Expansionism is also sometimes accused of requiring extraordinary conceptual tools for its formulation. In this paper, we put forward systematic characterisations of presentism and expansionism which involve neither the notion of being present nor unfamiliar conceptual tools. These characterisations are full blown logics, each logic comprising an axiomatic proof system and an intuitive semantics with respect to which the system is both sound and complete.
4 citations
TL;DR: In this paper, it was shown that Tlog of the asymptotic couple of the field of logarithmic transseries is distal, which is NIP.
Abstract: We show that the theory Tlog of the asymptotic couple of the field of logarithmic transseries is distal. As distal theories are NIP (have the non-independence property), this provides a new proof that Tlog is NIP.
3 citations
TL;DR: In this paper, the authors show that propositional intuitionistic logic is the maximal abstract logic satisfying a certain form of compactness, the Tarski union property (TUP), and preservation under asimulations.
Abstract: We show that propositional intuitionistic logic is the maximal (with respect to expressive power) abstract logic satisfying a certain form of compactness, the Tarski union property (TUP), and preservation under asimulations.
3 citations
TL;DR: In this article, the authors studied the problem of proving the paracompactness of the unit interval in higher-order reverse mathematics with respect to the notion of splittings.
Abstract: Reverse mathematics (RM hereafter) is a program in the foundations of mathematics founded by Friedman and developed extensively by Simpson and others. The aim of RM is to find the minimal axioms needed to prove a theorem of ordinary, that is, non-set-theoretic, mathematics. As suggested by the title, this paper deals with the study of the topological notions of dimension and paracompactness, inside Kohlenbach’s higher-order RM. As to splittings, there are some examples in RM of theorems A , B , C such that A ↔ ( B ∧ C ) , that is, A can be split into two independent (fairly natural) parts B and C , and the aforementioned topological notions give rise to a number of splittings involving highly natural A , B , C . Nonetheless, the higher-order picture is markedly different from the second-one: in terms of comprehension axioms, the proof in higher-order RM of, for example, the paracompactness of the unit interval requires full second-order arithmetic, while the second-order/countable version of paracompactness of the unit interval is provable in the base theory RCA 0 . We obtain similarly “exceptional” results for the Urysohn identity, the Lindelof lemma, and partitions of unity. We show that our results exhibit a certain robustness, in that they do not depend on the exact definition of cover, even in the absence of the axiom of choice.
TL;DR: In this paper, the authors investigated pseudofiniteness of certain Hrushovskiy constructions obtained via predimension functions in a relational language consisting of a single relation R and showed that they are decidable and pseudofinite.
Abstract: In a relational language consisting of a single relation R, we investigate pseudofiniteness of certain Hrushovski constructions obtained via predimension functions. It is notable that the arity of the relation R plays a crucial role in this context. When R is ternary, by extending the methods recently developed by Brody and Laskowski, we interpret 〈Q+,<〉 in the 〈K+,≤∗〉-generic and prove that this structure is not pseudofinite. This provides a negative answer to the question posed in an earlier work by Evans and Wong. This result, in fact, unfolds another aspect of complexity of this structure, along with undecidability and the strict order property proved in the mentioned earlier works. On the other hand, when R is binary, it can be shown that the 〈K+,≤∗〉-generic is decidable and pseudofinite.
TL;DR: In this article, it was shown that for all infinite sets M and all strongly almost disjoint families F⊆P(M), |F|<|P(m)| and there are no finite-to-one functions from P(M) into F, where P denotes the power set of M.
Abstract: For a set M, let |M| denote the cardinality of M. A family F is called strongly almost disjoint if there is an n∈ω such that |A∩B|
TL;DR: It is proven that the analogue of Sacks jump inversion fails for the bounded jump and the wtt-reducibility, and it is proved that no c.e. bounded high set can be low by showing that they all have to be Turing complete.
Abstract: We study the relationship between effective domination properties and the bounded jump. We answer two open questions about the bounded jump: (1) We prove that the analogue of Sacks jump inversion fails for the bounded jump and the wtt-reducibility. (2) We prove that no c.e. bounded high set can be low by showing that they all have to be Turing complete. We characterize the class of c.e. bounded high sets as being those sets computing the Halting problem via a reduction with use bounded by an ω-c.e. function. We define several notions of a c.e. set being effectively dominant, and show that together with the bounded high sets they form a proper hierarchy.
TL;DR: In this paper, it was shown that the classes of finitely generic and infinitely generic closure algebras are closed under finite products and bounded Boolean powers, extending part of Hausdorff's theory of reducible sets to existentially closed closures.
Abstract: The study of existentially closed closure algebras begins with Lipparini’s 1982 paper. After presenting new nonelementary axioms for algebraically closed and existentially closed closure algebras and showing that these nonelementary classes are different, this paper shows that the classes of finitely generic and infinitely generic closure algebras are closed under finite products and bounded Boolean powers, extends part of Hausdorff’s theory of reducible sets to existentially closed closure algebras, and shows that finitely generic and infinitely generic closure algebras are elementarily inequivalent. Special properties of algebraically closed (a.c.), existentially closed (e.c.), finitely generic (f.g.), and infinitely generic (i.g.) closure algebras are established along the way.
TL;DR: In this article, it was shown that the Henkin omitting types theorem fails in a very strong sense for L n, where L n is a countable theory that admits elimination of quantifiers, λ is a cardinal < 2 ℵ 0, and F = Γ i : i < λ 〉 is a family of complete nonprincipal types.
Abstract: Fix 2 < n < ω . Let CA n denote the class of cylindric algebras of dimension n , and let RCA n denote the variety of representable CA n ’s. Let L n denote first-order logic restricted to the first n variables. Roughly, CA n , an instance of Boolean algebras with operators, is the algebraic counterpart of the syntax of L n , namely, its proof theory, while RCA n algebraically and geometrically represents the Tarskian semantics of L n . Unlike Boolean algebras having a Stone representation theorem, RCA n ⊊ CA n . Using combinatorial game theory, we show that the existence of certain finite relation algebras RA , which are algebras whose domain consists of binary relations, implies that the celebrated Henkin omitting types theorem fails in a very strong sense for L n . Using special cases of such finite RA ’s, we recover the classical nonfinite axiomatizability results of Monk, Maddux, and Biro on RCA n and we re-prove Hirsch and Hodkinson’s result that the class of completely representable CA n ’s is not first-order definable. We show that if T is an L n countable theory that admits elimination of quantifiers, λ is a cardinal < 2 ℵ 0 , and F = 〈 Γ i : i < λ 〉 is a family of complete nonprincipal types, then F can be omitted in an ordinary countable model of T .
TL;DR: In this article, the authors study properties of definable sets and functions in power-bounded T -convex fields, proving that the latter have the multidimensional Jacobian property and that the theory of T -Convex Fields is b -minimal with centers.
Abstract: We study properties of definable sets and functions in power-bounded T -convex fields, proving that the latter have the multidimensional Jacobian property and that the theory of T -convex fields is b -minimal with centers. Through these results and work of I. Halupczok we ensure that a certain kind of geometrical stratifications exist for definable objects in said fields. We then discuss a number of applications of those stratifications, including applications to Archimedean o-minimal geometry.
TL;DR: In contrast to the model-theoretic analysis, the substitutional account of logical consequence features an intended interpretation, preserves non-relativized truth, and follows more closely traditional denitions of logical consequences.
Abstract: Logical consequence in first-order predicate logic is defined substitutionally in set theory augmented with a primitive satisfaction predicate: an argument is defined to be logically valid if and only if there is no substitution instance with true premises and a false conclusion. Substitution instances are permitted to contain parameters. Variants of this definition of logical consequence are given: logical validity can be defined with or without identity as a logical constant, and quantifiers can be relativized in substitution instances or not. It is shown that the resulting notions of logical consequence are extensionally equivalent to versions of first-order provability and model-theoretic consequence. Every model-theoretic interpretation has a substitutional counterpart, but not vice versa. In particular, in contrast to the model-theoretic account, there is a trivial intended interpretation on the substitutional account, namely, the homophonic interpretation that does not substitute anything. Applications to free logic, and theories and languages other than set theory are sketched.
TL;DR: In this article, the authors characterize the ultrafilters which are generic over the model L ( R ) for the poset of I -positive sets of natural numbers ordered by inclusion.
Abstract: Let I be an F σ -ideal on natural numbers. We characterize the ultrafilters which are generic over the model L ( R ) for the poset of I -positive sets of natural numbers ordered by inclusion.
TL;DR: In this paper, global dynamical properties of the isometry group of the Borel randomization of a separable complete structure were studied and it was shown that if properties such as the Rokhlin property, topometric generics, and extreme amenability hold for isometry groups of the structure, then they also hold in the randomization.
Abstract: We study global dynamical properties of the isometry group of the Borel randomization of a separable complete structure. We show that if properties such as the Rokhlin property, topometric generics, and extreme amenability hold for the isometry group of the structure, then they also hold in the isometry group of the randomization.
TL;DR: For any ω-r. degree d and n-r, there is a maximal incomplete (ω+1)-r as mentioned in this paper, i.e. degree a strictly between d and b.
Abstract: We show that for any ω-r.e. degree d and n-r.e. degree b with d
TL;DR: This note explicitly shows that after combining the standard (classical) values in {>,⊥} to get a space of four values, as FDE demands, the given process of combining values ‘all the way up’ to α many values, for any ordinal α, results in the same account of logical consequence (viz., FDE).
Abstract: A very natural and philosophically important subclassical logic is FDE (for first-degree entailment). This account of logical consequence can be seen as going beyond the standard two-valued account (of “just true” and “just false”) to a four-valued account (adding the additional values of “both true and false” and “neither true nor false”). A natural question arises: What account of logical consequence arises from considering further (positive) combinations of such values? A partial answer was given by Priest in 2014; Shramko and Wansing had also given a partial result some years earlier, although in a different (more algebraic) context. In this note we generalize Priest’s (and indirectly Shramko and Wansing’s) result to show that even if one considers ordinal-many (positive) combinations of the previous values, for any ordinal, the resulting consequence relation (the resulting logic) remains FDE.
TL;DR: In this article, it was shown that determinacy of all projective games with moves in R is equivalent to the statement that Mn(R) exists and satisfies AD for all n∈N. This generalizes a theorem of Martin and Steel for L(R), that is, the case n=0.
Abstract: Let Mn♯(R) denote the minimal active iterable extender model which has n Woodin cardinals and contains all reals, if it exists, in which case we denote by Mn(R) the class-sized model obtained by iterating the topmost measure of Mn(R) class-many times. We characterize the sets of reals which are Σ1-definable from R over Mn(R), under the assumption that projective games on reals are determined: 1. for even n, Σ1Mn(R)=⅁RΠn+11; 2. for odd n, Σ1Mn(R)=⅁RΣn+11. This generalizes a theorem of Martin and Steel for L(R), that is, the case n=0. As consequences of the proof, we see that determinacy of all projective games with moves in R is equivalent to the statement that Mn♯(R) exists for all n∈N, and that determinacy of all projective games of length ω2 with moves in N is equivalent to the statement that Mn♯(R) exists and satisfies AD for all n∈N.
TL;DR: In this paper, the authors axiomatize all combinations of these four-valued logics, for example, the logic of truth and exact truth or the logic for truth and material equivalence, which can express implications involving more than one of these features of propositions.
Abstract: The four-valued semantics of Belnap–Dunn logic, consisting of the truth values True, False, Neither, and Both, gives rise to several nonclassical logics depending on which feature of propositions we wish to preserve: truth, nonfalsity, or exact truth (truth and nonfalsity). Interpreting equality of truth values in this semantics as material equivalence of propositions, we can moreover see the equational consequence relation of this four-element algebra as a logic of material equivalence. In this paper, we axiomatize all combinations of these four-valued logics, for example, the logic of truth and exact truth or the logic of truth and material equivalence. These combined systems are consequence relations which allow us to express implications involving more than one of these features of propositions.
TL;DR: The semantics of the simple type assignment system λ→ is extended so that the completeness theorem holds and Tait's proof of the strong normalization theorem is generalized.
Abstract: Barendregt gave a sound semantics of the simple type assignment system λ → by generalizing Tait’s proof of the strong normalization theorem. In this paper, we aim to extend the semantics so that the completeness theorem holds.
TL;DR: In this article, the meaning of restricted quantification (RQ) when the embedded conditional (implication) is taken as the conditional of some first-order connexive logics is investigated.
Abstract: This paper investigates the meaning of restricted quantification (RQ) (also known as binary quantification) when the embedded conditional (implication) is taken as the conditional of some first-order connexive logics. The study is carried out by checking the suitability of RQ for defining a connexive class theory, in analogy to the definition of Boolean class theory by using RQ in classical logic (embedding the material implication). Negative results are obtained for Wansing’s first-order connexive logic QC and one variant of Priest’s first-order connexive logic QP (based on the null account for paraconsistent logical consequence). A positive result is obtained for another variant of QP (based on the partial account for paraconsistent logical consequence).
TL;DR: In this article, the weak stability principle is introduced for probabilistic measures of coherence, and it is shown that any correct, coherent, and complete PMC cannot satisfy it, and the argument offered in this paper can be applied to any coherence theory that uses a priori procedures.
Abstract: In recent years, some authors have proposed quantitative measures of the coherence of sets of propositions. Such probabilistic measures of coherence (PMCs) are, in general terms, functions that take as their argument a set of propositions (along with some probability distribution) and yield as their value a number that is supposed to represent the degree of coherence of the set. In this paper, I introduce a minimal constraint on PMC theories, the weak stability principle, and show that any correct, coherent, and complete PMC cannot satisfy it. As a matter of fact, the argument offered in this paper can be applied to any coherence theory that uses a priori procedures. I briefly explore some consequences of this fact.
TL;DR: In this article, a canonization scheme for Borel smooth equivalence relations on Rω modulo restriction to E0-large infinite products was proposed, and the authors showed that given a pair of Borel relations E, F on R ω, there is an infinite E 0-large perfect product P⊆R ω such that either E,F ⊆E on P, or for some l < ω the following for all x,y∈P: xEy implies x(l)=y(l), and x↾(
Abstract: We propose a canonization scheme for smooth equivalence relations on Rω modulo restriction to E0-large infinite products. It shows that, given a pair of Borel smooth equivalence relations E, F on Rω, there is an infinite E0-large perfect product P⊆Rω such that either F⊆E on P, or, for some l<ω, the following is true for all x,y∈P: xEy implies x(l)=y(l), and x↾(ω∖{l})=y↾(ω∖{l}) implies xFy.
TL;DR: The descriptive complexity of several $\Pi^1_1$ ranks from classical analysis which are associated to Denjoy integration are analyzed, answering questions of Walsh on the limsup rank on well-founded trees.
Abstract: We analyze the descriptive complexity of several Π11-ranks from classical analysis which are associated to Denjoy integration. We show that VBG, VBG*, ACG, and ACG* are Π11-complete, answering a question of Walsh in case of ACG*. Furthermore, we identify the precise descriptive complexity of the set of functions obtainable with at most α steps of the transfinite process of Denjoy totalization: if |⋅| is the Π11-rank naturally associated to VBG, VBG*, or ACG*, and if α<ω1ck, then {F∈C(I):|F|≤α} is Σ2α0-complete. These finer results are an application of the author’s previous work on the limsup rank on well-founded trees. Finally, {(f,F)∈M(I)×C(I):F∈ACG*andF'=fa.e.} and {f∈M(I):fis Denjoy integrable} are Π11-complete, answering more questions of Walsh.