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Nonstandard proof methods in toposes Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20240222
José Siqueira 
A New Model Construction by Making a Detour via Intuitionistic Theories IV: A Closer Connection between KPω and BI Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20240215
Kentaro SatoBy combining tree representation of sets with the method introduced in the previous papers in the series, we give a new interpretation of (Kripke–Platek set theory with the foundation schema restricted to augmented by ) in for any sentence , in such a way that sentences are preserved, where the language of second order arithmetic is considered as a sublanguage of that of set theory via the standard

Arboreal categories and equiresource homomorphism preservation theorems Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20240215
Samson Abramsky, Luca ReggioThe classical homomorphism preservation theorem, due to Łoś, Lyndon and Tarski, states that a firstorder sentence is preserved under homomorphisms between structures if, and only if, it is equivalent to an existential positive sentence . Given a notion of (syntactic) complexity of sentences, an “equiresource” homomorphism preservation theorem improves on the classical result by ensuring that can


Strong ergodicity phenomena for Bernoulli shifts of bounded algebraic dimension Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20240201
Aristotelis Panagiotopoulos, Assaf ShaniThe algebraic dimension of a Polish permutation group is the size of the largest with the property that the orbit of every under the pointwise stabilizer of is infinite. We study the Bernoulli shift for various Polish permutation groups and we provide criteria under which the shift is generically ergodic relative to the injective part of the shift, when has algebraic dimension ≤. We use this to show

The formal verification of the ctm approach to forcing Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20240130
Emmanuel Gunther, Miguel Pagano, Pedro Sánchez Terraf, Matías SteinbergWe discuss some highlights of our computerverified proof of the construction, given a countable transitive setmodel M of ZFC, of generic extensions satisfying ZFC+¬CH and ZFC+CH. Moreover, let R be the set of instances of the Axiom of Replacement. We isolated a 21element subset Ω⊆R and defined F:R→R such that for every Φ⊆R and Mgeneric G, M⊨ZC∪F“Φ∪Ω implies M[G]⊨ZC∪Φ∪{¬CH}, where ZC is Zermelo

Sharp Vaught's conjecture for some classes of partial orders Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20240112
Miloš S. KurilićMatatyahu Rubin has shown that a sharp version of Vaught's conjecture, I(T,ω)∈{0,1,c}, holds for each complete theory of linear order T. We show that the same is true for each complete theory of partial order having a model in the minimal class of partial orders containing the class of linear orders and which is closed under finite products and finite disjoint unions. The same holds for the extension

Towards characterizing the >ω2fickle recursively enumerable Turing degrees Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20240105
Liling KoGiven a finite lattice L that can be embedded in the recursively enumerable (r.e.) Turing degrees 〈RT,≤T〉, it is not known how one can characterize the degrees d∈RT below which L can be embedded. Two important characterizations are of the L7 and M3 lattices, where the lattices are embedded below d if and only if d contains sets of “fickleness” >ω and ≥ωω respectively. We work towards finding a lattice


Structural and universal completeness in algebra and logic Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20231216
Paolo Aglianò, Sara UgoliniIn this work we study the notions of structural and universal completeness both from the algebraic and logical point of view. In particular, we provide new algebraic characterizations of quasivarieties that are actively and passively universally complete, and passively structurally complete. We apply these general results to varieties of bounded lattices and to quasivarieties related to substructural

SOP1, SOP2, and antichain tree property Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20231211
JinHoo Ahn, Joonhee KimIn this paper, we study some tree properties and their related indiscernibilities. First, we prove that SOP2 can be witnessed by a formula with a tree of tuples holding ‘arbitrary homogeneous inconsistency’ (e.g., weak kTP1 conditions or other possible inconsistency configurations). And we introduce a notion of treeindiscernibility, which preserves witnesses of SOP1, and by using this, we investigate

Constructing the constructible universe constructively Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20231207
Richard Matthews, Michael RathjenWe study the properties of the constructible universe, L, over intuitionistic theories. We give an extended set of fundamental operations which is sufficient to generate the universe over Intuitionistic KripkePlatek set theory without Infinity. Following this, we investigate when L can fail to be an inner model in the traditional sense. Namely, we show that over Constructive ZermeloFraenkel (even

On categorical structures arising from implicative algebras: From topology to assemblies Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20231120
Samuele Maschio, Davide TrottaImplicative algebras have been recently introduced by Miquel in order to provide a unifying notion of model, encompassing the most relevant and used ones, such as realizability (both classical and intuitionistic), and forcing. In this work, we initially approach implicative algebras as a generalization of locales, and we extend several topologicallike concepts to the realm of implicative algebras

Probabilistic temporal logic with countably additive semantics Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20231114
Dragan Doder, Zoran OgnjanovićThis work presents a prooftheoretical and modeltheoretical approach to probabilistic temporal logic. We present two novel logics; each of them extends both the language of linear time logic (LTL) and the language of probabilistic logic with polynomial weight formulas. The first logic is designed for reasoning about probabilities of temporal events, allowing statements like “the probability that A

On duality and model theory for polyadic spaces Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20231107
Sam van Gool, Jérémie MarquèsThis paper is a study of firstorder coherent logic from the point of view of duality and categorical logic. We prove a duality theorem between coherent hyperdoctrines and open polyadic Priestley spaces, which we subsequently apply to prove completeness, omitting types, and Craig interpolation theorems for coherent or intuitionistic logic. Our approach emphasizes the role of interpolation and openness

Pathologies in satisfaction classes Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20231020
Athar AbdulQuader, Mateusz ŁełykWe study subsets of countable recursively saturated models of PA which can be defined using pathologies in satisfaction classes. More precisely, we characterize those subsets X such that there is a satisfaction class S where S behaves correctly on an idempotent disjunction of length c if and only if c∈X. We generalize this result to characterize several types of pathologies including double negations

On the geometric equivalence of algebras Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20231006
M. ShahryariIt is known that an algebra is geometrically equivalent to any of its filterpowers if it is qωcompact. We present an explicit description for the radicals of systems of equation over an algebra A and then we prove the above assertion by an elementary new argument. Then we define qκcompact algebras and κfilterpowers for any infinite cardinal κ. We show that any qκcompact algebra is geometric equivalent


Towards a finer classification of strongly minimal sets Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230928
John T. Baldwin, Viktor V. VerbovskiyLet M be strongly minimal and constructed by a ‘Hrushovski construction’ with a single ternary relation. If the Hrushovski algebraization function μ is in a certain class T (μ triples) we show that for independent I with I>1, dcl⁎(I)=∅ (* means not in dcl of a proper subset). This implies the only definable truly nary functions f (f ‘depends’ on each argument), occur when n=1. We prove for Hrushovski's

Classification of ℵ0categorical Cminimal pure Csets Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230927
Françoise Delon, MarieHélène MourguesWe classify all ℵ0categorical and Cminimal Csets up to elementary equivalence. As usual the RyllNardzewski Theorem makes the classification of indiscernible ℵ0categorical Cminimal sets as a first step. We first define solvable good trees, via a finite induction. The trees involved in initial and induction steps have a set of nodes, either consisting of a singleton, or having dense branches without

Positive modal logic beyond distributivity Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230926
Nick Bezhanishvili, Anna Dmitrieva, Jim de Groot, Tommaso MoraschiniWe develop a duality for (modal) lattices that need not be distributive, and use it to study positive (modal) logic beyond distributivity, which we call weak positive (modal) logic. This duality builds on the Hofmann, Mislove and Stralka duality for meetsemilattices. We introduce the notion of Π1persistence and show that every weak positive modal logic is Π1persistent. This approach leads to a new

Weak saturation properties and side conditions Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230906
Monroe EskewTowards combining “compactness” and “hugeness” properties at ω2, we investigate the relevance of sideconditions forcing. We reduce the upper bound on the consistency strength of the weak Chang's Conjecture at ω2 using Neeman's forcing. On the other hand, we find a barrier to the applicability of these methods to our problem and give a counterexample to a claim of Neeman about the effects of iterating

Generalized fusible numbers and their ordinals Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230901
Alexander I. Bufetov, Gabriel Nivasch, Fedor PakhomovErickson defined the fusible numbers as a set F of reals generated by repeated application of the function x+y+12. Erickson, Nivasch, and Xu showed that F is well ordered, with order type ε0. They also investigated a recursively defined function M:R→R. They showed that the set of points of discontinuity of M is a subset of F of order type ε0. They also showed that, although M is a total function on

Extensions of Solovay's system S without independent sets of axioms Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230901
Igor Gorbunov, Dmitry ShkatovChagrov and Zakharyaschev posed the problem of existence of extensions of Solovay's system S, which is a nonnormalizable quasinormal modal logic, that do not admit deductively independent sets of axioms. This paper gives a solution by exhibiting countably many extensions of S without deductively independent sets of axioms.

Absoluteness for the theory of the inner model constructed from finitely many cofinality quantifiers Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230901
Ur Ya'arWe prove that the theory of the models constructible using finitely many cofinality quantifiers – Cλ1,…,λn⁎ and C<λ1,…,<λn⁎ for λ1,…,λn regular cardinals – is setforcing absolute under the assumption of class many Woodin cardinals, and is independent of the regular cardinals used. Towards this goal we prove some properties of the generic embedding induced from the stationary tower restricted to <μclosed

Indestructibility of some compactness principles over models of PFA Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230830
Radek Honzik, Chris LambieHanson, Šárka StejskalováWe show that PFA (Proper Forcing Axiom) implies that adding any number of Cohen subsets of ω will not add an ω2Aronszajn tree or a weak ω1Kurepa tree, and moreover no σcentered forcing can add a weak ω1Kurepa tree (a tree of height and size ω1 with at least ω2 cofinal branches). This partially answers an open problem whether ccc forcings can add ω2Aronszajn trees or ω1Kurepa trees (with ¬□ω1

On countably perfectly meager and countably perfectly null sets Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230829
Tomasz Weiss, Piotr ZakrzewskiWe study a strengthening of the notion of a universally meager set and its dual counterpart that strengthens the notion of a universally null set. We say that a subset A of a perfect Polish space X is countably perfectly meager (respectively, countably perfectly null) in X, if for every perfect Polish topology τ on X, giving the original Borel structure of X, A is covered by an Fσset F in X with the

Primitive recursive reverse mathematics Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230825
Nikolay Bazhenov, Marta FioriCarones, Lu Liu, Alexander MelnikovWe use a secondorder analogy PRA2 of PRA to investigate the prooftheoretic strength of theorems in countable algebra, analysis, and infinite combinatorics. We compare our results with similar results in the fastdeveloping field of primitive recursive (‘punctual’) algebra and analysis, and with results from ‘online’ combinatorics. We argue that PRA2 is sufficiently robust to serve as an alternative

Large cardinals at the brink Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230810
W. Hugh WoodinKunen's theorem that assuming the Axiom of Choice there are no Reinhardt cardinals is a key milestone in the program to understand large cardinal axioms. But this theorem is not the end of a story, rather it is the beginning.

Counterfactual and seeingtoit responsibilities in strategic games Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230805
Pavel Naumov, Jia TaoThe article studies two forms of responsibility in the setting of strategic games with imperfect information. They are referred to as seeingtoit responsibility and counterfactual responsibility. It shows that counterfactual responsibility is definable through seeingtoit, but not the other way around. The article also proposes a sound and complete bimodal logical system that describes the interplay

The comparison lemma Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230805
John R. SteelThe standard comparison lemma of inner model theory is deficient, in that it does not in general produce a comparison of all the relevant inputs. How two mice compare can depend upon which iteration strategies are used to compare them. We shall outline here a method for comparing iteration strategies that removes this defect.

Towards Logical Foundations for Probabilistic Computation Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230727
Melissa Antonelli, Ugo Dal Lago, Paolo PistoneThe overall purpose of the present work is to lay the foundations for a new approach to bridge logic and probabilistic computation. To this aim we introduce extensions of classical and intuitionistic propositional logic with counting quantifiers, that is, quantifiers that measure to which extent a formula is true. The resulting systems, called cCPL and iCPL, respectively, admit a natural semantics

A Lindström theorem for intuitionistic firstorder logic Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230726
Grigory Olkhovikov, Guillermo Badia, Reihane ZoghifardWe extend the main result of [1] to the firstorder intuitionistic logic (with and without equality), showing that it is a maximal (with respect to expressive power) abstract logic satisfying a certain form of compactness, the Tarski union property and preservation under asimulations. A similar result is also shown for the intuitionistic logic of constant domains.

Measurable cardinals and choiceless axioms Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230725
Gabriel GoldbergKunen refuted the existence of an elementary embedding from the universe of sets to itself assuming the Axiom of Choice. This paper concerns the ramifications of this hypothesis when the Axiom of Choice is not assumed. For example, the existence of such an embedding implies that there is a proper class of cardinals λ such that λ+ is measurable.

Locally compact, ω1compact spaces Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230725
Peter Nyikos, Lyubomyr ZdomskyyAn ω1compact space is a space in which every closed discrete subspace is countable. We give various general conditions under which a locally compact, ω1compact space is σcountably compact, i.e., the union of countably many countably compact spaces. These conditions involve very elementary properties. Many results shown here are independent of the usual (ZFC) axioms of set theory, and the consistency

Some variations on the splitting number Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230724
Saharon Shelah, Juris SteprānsVariations on the splitting number s are examined by localizing the splitting property to finite sets. To be more precise, rather than considering families of subsets of the integers that have the property that every infinite set is split into two infinite sets by some member of the family a stronger property is considered: Whenever an subset of the integers is represented as the disjoint union of

Some simple theories from a Boolean algebra point of view Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230724
M. Malliaris, S. ShelahWe find a strong separation between two natural families of simple rank one theories in Keisler's order: the theories Tm reflecting graph sequences, which witness that Keisler's order has the maximum number of classes, and the theories Tn,k, which are the higherorder analogues of the trianglefree random graph. The proof involves building Boolean algebras and ultrafilters “by hand” to satisfy certain

Alternating (In)DependenceFriendly Logic Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230722
Dylan Bellier, Massimo Benerecetti, Dario Della Monica, Fabio MogaveroHintikka and Sandu originally proposed Independence Friendly Logic ( ) as a firstorder logic of imperfect information to describe gametheoretic phenomena underlying the semantics of natural language. The logic allows for expressing independence constraints among quantified variables, in a similar vein to Henkin quantifiers, and has a nice gametheoretic semantics in terms of imperfect information

Causal Modeling Semantics for Counterfactuals with Disjunctive Antecedents Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230720
Causal Modeling Semantics (CMS, e.g., Galles and Pearl, 1998; Pearl, 2000; Halpern, 2000) is a powerful framework for evaluating counterfactuals whose antecedent is a conjunction of atomic formulas. We extend CMS to an evaluation of the probability of counterfactuals with disjunctive antecedents, and more generally, to counterfactuals whose antecedent is an arbitrary Boolean combination of atomic formulas

Probing the quantitative–qualitative divide in probabilistic reasoning Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230720
This paper explores the space of (propositional) probabilistic logical languages, ranging from a purely ‘qualitative’ comparative language to a highly ‘quantitative’ language involving arbitrary polynomials over probability terms. While talk of qualitative vs. quantitative may be suggestive, we identify a robust and meaningful boundary in the space by distinguishing systems that encode (at most) additive

Editorial Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230720
Dilip RaghavanAbstract not available

Subadditive families of hypergraphs Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230720
Jindřich ZapletalI analyze a natural class of proper forcings associated with actions of countable groups on Polish spaces, providing a practical and informative characterization as to when these forcings add no independent reals.

Kunen the expositor Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230718
Akihiro KanamoriKunen's expository work is described, bringing out both his way of assimilating and thinking about set theory and how it had a meaningful hand in its promulgation into the next generations.

Mutually embeddable models of ZFC Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230718
Monroe Eskew, SyDavid Friedman, Yair Hayut, Farmer SchlutzenbergWe investigate systems of transitive models of ZFC which are elementarily embeddable into each other and the influence of definability properties on such systems.

Boolean valued semantics for infinitary logics Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230718
Juan M. Santiago Suárez, Matteo VialeIt is well known that the completeness theorem for Lω1ω fails with respect to Tarski semantics. Mansfield showed that it holds for L∞∞ if one replaces Tarski semantics with Boolean valued semantics. We use forcing to improve his result in order to obtain a stronger form of Boolean completeness (but only for L∞ω). Leveraging on our completeness result, we establish the Craig interpolation property and

A logicogeometric comparison of coherence for nonadditive uncertainty measures Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230717
We investigate the notion of coherence for (non)additive uncertainty measures from a logicogeometric point of view. Our main result is to the effect that distinct criteria for coherence are not always matched by axiomatically distinct measures of uncertainty. In addition we introduce a metalogic within which this kind of result can be captured formally.

Asymptotic conditional probabilities for binary probability functions Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230717
The paper investigates the class of probability functions defined on sentences of a predicate language containing binary and possibly also unary predicates which satisfy the Principle of Binary Exchangeability (a strengthening of the Principle of Exchangeability). Following a survey of relevant properties of such functions we prove the main theorems of this paper showing that under some mild conditions

Probability propagation rules for Aristotelian syllogisms Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230717
We present a coherencebased probability semantics and probability propagation rules for (categorical) Aristotelian syllogisms. For framing the Aristotelian syllogisms as probabilistic inferences, we interpret basic syllogistic sentence types A, E, I, O by suitable precise and imprecise conditional probability assessments. Then, we define validity of probabilistic inferences and probabilistic notions

Reasoning with belief functions over Belnap–Dunn logic Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230717
We design an expansion of Belnap–Dunn logic with belief and plausibility functions that allows nontrivial reasoning with contradictory and incomplete probabilistic information. We also formalise reasoning with nonstandard probabilities and belief functions in two ways. First, using a calculus of linear inequalities, akin to the one presented in [23]. Second, as a twolayered modal logic wherein reasoning

Encoding de Finetti's coherence within Łukasiewicz logic and MValgebras Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230716
The present paper investigates prooftheoretical and algebraic properties for the probability logic FP(Ł,Ł), meant for reasoning on the uncertainty of Łukasiewicz events. Methodologically speaking, we will consider a translation function between formulas of FP(Ł,Ł) to the propositional language of Łukasiewicz logic that allows us to apply the latter and the welldeveloped theory of MValgebras directly

HL ideals and Sacks indestructible ultrafilters Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230717
David Chodounský, Osvaldo Guzmán, Michael HrušákWe study ultrafilters on countable sets and reaping families which are indestructible by Sacks forcing. We deal with the combinatorial characterization of such families and we prove that every reaping family of size smaller than the continuum is Sacks indestructible. We prove that complements of many definable ideals are Sacks reaping indestructible, with one notable exception, the complement of the

On middle box products and paracompact cardinals Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230717
David Buhagiar, Mirna DžamonjaThe paper gives several sufficient conditions on the paracompactness of box products with an arbitrary number of factors and boxes of arbitrary size. The former include results on generalised metrisability and Sikorski spaces. Of particular interest are products of the type □<κ2λ, where we prove that for a regular uncountable cardinal κ, if □<κ2λ is paracompact for every λ≥κ, then κ is at least inaccessible

Zerodimensional σhomogeneous spaces Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230716
Andrea Medini, Zoltán VidnyánszkyAll spaces are assumed to be separable and metrizable. Ostrovsky showed that every zerodimensional Borel space is σhomogeneous. Inspired by this theorem, we obtain the following results: • Assuming AD, every zerodimensional space is σhomogeneous, • Assuming AC, there exists a zerodimensional space that is not σhomogeneous, • Assuming V=L, there exists a coanalytic zerodimensional space that

Dense metrizability Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230711
Stevo TodorcevicIf a compact space K has a dense metrizable subspace its regular open algebra forces that the generic ultrafilter is countably generated. We investigate the class of compact spaces K for which the converse of this implication is true and give some applications of this. More precisely, we shall show that this gives us a rather powerful method for proving that a given compact space has a dense metrizable

Two chain conditions and their Todorčević's fragments of Martin's Axiom Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230710
Teruyuki YoriokaIn this paper, we investigate two chain conditions of forcing notions, called the rectangle refining property and property R1,ℵ1. They are stronger than the countable chain condition. Some of their typical examples are forcing notions about indestructible gaps introduced by Kunen. Both chain conditions are similar and have common examples, however, no distinction between them is known so far. In this

A Borel maximal eventually different family Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230705
Haim Horowitz, Saharon ShelahWe construct a Borel maximal eventually different family.

Choice and independence of premise rules in intuitionistic set theory Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230705
Emanuele Frittaion, Takako Nemoto, Michael RathjenChoice and independence of premise principles play an important role in characterizing Kreisel's modified realizability and Gödel's Dialectica interpretation. In this paper we show that a great many intuitionistic set theories are closed under the corresponding rules for finite types over N. It is also shown that the existence property (or existential definability property) holds for statements of

The relative strengths of fragments of Martin's axiom Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230705
Joan BagariaWe give a survey of results on the relative strengths of different fragments of Martin's Axiom, as well as a list of the main remaining open questions.

An undecidable extension of Morley's theorem on the number of countable models Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230704
Christopher J. Eagle, Clovis Hamel, Sandra Müller, Franklin D. TallWe show that Morley's theorem on the number of countable models of a countable firstorder theory becomes an undecidable statement when extended to secondorder logic. More generally, we calculate the number of equivalence classes of equivalence relations obtained by countable intersections of projective sets in several models of set theory. Our methods include random and Cohen forcing, Woodin cardinals

Burden in Henselian valued fields Ann. Pure Appl. Logic (IF 0.8) Pub Date : 20230704
Pierre TouchardIn the spirit of the AxKochenErshov principle, we show that in certain cases the burden of a Henselian valued field can be computed in terms of the burden of its residue field and that of its value group. To do so, we first see that the burden of such a field is equal to the burden of its leading term structure. These results are generalisations of Chernikov and Simon's work in [11].