Continuum Hypothesis Research Articles (Page 1)

Covarying gray and white matter networks characterize schizophrenia and bipolar disorders on a continuum: A data fusion machine learning approach and a brain network analysis.

Abstract Probabilism holds that rational credence functions are probability functions defined over some probability space $$(\Omega, \mathcal{F}, P)$$ . According to some recent philosophical arguments, in some situations, rational credence function must be total, i.e. $$\mathcal{F}=2^\Omega$$ , a view which I call credence totalism. Arguments for credence totalism are based on the premise that non-Lebesgue measurable subsets of $$\mathbb{R}$$ are epistemically significant, in the sense that an agent has reasons to assign probability to these sets. This paper argues that nonmeasurable sets are not epistemically significant in this sense. Consequently, the arguments for credence totalism are not successful. My argument is based on a careful consideration of the role of the Axiom of Choice in probabilistic practice. I also discuss some topics considered closely related, viz. the existence of total chance functions and the truth value of the Continuum Hypothesis. I argue that the role of nonmeasurability in epistemology does not shed light on these issues.

Gödel's first steps in set theory, from the summer of 1935 to the end of his stay in Princeton half a year later, are described in the light of his shorthand notebooks. The notes end with an English manuscript titled ‘The freedom from contradiction of the axiom of choice’ that is analyzed in detail. Gödel works out a logical hierarchical construction that systematically incorporates well-orderings, thereby affirming the title of his paper. He also sees an avenue to having his construction affirm the relative consistency of the Continuum Hypothesis, even to a key lemma about condensation along the hierarchy. However, he could not see a way to establishing the lemma until two years later.

Exploring the links between functional activation and hallucination proneness.

The influence of sound on joint attention in individuals with high autistic traits: Evidence from eye movements and fNIRS.

ABSTRACT We give a characterization of finitely ramified $\omega$-pseudo complete valued fields of mixed characteristic $(0,p)$, with fixed residue field k and value group G of cardinality $\aleph _{1}$ in terms of a Hahn-like construction over the Cohen field $C(k)$, modulo the Continuum Hypothesis. This is a generalization of results due to Ax and Kochen in ’65 for formally p-adic fields and by Kochen in ’74 for unramified valued fields with perfect residue field. We consider a more general context of finitely ramified valued fields of mixed characteristic with arbitrary residue field and a cross-section.

We provide a new characterization of quasi-analyticity of Denjoy–Carleman classes, related to Wetzel’s Problem. We also completely resolve which Denjoy–Carleman classes carry sparse systems: if the Continuum Hypothesis (CH) holds, all Denjoy–Carleman classes carry sparse systems; but if CH fails, a Denjoy–Carleman class carries a sparse system if and only if it is not quasi-analytic. As corollaries, we extend results of Bajpai–Velleman (2016) and Cody–Cox–Lee (2023) about non-existence of “anonymous predictors” for real functions.

If the Continuum Hypothesis is false, it implies the existence of cardinalities between the integers and the real numbers. In studying these “cardinal characteristics of the continuum,” it was discovered that many of the associated inequalities can be interpreted as morphisms within the “Galois-Tukey” category. This paper aims to reformulate traditional direct proofs of cardinal characteristic inequalities by making the underlying morphisms explicit. Purely categorical results are also discussed.

This essay deals with one of the most basic questions that concern the philosophy of mathematics, which has come to be known as the Continuum Hypothesis. First put forth by Georg Cantor in 1878, the Continuum Hypothesis is a postulate on whether there exists an infinite set of real numbers whose cardinality lies strictly between that of the natural numbers and that of the real numbers themselves. The independence of Continuum Hypothesis from the standard axiomatic system of Zermelo-Fraenkel set theory with the Axiom of Choice was shown by Kurt Gödel and Paul Cohen; the independence has given rise to much interesting philosophical debate about the nature of mathematical truth. This essay argues from a Platonist perspective, maintaining that Continuum Hypothesis must have a determinate truth value, independent of the limitations of formal systems. The essay contrasts this view with formalism, which sees mathematical truths as dependent on the choice of axioms. By drawing historical analogies and examining both Platonist and formalist viewpoints, the paper advocates for the pursuit of new axioms and alternative frameworks—such as large cardinal and forcing axioms—that might ultimately resolve the Continuum Hypothesis. The discussion highlights the broader implications of Continuum Hypothesis for understanding the nature of infinity, the completeness of mathematical systems, and the foundations of mathematics itself.

Abstract Assuming the Generalized Continuum hypothesis, this paper answers the question: when is the tensor product of two ultrafilters equal to their Cartesian product? It is necessary and sufficient that their Cartesian product is an ultrafilter; that the two ultrafilters commute in the tensor product; that for all cardinals $\lambda $ , one of the ultrafilters is both $\lambda $ -indecomposable and $\lambda ^+$ -indecomposable; that the ultrapower embedding associated with each ultrafilter restricts to a definable embedding of the ultrapower of the universe associated with the other.

Abstract We give a unified overview of the study of the effects of additional set theoretic axioms on quotient structures. Our focus is on rigidity, measured in terms of existence (or rather non-existence) of suitably non-trivial automorphisms of the quotients in question. A textbook example for the study of this topic is the Boolean algebra $\mathcal {P}({\mathbb N})/\operatorname {\mathrm {Fin}}$ , whose behavior is the template around which this survey revolves: Forcing axioms imply that all of its automorphisms are trivial, in the sense that they are induced by almost permutations of ${\mathbb N}$ , while under the Continuum Hypothesis this rigidity fails and $\mathcal {P}({\mathbb N})/\operatorname {\mathrm {Fin}}$ admits uncountably many non-trivial automorphisms. We consider far-reaching generalisations of this phenomenon and present a wide variety of situations where analogous patterns persist, focusing mainly (but not exclusively) on the categories of Boolean algebras, Čech–Stone remainders, and $\mathrm {C}^{*}$ -algebras. We survey the state of the art and the future prospects of this field, discussing the major open problems and outlining the main ideas of the proofs whenever possible.

Cannabis use and psychotic-like experiences: A systematic review of biological vulnerability, potency effects, and clinical trajectories.

Neurocognitive dysfunctions in childhood-onset schizophrenia: A systematic review.

BackgroundIt is becoming widely recognized that emotion regulation difficulties are an essential feature present along the continuum from subclinical to clinical Attention Deficit/Hyperactivity Disorder (ADHD). Yet, it remains unclear whether and how specific processes related to emotion regulation contribute to daily life impairments, across different domains of functioning. The aim of this cross-sectional study in community adolescents was to investigate whether three processes commonly implicated in adaptive emotion regulation—emotion recognition, emotion reactivity and use of cognitive emotion regulation strategies—uniquely contribute to adolescent-rated functional impairment, above and beyond the effects of age and gender, ADHD symptoms, and individual differences in verbal ability and executive functions.Methods161 adolescents from the general population (mean age = 15.57; SD = 1.61) completed the Weiss Functional Impairment Scale, the Emotion Reactivity Scale, the Cognitive Emotion Regulation Questionnaire and the Geneva Emotion Recognition Test. Hierarchical regression analysis examined the unique contributions of candidate predictors to impairment scores.ResultsTotal impairment scores were best predicted by older age, inattention symptoms, higher emotion reactivity, and higher use of maladaptive cognitive emotion regulation strategies. Emotion regulation processes were associated with interpersonal difficulties and self-concept impairments, whereas inattention symptoms were associated with school and life skills impairments.ConclusionsThis study stresses that emotion reactivity and maladaptive cognitive emotion regulation represent major sources of perceived social and emotional difficulties in community adolescents. Our results also support the continuum hypothesis of attention difficulties, where emotion regulation abilities may at least partially explain the association between ADHD symptoms and social impairments. Together, these findings highlight the vital importance of targeting emotion regulation in psychotherapeutic interventions aiming to improve socio-emotional outcomes in adolescents.

Abstract Gödel proved in the 1930 s in his famous Incompleteness Theorems that not all statements in mathematics can be proven or disproven from the accepted $$\operatorname {ZFC} $$ ZFC axioms. A few years later he showed the celebrated result that Cantor’s Continuum Hypothesis is consistent. Afterwards, Gödel raised the question whether, despite the fact that there is no reasonable axiomatic framework for all mathematical statements, natural statements, such as Cantor’s Continuum Hypothesis, can be decided via extending $$\operatorname {ZFC} $$ ZFC by large cardinal axioms. While this question has been answered negatively, the problem of finding good axioms that decide natural mathematical statements remains open. There is a compelling candidate for an axiom that could solve Gödel’s problem: $$\mathsf {V = Ultimate-L}$$ V = Ultimate - L . In addition, due to recent results the $$\textsf{Sealing}$$ Sealing scenario has gained a lot of attention. We describe these candidates as well as their impact and relationship.

ObjectivesThe Bipolar Continuum Hypothesis suggests that compassionate self-responding (CS) and uncompassionate self-responding (UCS) operate as opposing ends of a dynamic continuum. While this aligns with the view of self-compassion as a synergistic system, some researchers argue CS and UCS may function independently, raising questions about their relationship. This study examined real-time fluctuations in CS and UCS in response to contextual factors, addressing these theoretical and methodological complexities.MethodsAcross two longitudinal field studies (Study 1, n = 326; Study 2, n = 168), 494 participants provided weekly Ecological Momentary Assessment (EMA) data over 3 months. We assessed how immediate emotional states, decentering (a mindfulness-related skill), and event unpleasantness influenced CS and UCS in daily life.ResultsPartial support was found for the Bipolar Continuum Hypothesis, with CS and UCS generally showing inverse fluctuations in response to negative affect and decentering. Negative affect was the strongest predictor, linked to higher UCS and lower CS. Decentering showed a stronger association with reducing UCS than increasing CS, suggesting an asymmetry in their interaction. An idionomic analysis revealed individual variability, with a subset of participants displaying no clear inverse relationship, or even a positive association, between CS and UCS. Event unpleasantness had a minor impact.ConclusionsThese findings partially support the Bipolar Continuum Hypothesis, particularly regarding responses to emotional states, while also highlighting individual differences. Future research could explore the potential benefits of refining interventions and tailoring approaches to account for individual variations in CS and UCS dynamics.PreregistrationThis study is not preregistered.

Abstract Hamkins’ multiverse view is a prominent position on the nature of set theory. It is posited against the universe view and proposed as a philosophical theory explaining current set-theoretic practice. This paper confronts the multiverse view with the results of an interview study investigating current set-theoretic practice. The study reveals a heterogeneity of set-theoretic research practices. The multiverse view is found to align well with pluralist research practices but not with absolutist practices. The generalisation claim of the multiverse view fails because of this heterogeneity; only a reduced interpretation of the multiverse view might hold generally. Furthermore, Hamkins’ prediction that the community as a whole will not adopt axioms deciding the continuum hypothesis is found probably true, but for different reasons: although part of the community has already accepted axioms beyond ZFC, another part of the community will withdraw from doing so. In conclusion, I argue that Hamkins’ multiverse view is best interpreted as a valuable perspective on pluralist set-theoretic practices.

A numerical study supplemented with theoretical analysis is made, to analyse the electrophoresis of highly charged soft particles in electrolytes with trivalent counterions. The electrokinetic model is devised under the continuum hypothesis, which incorporates the ion–ion electrostatic correlations, hydrodynamic steric interactions of finite sized ions and ion–solvent interactions. The governing equations for ion transport and electric field are derived from the volumetric free energy of the system, which includes the first-order correction for the non-local electrostatic correlations of interacting ions, excess electrochemical potential due to finite ion size as well as the Born energy difference of ions due to dielectric permittivity variation. The electrolyte viscosity is considered to be a function of the local volume fraction of finite-sized ions, which causes the diffusivity of ions to vary locally. The occurrence of mobility reversal of a soft particle having the same polarity of its core and soft shell charge and formation of a coion-dominated zone in the soft layer is elaborated through this study. This can explain the mechanisms for the attraction between like-charged soft particles, as seen in the condensation of DNAs. The impact of ion–ion correlations and ion–solvent interactions of finite-sized ions are analysed by comparing them with the results based on the standard model. At a higher range of the core charge density, the ion–ion correlations induce a condensed layer of counterions on the outer surface of the core, which draws coions in the electric double layer, leading to an inversion in polarity of the charge density and mobility reversal. The dielectric decrement and ion steric interactions create a saturation in ion distribution and hence, modify the condensed layer of counterions. The enhanced fixed charge density of the polyelectrolyte layer diminishes the ion correlations due to the stronger screening effects and prevents the formation of a coion dominated zone in the Debye layer. The impact of the counterion size and the mixture of monovalent and trivalent counterions on mobility is analysed.

The continuum hypothesis proposes that binge drinking and severe alcohol use disorder (SAUD) share qualitatively similar cognitive and emotional impairments. In SAUD, these deficits have a demonstrated impact on social decision making, resulting in a utilitarian bias. Namely, when confronted with moral dilemmas, patients with SAUD tend to focus on the consequences of their actions rather than on social norms. However, social decision-making abilities remain unexplored in binge drinking. We offered the first insights on the generalization of the continuum hypothesis to social decision making, through a multinomial processing tree model applied to moral dilemmas, the "CNI model" of moral decision making. We compared 35 binge drinkers (20 females) and 36 light drinkers (21 females) on a battery of 48 moral dilemmas involving interpersonal relations from the CNI model, through multinomial modeling analyses. In each dilemma, participants were asked if they would perform the described action, generating individual scores for sensitivity to consequences, sensitivity to norms, and inaction tendency. The statistical model related to the CNI approach fits the data well. Binge drinkers and controls did not differ regarding their sensitivity to consequences nor their sensitivity to moral norms, and both groups displayed an equal inaction tendency in response to moral dilemmas. We provided insights to better understand the specific (socio)cognitive domains impaired in subclinical populations with alcohol use disorder. We showed preserved social decision making in binge drinking, which suggests that the continuum hypothesis documented for classical neurocognitive functions does not extend to complex social abilities. (PsycInfo Database Record (c) 2025 APA, all rights reserved).

Abstract We provide partial solutions to two problems posed by Shehtman concerning the modal logic of the Čech–Stone compactification of an ordinal space. We use the Continuum Hypothesis to give a finite axiomatization of the modal logic of , thus resolving Shehtman's first problem for . We also characterize modal logics arising from the Čech–Stone compactification of an ordinal provided the Cantor normal form of satisfies an additional condition. This gives a partial solution of Shehtman's second problem.