Continuum Hypothesis Research Articles

Monocolored topological complete graphs in colorings of uncountable complete graphs

If $$\kappa > \aleph_0$$ then $$\kappa \to(\kappa,\operatorname{Top}K_\kappa)^2$$ , i.e., every graph on $$\kappa$$ vertices contains either an independent set of $$\kappa$$ vertices, or a topological $$K_\kappa$$ , iff $$\kappa$$ is regular and there is no $$\kappa$$ -Suslin tree. Concerning the statement $$\omega_2\to(\operatorname{Top}K_{\omega_2})^2_\omega$$ , i.e., in every coloring of the edges of $$K_{\omega_2}$$ with countably many colors, there is a monochromatic topological $$K_{\omega_2}$$ , both the statement and its negation are consistent with the Generalized Continuum Hypothesis.

The Axiom of Choice in computability theory and Reverse Mathematics with a cameo for the Continuum Hypothesis

Abstract The Axiom of Choice (${\textsf{AC}}$ for short) is the most (in)famous axiom of the usual foundations of mathematics, ${\textsf{ZFC}}$ set theory. The (non-)essential use of ${\textsf{AC}}$ in mathematics has been well-studied and thoroughly classified. Now, fragments of countable ${\textsf{AC}}$ not provable in ${\textsf{ZF}}$ have recently been used in Kohlenbach’s higher-order Reverse Mathematics to obtain equivalences between closely related compactness and local–global principles. We continue this study and show that ${\textsf{NCC}}$, a weak choice principle provable in ${\textsf{ZF}}$ and much weaker systems, suffices for many of these results. In light of the intimate connection between Reverse Mathematics and computability theory, we also study realisers for ${\textsf{NCC}}$, i.e. functionals that produce the choice functions claimed to exist by the latter, from the other data. Our hubris of undertaking the hitherto underdeveloped study of the computational properties of (choice functions from) ${\textsf{AC}}$ leads to interesting results. For instance, using Kleene’s S1-S9 computation schemes, we show that various total realisers for ${\textsf{NCC}}$ compute Kleene’s $\exists ^{3}$, a functional that gives rise to full second-order arithmetic, and vice versa. By contrast, partial realisers for ${\textsf{NCC}}$ should be much weaker, but establishing this conjecture remains elusive. By way of catharsis, we show that the Continuum Hypothesis (${\textsf{CH}}$ for short) is equivalent to the existence of a countably based partial realiser for ${\textsf{NCC}}$. The latter kind of realiser does not compute Kleene’s $\exists ^{3}$ and is therefore strictly weaker than a total one.

The Chalmers Trilemma Re-examined

The Continuum Hypothesis seems to be a counterexample to David Chalmers’s A Priori Scrutability thesis, according to which there is a compact class of truths (the scrutability base) from which all truths are a priori scrutable. Chalmers’s three-part answer to this problem (which I call the “Chalmers trilemma”) runs as follows: either the Continuum Hypothesis is indeterminate; or adding a new axiom will settle the issue; or, if these two options do not work, we should add the Continuum Hypothesis (or its negation) to the scrutability base. I argue that Chalmers’s answer is unsatisfactory: the first horn of the trilemma can be interpreted in several ways, and either it departs from common mathematical practice and rests on weak analogies, or it shares the same problems with two other horns; the second horn does not provide good reasons to believe that from a fixed system of axioms all truths about our world are scrutable; the third horn of the trilemma renders Chalmers’s project empty.

A continuum hypothesis of psychotomimetic rapid antidepressants.

Ketamine, classical psychedelics and sleep deprivation are associated with rapid effects on depression. Interestingly, these interventions also have common psychotomimetic actions, mirroring aspects of psychosis such as an altered sense of self, perceptual distortions and distorted thinking. This raises the question whether these interventions might be acute antidepressants through the same mechanisms that underlie some of their psychotomimetic effects. That is, perhaps some symptoms of depression can be understood as occupying the opposite end of a spectrum where elements of psychosis can be found on the other side. This review aims at reviewing the evidence underlying a proposed continuum hypothesis of psychotomimetic rapid antidepressants, suggesting that a range of psychotomimetic interventions are also acute antidepressants as well as trying to explain these common features in a hierarchical predictive coding framework, where we hypothesise that these interventions share a common mechanism by increasing the flexibility of prior expectations. Neurobiological mechanisms at play and the role of different neuromodulatory systems affected by these interventions and their role in controlling the precision of prior expectations and new sensory evidence will be reviewed. The proposed hypothesis will also be discussed in relation to other existing theories of antidepressants. We also suggest a number of novel experiments to test the hypothesis and highlight research areas that could provide further insights, in the hope to better understand the acute antidepressant properties of these interventions.

Inner models from extended logics: Part 1

If we replace first-order logic by second-order logic in the original definition of Gödel’s inner model [Formula: see text], we obtain the inner model of hereditarily ordinal definable (HOD) sets [33]. In this paper, we consider inner models that arise if we replace first-order logic by a logic that has some, but not all, of the strength of second-order logic. Typical examples are the extensions of first-order logic by generalized quantifiers, such as the Magidor–Malitz quantifier [24], the cofinality quantifier [35], or stationary logic [6]. Our first set of results show that both [Formula: see text] and HOD manifest some amount of formalism freeness in the sense that they are not very sensitive to the choice of the underlying logic. Our second set of results shows that the cofinality quantifier gives rise to a new robust inner model between [Formula: see text] and HOD. We show, among other things, that assuming a proper class of Woodin cardinals the regular cardinals [Formula: see text] of [Formula: see text] are weakly compact in the inner model arising from the cofinality quantifier and the theory of that model is (set) forcing absolute and independent of the cofinality in question. We do not know whether this model satisfies the Continuum Hypothesis, assuming large cardinals, but we can show, assuming three Woodin cardinals and a measurable above them, that if the construction is relativized to a real, then on a cone of reals, the Continuum Hypothesis is true in the relativized model.

Connectedness properties of the Higson corona of the half line

It is proved that the Higson corona of the half line satisfies the following connectedness properties: The Higson corona of the half line is hereditarily unicoherent; under the Continuum Hypothesis, there are 2c composants of the Higson corona of the half line; the principle of Near Coherence Filters is equivalent to the assertion that the Higson corona of the half line has only one composant; no non-degenerate subcontinuum of the Higson corona of the half line is hereditarily indecomposable; and every non-empty open subset of the Higson corona of the half line contains a non-degenerate indecomposable continuum.

The Strength of Cross‐Language Interdependence for Listening Comprehension Proficiency in Turkish–Dutch Emergent Bilinguals: Testing Three Hypotheses

AbstractThis study aimed to test three competing hypotheses concerning the strength of the cross‐language relationship in listening comprehension proficiency in emergent bilinguals: Cummins's developmental linguistic interdependence hypothesis, Proctor, August, Snow, and Barr's interdependence continuum hypothesis, and Goodrich, Lonigan, Kleuver, and Farver's language independence hypothesis. We assessed 75 Turkish–Dutch bilingual children attending Dutch‐medium schools using a two‐wave design. This involved an assessment of sentence comprehension when the children were in preschool, followed by an assessment of their discourse comprehension in 1st grade. Correlational, multiple regression, and path analyses indicated significant positive and medium‐sized L1–L2 relationships between listening comprehension proficiency at the two points in time yet not over time. These findings were consistent with the developmental linguistic interdependence hypothesis for sentence comprehension and with the developmental linguistic interdependence and the language independence hypotheses for discourse comprehension. We advocate a multidimensional explanatory framework to explore the moderating role of type of skill in determining the strength of the cross‐language relationship.

An Algorithmic Approach to Solve Continuum Hypothesis

The continuum hypothesis has been unsolved for hundreds of years. In other words, can I answer it completely? By refuting the culturally responsible continuum [1], one can link the problem to the mathematical continuum, and it is possible to disproof the continuum hypothesis [2] . To go ahead a step, one may extend our mathematical system (by employing a more powerful set theory) and solve the continuum problem by three conditional cases. This event is sim-ilar to the status cases in the discriminant of solving a quadratic equation. Hence, my proposed al-gorithmic flowchart can best settle and depict the problem. From the above, one can further con-clude that when people extend mathematics (like set theory — ZFC) into new systems (such as Force Axioms), experts can solve important mathematical problems (CH). Indeed, there are differ-ent types of such mathematical systems, similar to ancient mathematical notation. Hence, different cultures have different ways of representation, which is similar to a Chinese saying: “different vil-lages have different laws.” However, the primary purpose of mathematical notation was initially to remember and communicate. This event indicates that the basic purpose of developing any new mathematical system is to help solve a natural phenomenon in our universe.

Small-Scale Environmental Drivers of Plant Community Structure and Diversity in Neotropical Montane Cloud Forests Harboring Threatened Magnolia dealbata in Southern Mexico

Gradient analysis was used to determine factors driving small-scale variation of cloud forest communities harboring Magnolia dealbata, a threatened species and bioculturally relevant tree for the Chinantecan, Mazatecan, Nahuan, and Zapotecan ethnicities in southern Mexico. Particularly, we aimed to: (a) determine factors explaining major community gradients at different heterogeneity scales along a small-scale elevational gradient, (b) test the Decreasing and the Continuum hypotheses along elevation, and (c) classify vegetation to assist in identifying conservation priorities. We used a stratified random sampling scheme for 21 woody stands along a small-scale (352 m) elevational transect. Four main data matrices were used (presence-absence, density, basal area, and guild data). Through Non-metric Multidimensional Scaling (NMS), Principal Coordinates Analysis (PCoA), and distance-based Redundancy Analysis (db-RDA), we found that major community variation was explained by soil pH, displaying an outstanding vegetation discontinuity, separating the species-rich relic Oreomunnea-Ticodendron-stands from stands with higher importance values for M. dealbata. The high species richness observed was explained by a combination of the windward effect of dry-seasonal maximum cloud condensation gain and habitat differentiation-specialization, a phenomenon that may also explain the mid-peak hypothesis and ensure the survival of relic species. Sampling-truncation and conservation status also played a role in this. Our results do not support the Decreasing and Continuum hypotheses along elevation.

Abolishing Platonism in Multiverse Theories

A debated issue in the mathematical foundations in at least the last two decades is whether one can plausibly argue for the merits of treating undecidable questions of mathematics, e.g., the Continuum Hypothesis (CH), by relying on the existence of a plurality of set-theoretical universes except for a single one, i.e., the well-known set-theoretical universe V associated with the cumulative hierarchy of sets. The multiverse approach has some varying versions of the general concept of multiverse yet my intention is to primarily address ontological multiversism as advocated, for instance, by Hamkins or Vaatanen, precisely for the reason that they proclaim, to the one or the other extent, ontological preoccupations for the introduction of respective multiverse theories. Taking also into account Woodin’s and Steel’s multiverse versions, I take up an argumentation against multiversism, and in a certain sense against platonism in mathematical foundations, mainly on subjectively founded grounds, while keeping an eye on Clarke-Doane’s concern with Benacerraf’s challenge. I note that even though the paper is rather technically constructed in arguing against multiversism, the non-negligible philosophical part is influenced to a certain extent by a phenomenologically motivated view of the matter.

On the applicability of continuum scale models for ultrafast nanoscale liquid-vapor phase change

Continuum methods are efficient in modeling multi-phase flow at large time and length scales, however, their applicability to nanoscale systems and processes is questionable. When mean free path and average time between atomic collisions are comparable to the characteristic length and time scales of interest, the continuum hypothesis approaches its spatial and temporal limit. Here we discuss the implications of modeling such a limiting problem involving liquid-vapor phase change using continuum equations of mass, momentum, and energy conservation. Our results indicate that, continuum conservation laws can correctly represent the dynamics of the specific problem of interest provided appropriate constitutive relations are used at liquid-vapor interfaces. We show that with the Schrage relation for phase change rates and a physically motivated expression for temperature jump, interfacial phenomena can be described quite accurately.

A new exponential wear model to analyze precision retention of guideway based on macro-micro multiscale method

At present, wear models and accuracy prediction of guideway are established based on elastoplastic mechanics of continuous media. These methods are limited to describe the process of accuracy degradation by using the material performance defined based on macroscopic hypothesis condition. In this paper, the multiscale method based on quasicontinuum medium(QM) principle is proposed to describe the degradation process of guideway linear precision with an exponential model. According to the wear distribution of the guideway surface with the micro-morphology evolution process, the measurement value of the guideway linear precision is modeled systematically. Then, using the quasicontinuum medium method instead of the continuum hypothesis, the exponential wear model of guideway is established. The exponential wear model uses the wear exponent to describe the wear status based on linear precision measurement rather than tremendous wear volume tests. By means of Depth From Defocus (DFD) method, the micro-topography information of the guideway surface is obtained. Therefore, the wear state of the guideway is verified under different loading conditions, and the validity of using the QM method to establish the guideway exponential wear model is verified.

Asymptotic properties of mixing length closures for turbulent pipe flow

Turbulence is investigated as the Reynolds number approaches infinity for the flow of an incompressible fluid through straight pipes with a circular cross section under the assumptions that the continuum hypothesis holds, the pipe wall is smooth, and the mixing length closure constructed by Cantwell [“A universal velocity profile for smooth wall pipe flow,” J. Fluid Mech. 878, 834–874 (2019)] is sufficiently accurate to allow the extrapolation to Reynolds numbers beyond the range of the Princeton superpipe data used as a foundation for the closure model. Two sets of scales are introduced to set up two sets of dimensionless equations and two Reynolds numbers for the near wall region (Re, inner scaling) and the center part of the pipe flow (Rτ, outer scaling). It is shown analytically that the turbulent flow asymptotically approaches a two-layer structure: The core of the pipe flow becomes uniform with constant mean velocity and zero shear stress in the outer scaling and the near wall region (inner scaling) with the mean velocity satisfying the law of the wall and non-zero shear stress. The core part of the flow pushes, as Rτ → ∞, the near wall layer to the boundary restricting it to a cylindrical subdomain with zero volume.

Chance and the Continuum Hypothesis

AbstractThis paper presents and defends an argument that the continuum hypothesis is false, based on considerations about objective chance and an old theorem due to Banach and Kuratowski. More specifically, I argue that the probabilistic inductive methods standardly used in science presuppose that every proposition about the outcome of a chancy process has a certain chance between 0 and 1. I also argue in favour of the standard view that chances are countably additive. Since it is possible to randomly pick out a point on a continuum, for instance using a roulette wheel or by flipping a countable infinity of fair coins, it follows, given the axioms of ZFC, that there are many different cardinalities between countable infinity and the cardinality of the continuum.

Playing with Mathias

We present a series of axioms that describe the combinatorial core of the Mathias model (countable support iteration of ω2 Mathias reals over the Continuum Hypothesis). Our axioms are formulated in terms of games with Borel sets and functions and are strong enough to imply most of the propositions usually proved by means of the iterated Mathias forcing.

INFINITARY GENERALIZATIONS OF DELIGNE’S COMPLETENESS THEOREM

AbstractGiven a regular cardinal $\kappa $ such that $\kappa ^{<\kappa }=\kappa $ (or any regular $\kappa $ if the Generalized Continuum Hypothesis holds), we study a class of toposes with enough points, the $\kappa $ -separable toposes. These are equivalent to sheaf toposes over a site with $\kappa $ -small limits that has at most $\kappa $ many objects and morphisms, the (basis for the) topology being generated by at most $\kappa $ many covering families, and that satisfy a further exactness property T. We prove that these toposes have enough $\kappa $ -points, that is, points whose inverse image preserve all $\kappa $ -small limits. This generalizes the separable toposes of Makkai and Reyes, that are a particular case when $\kappa =\omega $ , when property T is trivially satisfied. This result is essentially a completeness theorem for a certain infinitary logic that we call $\kappa $ -geometric, where conjunctions of less than $\kappa $ formulas and existential quantification on less than $\kappa $ many variables is allowed. We prove that $\kappa $ -geometric theories have a $\kappa $ -classifying topos having property T, the universal property being that models of the theory in a Grothendieck topos with property T correspond to $\kappa $ -geometric morphisms (geometric morphisms the inverse image of which preserves all $\kappa $ -small limits) into that topos. Moreover, we prove that $\kappa $ -separable toposes occur as the $\kappa $ -classifying toposes of $\kappa $ -geometric theories of at most $\kappa $ many axioms in canonical form, and that every such $\kappa $ -classifying topos is $\kappa $ -separable. Finally, we consider the case when $\kappa $ is weakly compact and study the $\kappa $ -classifying topos of a $\kappa $ -coherent theory (with at most $\kappa $ many axioms), that is, a theory where only disjunction of less than $\kappa $ formulas are allowed, obtaining a version of Deligne’s theorem for $\kappa $ -coherent toposes from which we can derive, among other things, Karp’s completeness theorem for infinitary classical logic.

Validation of the Launay-Slade Hallucination Scale among Indian Healthy Adults.

Psychotic Like Experiences (PLEs) have been reported in several cultures. The 16 item Launay-Slade Hallucination Scale-Extended (LSHS-E) measures hallucination like experiences (HLEs) in the general population. This study investigated the psychometric properties and the factor structure of LSHS-E Hindi among healthy adults of Delhi. LSHS-E was translated from English to Hindi and then back to English. It was administered as a paper pencil questionnaire to 182 adults from the general population. Reliability of LSHS-E Hindi was measured using Cronbach's alpha and factor structure was established using confirmatory factor analysis (CFA). It was tested against the Community Assessment of Psychic Experiences (CAPE-42) for convergent and divergent validity. Latent Class Analysis (LCA) was performed to identify subgroups with different endorsement of HLEs. Among 182, 18 participants reporting mental and neurological disorders were excluded. LSHS-E Hindi had good reliability (0.85; 95% CI: 0.82 to 0.88). CFA of Hindi LSHS-E revealed the a priori four-factor solution to be best, namely: 'intrusive thoughts', 'vivid daydreams', 'multisensory HLEs', 'auditory and visual HLEs'. LSHS-E Hindi showed stronger correlation with positive domain of CAPE than with negative and depression domains. LCA revealed three classes: low, intermediate and high endorsement of HLEs. Participants with highest endorsement of HLEs were less educated and had highest endorsement on all CAPE dimensions. LSHS-E Hindi has good psychometric properties and can be used to study HLEs in Indians. The four-factor structure model depicts the multidimensionality of HLEs, with 'intrusive thoughts' being the most commonly reported HLE in the sample. LCA supports the continuum hypothesis of HLEs.

The weight of lexicographic products

We will calculate the weight of lexicographic products of GO-spaces, using this we will see:•the assertion that the weight of the lexicographic product 2ω1 is ℵ1 is equivalent to the Continuum Hypothesis (CH), that is, 2ℵ0=ℵ1,•the assertion that the weight of both lexicographic products 2ω1 and 2ω1+1 coincide is equivalent to the assertion 2ℵ0=2ℵ1,•the assertion that the lexicographic product 2γ is homeomorphic to the usual Tychonoff product 2γ is equivalent to γ≤ω.

Physical and Mathematical Fluid Mechanics

Fluid mechanics has emerged as a basic concept for nearly every field of technology. Despite there being a well-developed mathematical theory and available commercial software codes, the computation of solutions of the governing equations of motion is still challenging, especially due to the nonlinearity involved, and there are still open questions regarding the underlying physics of fluid flow, especially with respect to the continuum hypothesis and thermodynamic local equilibrium. The aim of this Special Issue is to reference recent advances in the field of fluid mechanics both in terms of developing sophisticated mathematical methods for finding solutions of the equations of motion, on the one hand, and on novel approaches to the physical modelling beyond the continuum hypothesis and thermodynamic local equilibrium, on the other.

Quasi-normality of Mrowka spaces

A topological space X is called quasi-normal if X is regular and any two disjoint π-closed subsets A and B of X are separated. We give a Mr ́owka space which is not quasi-normal and use the continuum hypothesis (CH) and truly cardinality c to present Mr ́owka spaces which are quasi-normal.