Cardinality Of The Continuum Research Articles

On the intermediate value property of spectra for a class of Moran spectral measures

We prove that the Beurling dimensions of the spectra for a class of Moran spectral measures are in 0 and their upper entropy dimensions. Moreover, for such a Moran spectral measure μ, we show that the Beurling dimension for the spectra of μ has the intermediate value property: let t be any value in 0 and the upper entropy dimension of μ, then there exists a spectrum whose Beurling dimension is t. In particular, this result settles affirmatively a conjecture involving spectral Bernoulli convolution in Fu et al. (2018) [20]. Furthermore, we prove that the set of the spectra whose Beurling dimensions are equal to any fixed value in 0 and dim‾eμ has the cardinality of the continuum.

A Minimal Probability Space for Conditionals

One of central problems in the theory of conditionals is the construction of a probability space, where conditionals can be interpreted as events and assigned probabilities. The problem has been given a technical formulation by van Fraassen (23), who also discussed in great detail the solution in the form of Stalnaker Bernoulli spaces. These spaces are very complex – they have the cardinality of the continuum, even if the language is finite. A natural question is, therefore, whether a technically simpler (in particular finite) partial construction can be given. In the paper we provide a new solution to the problem. We show how to construct a finite probability space mathrm {S}^#=left(mathrmOmega^#,mathrmSigma^#,mathrm P^#right) in which simple conditionals and their Boolean combinations can be interpreted. The structure is minimal in terms of cardinality within a certain, naturally defined class of models – an interesting side-effect is an estimate of the number of non-equivalent propositions in the conditional language. We demand that the structure satisfy certain natural assumptions concerning the logic and semantics of conditionals and also that it satisfy PCCP. The construction can be easily iterated, producing interpretations for conditionals of arbitrary complexity.

Lower and Upper Estimates of the Quantity of Algebraic Numbers

It is well known that the set of algebraic numbers (let us call it A) is countable. In this paper, instead of the usage of the classical terminology of cardinals proposed by Cantor, a recently introduced methodology using textcircled {1}-based infinite numbers is applied to measure the set A (where the number textcircled {1} is called grossone). Our interest to this methodology is explained by the fact that in certain cases where cardinals allow one to say only whether a set is countable or it has the cardinality of the continuum, the textcircled {1}-based methodology can provide a more accurate measurement of infinite sets. In this article, lower and upper estimates of the number of elements of A are obtained. Both estimates are expressed in textcircled {1}-based numbers.

ERDŐS–LIOUVILLE SETS

AbstractIn 1844, Joseph Liouville proved the existence of transcendental numbers. He introduced the set $\mathcal L$ of numbers, now known as Liouville numbers, and showed that they are all transcendental. It is known that $\mathcal L$ has cardinality $\mathfrak {c}$ , the cardinality of the continuum, and is a dense $G_{\delta }$ subset of the set $\mathbb {R}$ of all real numbers. In 1962, Erdős proved that every real number is the sum of two Liouville numbers. In this paper, a set W of complex numbers is said to have the Erdős property if every real number is the sum of two numbers in W. The set W is said to be an Erdős–Liouville set if it is a dense subset of $\mathcal {L}$ and has the Erdős property. Each subset of $\mathbb {R}$ is assigned its subspace topology, where $\mathbb {R}$ has the euclidean topology. It is proved here that: (i) there exist $2^{\mathfrak {c}}$ Erdős–Liouville sets no two of which are homeomorphic; (ii) there exist $\mathfrak {c}$ Erdős–Liouville sets each of which is homeomorphic to $\mathcal {L}$ with its subspace topology and homeomorphic to the space of all irrational numbers; (iii) each Erdős–Liouville set L homeomorphic to $\mathcal {L}$ contains another Erdős–Liouville set $L'$ homeomorphic to $\mathcal {L}$ . Therefore, there is no minimal Erdős–Liouville set homeomorphic to $\mathcal {L}$ .

On 𝑇1- and 𝑇2-productable compact spaces

Abstract We prove that if there exists a continuous surjection from a metric compact space 𝑋 onto a product X × T X\times T , where 𝑇 is a T 1 T_{1} second countable topological space which has the cardinality of the continuum, then there exists a surjection from 𝑋 onto the product X × [ 0 , 1 ] X\times[0,1] , where the interval [ 0 , 1 ] [0,1] is equipped with the usual Euclidean topology.

Gap‐2 morass‐definable η1‐orderings

AbstractWe prove that in the Cohen extension adding ℵ3 generic reals to a model of containing a simplified (ω1, 2)‐morass, gap‐2 morass‐definable η1‐orderings with cardinality ℵ3 are order‐isomorphic. Hence it is consistent that and that morass‐definable η1‐orderings with cardinality of the continuum are order‐isomorphic. We prove that there are ultrapowers of over ω that are gap‐2 morass‐definable. The constructions use a simplified gap‐2 morass, and commutativity with morass‐maps and morass‐embeddings, to extend a transfinite back‐and‐forth construction of order‐type ω1 to an order‐preserving bijection between objects of cardinality ℵ3.

Surviving ends in Bernoulli percolation on graphs roughly isometric to a tree

Let G be an infinite locally-finite connected graph roughly isometric to a tree, and o a fixed vertex of G. Given any p∈(0,1). Then under a mild condition, the number of surviving ends under Bernoulli-p bond percolation ω on G a.s. either is 0 or has the cardinality of the continuum; which generalizes Proposition 5.27 in Lyons and Peres (2016) from a viewpoint of rough isometry. Here a surviving end is an end of G induced by a surviving ray from o in the ω.

Strong downward Löwenheim–Skolem theorems for stationary logics, II: reflection down to the continuum

Continuing (Fuchino et al. in Arch Math Log, 2020. https://doi.org/10.1007/s00153-020-00730-x), we study the Strong Downward Lowenheim–Skolem Theorems (SDLSs) of the stationary logic and their variations. In Fuchino et al. (2020) it has been shown that the SDLS for the ordinary stationary logic with weak second-order parameters $$\textsf {SDLS}({\mathcal {L}}^{\aleph _0}_{stat},{<}\,\aleph _2)$$ down to $${<}\,\aleph _2$$ is equivalent to the conjunction of CH and Cox’s Diagonal Reflection Principle for internally clubness. We show that the SDLS for the stationary logic without weak second-order parameters $$\textsf {SDLS}^-({\mathcal {L}}^{\aleph _0}_{stat},{<}\,2^{\aleph _0})$$ down to $${<}\,2^{\aleph _0}$$ implies that the size of the continuum is $$\aleph _2$$ . In contrast, an internal interpretation of the stationary logic can satisfy the SDLS down to $${<}\,2^{\aleph _0}$$ under the continuum being of size $$>\aleph _2$$ . This SDLS is shown to be equivalent to an internal version of the Diagonal Reflection Principle down to an internally stationary set of size $${<}\,2^{\aleph _0}$$ . We also consider a version of the stationary logic and show that the SDLS for this logic in internal interpretation $$\textsf {SDLS}^{int}_+({\mathcal {L}}^{PKL}_{stat},{<}\,2^{\aleph _0})$$ for reflection down to $${<}\,2^{\aleph _0}$$ is consistent under the assumption of the consistency of ZFC $$+$$ “the existence of a supercompact cardinal” and this SDLS implies that the continuum is (at least) weakly Mahlo. These three “axioms” in terms of SDLS are consequences of three instances of a strengthening of generic supercompactness which we call Laver-generic supercompactness. Existence of a Laver-generic supercompact cardinal in each of these three instances also fixes the cardinality of the continuum to be $$\aleph _1$$ or $$\aleph _2$$ or very large respectively. We also show that the existence of one of these generic large cardinals implies the “ $$++$$ ” version of the corresponding forcing axiom.

Cantor’s "Cardinality of the Continuum" is False in Non-Euclidean Geometries

Cantor’s "Cardinality of the Continuum" is False in Non-Euclidean Geometries

Banach spaces for which the space of operators has 2𝔠closed ideals

AbstractWe formulate general conditions which imply that${\mathcal L}(X,Y)$, the space of operators from a Banach spaceXto a Banach spaceY, has$2^{{\mathfrak {c}}}$closed ideals, where${\mathfrak {c}}$is the cardinality of the continuum. These results are applied to classical sequence spaces and Tsirelson-type spaces. In particular, we prove that the cardinality of the set ofclosed ideals in${\mathcal L}\left (\ell _p\oplus \ell _q\right )$is exactly$2^{{\mathfrak {c}}}$for all$1&lt;p&lt;q&lt;\infty $.

On the generalized nonmeasurability of Vitali sets and Bernstein sets

Abstract It is shown that the cardinality of the continuum is not real-valued measurable if and only if there exists no nonzero σ-finite diffused measure μ on the real line such that all Vitali sets (respectively all Bernstein sets) are μ-measurable.

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.

On semidirectly closed non-aperiodic pseudovarieties of finite monoids

AbstractIt is shown that, for every prime number p, the complete lattice of all semidirectly closed pseudovarieties of finite monoids whose intersection with the pseudovariety G of all finite groups is equal to the pseudovariety Gp of all finite p-groups has the cardinality of the continuum. Furthermore, it is shown, in addition, that the complete lattice of all semidirectly closed pseudovarieties of finite monoids whose intersection with the pseudovariety G of all finite groups is equal to the pseudovariety Gsol of all finite solvable groups has also the cardinality of the continuum.

THE SET OF 2-GENERETED $C^*$-SIMPLE RELATIVELY FREE GROUPS HAS THE CARDINALITY OF THE CONTINUUM

In this paper we prove that the set of non-isomorphic 2-generated $C^*$-simple relatively free groups has the cardinality of the continuum. A non-trivial identity is satisfied in any (not absolutely free) relatively free group. Hence, they cannot contain a non-abelian absolutely free subgroups. The question of the existence of $C^*$-simple groups without free subgroups of rank 2 was posed by de la Harpe in 2007.

Hausdorff dimension of non-conical limit sets

Geometrically infinite Kleinian groups have non-conical limit sets with the cardinality of the continuum. In this paper, we construct a geometrically infinite Fuchsian group such that the Hausdorff dimension of the non-conical limit set equals zero. For finitely generated, geometrically infinite Kleinian groups, we prove that the Hausdorff dimension of the non-conical limit set is positive.

Closed ideals of operators on the Tsirelson and Schreier spaces

Let B(X) denote the Banach algebra of bounded operators on X, where X is either Tsirelson's Banach space or the Schreier space of order n for some n∈N. We show that the lattice of closed ideals of B(X) has a very rich structure; in particular B(X) contains at least continuum many maximal ideals.Our approach is to study the closed ideals generated by the basis projections. Indeed, the unit vector basis is an unconditional basis for each of the above spaces, so there is a basis projection PN∈B(X) corresponding to each non-empty subset N of N. A closed ideal of B(X) is spatial if it is generated by PN for some N. We can now state our main conclusions as follows:•the family of spatial ideals lying strictly between the ideal of compact operators and B(X) is non-empty and has no minimal or maximal elements;•for each pair I⫋J of spatial ideals, there is a family {ΓL:L∈Δ}, where the index set Δ has the cardinality of the continuum, such that ΓL is an uncountable chain of spatial ideals, ⋃ΓL is a closed ideal that is not spatial, andI⫋L⫋JandL+M‾=J whenever L,M∈Δ are distinct and L∈ΓL, M∈ΓM.

The ‘Cardinality of the Continuum’ Is False in Non-Euclidean Geometries

According Georg Cantor's Cardinality of the Continuum, as proved in Euclidean geometry, any segment (not inclusive of its two end points) has as many points as an entire line. This theorem is an underlying assumption of the Hypothesis, that conjectures that every infinite subset of the continuum of real numbers $\mathbb{R}$ is either equivalent to the set of natural numbers $\mathbb{N}$, or to $\mathbb{R}$ itself. However, the proof in Euclidean geometry uses parallel lines, and therefore assumes that Euclid's Parallel Postulate is true. But in Non-Euclidean geometries, Euclid's Parallel Postulate is false. In Hyperbolic geometry, at any point off of a given line, there are a plurality of lines parallel to the given line. In Elliptic geometry (which includes Spherical geometry), no lines are parallel, so two lines always intersect. Absolute geometry has neither Euclid's parallel postulate nor either of its alternatives. We provide examples in Spherical geometry and in Hyperbolic geometry where the Cardinality of the Continuum is false. Therefore it is also false in Absolute geometry. So it is a false proposition, as is the Hypothesis, that falsely assumes that the Cardinality of the Continuum is true. Concepts in theory, such as the number line of real numbers, cannot be limited to Euclidean geometry.

Kernel classes of varieties of completely regular semigroups I

Several complete congruences on the lattice $${\mathcal {L}}({{\mathcal {C}}}{{\mathcal {R}}})$$ of varieties of completely regular semigroups have been fundamental to studies of the structure of $${\mathcal {L}}({{\mathcal {C}}}{{\mathcal {R}}})$$. These are the kernel relation K, the left trace relation $$T_{\ell }$$, the right trace relation $$T_r$$ and their intersections $$K\cap T_{\ell }, K\cap T_r$$. However, with the exception of the lattice of all band varieties which happens to coincide with the kernel class of the trivial variety, almost nothing is known about the internal structure of individual K classes beyond the fact that they are intervals in $${\mathcal {L}}({{\mathcal {C}}}{{\mathcal {R}}})$$ (with the notable exception of the remarkable results in the very recent article by Kad’ourek (Int J Algebra Comput, https://doi.org/10.1142/S0218196719500541, 2019)). Here we present a number of general results that are pertinent to the study of K classes. This includes a variation of the renowned Polak Theorem and its relationship to the complete retraction $${\mathcal {V}} \longrightarrow {\mathcal {V}}\cap {\mathcal {B}}$$, where $${\mathcal {B}}$$ denotes the variety of bands. These results are then applied, here and in a sequel, to the detailed analysis of certain families of K classes. The paper concludes with results hinting at the complexity of K classes in general, such as that the classes of relation $$K/(K\cap T_{\ell })$$ may have the cardinality of the continuum.

Kernel classes of varieties of completely regular semigroups II

Several complete congruences on the lattice $${\mathcal {L}}({{\mathcal {C}}}{{\mathcal {R}}})$$ of varieties of completely regular semigroups have been fundamental to studies of the structure of $${\mathcal {L}}({{\mathcal {C}}}{{\mathcal {R}}})$$. These are the kernel relation K, the left trace relation $$T_{\ell }$$, the right trace relation $$T_r$$ and their intersections $$K\cap T_{\ell }, K\cap T_r$$. However, with the exception of the lattice of all band varieties which happens to coincide with the kernel class of the trivial variety, almost nothing is known about the internal structure of individual K classes beyond the fact that they are intervals in $${\mathcal {L}}({{\mathcal {C}}}{{\mathcal {R}}})$$ (with the notable exception of the remarkable results in the very recent article by Kad’ourek (Int J Algebra Comput, https://doi.org/10.1142/S0218196719500541, 2019)). Here we present a number of general results that are pertinent to the study of K classes. This includes a variation of the renowned Polak Theorem and its relationship to the complete retraction $${\mathcal {V}} \longrightarrow {\mathcal {V}}\cap {\mathcal {B}}$$, where $${\mathcal {B}}$$ denotes the variety of bands. These results are then applied, here and in a sequel, to the detailed analysis of certain families of K classes. The paper concludes with results hinting at the complexity of K classes in general, such as that the classes of relation $$K/(K\cap T_{\ell })$$ may have the cardinality of the continuum.

Cardinality of the Continuum of Closed Superclasses of Some Minimal Classes in the Partially Ordered Set $${\cal L}_2^3$$

It is proved that the set of closed classes containing some minimal classes in the partially ordered set $${\cal L}_2^3$$ of closed classes in the three-valued logic that can be homomorphically mapped onto the two-valued logic has the cardinality of continuum.