Hilbert’s First Problem and the New Progress of Infinity Theory

In the 19th century, Cantor created the infinite cardinal number theory based on the "1-1 correspondence" principle. The continuum hypothesis is proposed under this theoretical framework. In 1900, Hilbert made it the first problem in his famous speech on mathematical problems, which shows the importance of this question. We know that the infinitesimal problem triggered the second mathematical crisis in the 17-18th centuries. The Infinity problem is no less important than the infinitesimal problem. In the 21st century, Sergeyev introduced the Grossone method from the principle of "whole is greater than part", and created another ruler for measuring infinite sets. At the same time, 16 the development of the infinity theory provides new ideas for solving Hilbert's first problem, and provides a new mathematical foundation for Cosmic Continuum Theory

Measuring Emerging Number Knowledge in Toddlers

Recent evidence suggests that infants and toddlers may recognize counting as numerically relevant long before they are able to count or understand the cardinal meaning of number words. The Give-N task, which asks children to produce sets of objects in different quantities, is commonly used to test children’s cardinal number knowledge and understanding of exact number words but does not capture children’s preliminary understanding of number words and is difficult to administer remotely. Here, we asked whether toddlers correctly map number words to the referred quantities in a two-alternative forced choice Point-to-X task (e.g., “Which has three?”). Two- to three-year-old toddlers (N = 100) completed a Give-N task and a Point-to-X task through in-person testing or online via videoconferencing software. Across number-word trials in Point-to-X, toddlers pointed to the correct image more often than predicted by chance, indicating that they had some understanding of the prompted number word that allowed them to rule out incorrect responses, despite limited understanding of exact cardinal values. No differences in Point-to-X performance were seen for children tested in-person versus remotely. Children with better understanding of exact number words as indicated on the Give-N task also answered more trials correctly in Point-to-X. Critically, in-depth analyses of Point-to-X performance for children who were identified as 1- or 2-knowers on Give-N showed that 1-knowers do not show a preliminary understanding of numbers above their knower-level, whereas 2-knowers do. As researchers move to administering assessments remotely, the Point-to-X task promises to be an easy-to-administer alternative to Give-N for measuring children’s emerging number knowledge and capturing nuances in children’s number-word knowledge that Give-N may miss.

On a Rigidity Problem of Beardon and Minda

In this paper, we give a positive answer to a rigidity problem of maps on the Riemann sphere related to cross-ratios, posed by Beardon and Minda (Proc Am Math Soc 130(4):987–998, 2001). Our main results are: (I) Let $$E\not \subset {\hat{\mathbb {R}}}$$ E ⊄ R ^ be an arc or a circle. If a map $$f:{\hat{\mathbb {C}}}\mapsto {\hat{\mathbb {C}}}$$ f : C ^ ↦ C ^ preserves cross-ratios in E, then f is a Möbius transformation when $${\bar{E}}\not =E$$ E ¯ ≠ E and f is a Möbius or conjugate Möbius transformation when $${\bar{E}}=E$$ E ¯ = E , where $${\bar{E}}=\{{\bar{z}}|z\in E\}$$ E ¯ = { z ¯ | z ∈ E } . (II) Let $$E\subset {\hat{\mathbb {R}}}$$ E ⊂ R ^ be an arc satisfying the condition that the cardinal number $$\#(E\cap \{0,1,\infty \})<2$$ # ( E ∩ { 0 , 1 , ∞ } ) < 2 . If f preserves cross-ratios in E, then f is a Möbius or conjugate Möbius transformation. Examples are provided to show that the assumption $$\#(E\cap \{0,1,\infty \})<2$$ # ( E ∩ { 0 , 1 , ∞ } ) < 2 is necessary.

A note on 3×3-valued Łukasiewicz algebras with negation

In 2004, C. Sanza, with the purpose of legitimizing the study of $$n\times m$$-valued Łukasiewicz algebras with negation (or $$NS_{n\times m}$$-algebras) introduced $$3\times 3$$-valued Łukasiewicz algebras with negation. Despite the various results obtained about $$NS_{n\times m}$$-algebras, the structure of the free algebras for this variety has not been determined yet. She only obtained a bound for their cardinal number with a finite number of free generators. In this note we describe the structure of the free finitely generated $$NS_{3 \times 3}$$-algebras and we determine a formula to calculate its cardinal number in terms of the number of free generators. Moreover, we obtain the lattice $$\Lambda$$(NS$$_{3\times 3}$$) of all subvarieties of NS$$_{3\times 3}$$ and we show that the varieties of Boolean algebras, three-valued Łukasiewicz algebras and four-valued Łukasiewicz algebras are proper subvarieties of NS_$${3\times 3}$$.

Hilbert's First Problem and the New Progress of Infinity Theory

In the 19th century, Cantor created the infinite cardinal number theory based on the "1-1 correspondence" principle. The continuum hypothesis is proposed under this theoretical framework. In 1900, Hilbert made it the first problem in his famous speech on mathematical problems, which shows the importance of this question. We know that the infinitesimal problem triggered the second mathematical crisis in the 17-18th centuries. The Infinity problem is no less important than the infinitesimal problem. In the 21st century, Sergeyev introduced the Grossone method from the principle of "whole is greater than part", and created another ruler for measuring infinite sets. The development of the infinity theory provides new ideas for solving Hilbert's first problem, and provides a new mathematical foundation for Cosmic Continuum Theory.

Hilbert's First Problem and the New Progress of Infinity Theory

In the 19th century, Cantor created the infinite cardinal number theory based on the "1-1 correspondence" principle. The continuum hypothesis is proposed under this theoretical framework. In 1900, Hilbert made it the first problem in his famous speech on mathematical problems, which shows the importance of this question. We know that the infinitesimal problem triggered the second mathematical crisis in the 17-18th centuries. The Infinity problem is no less important than the infinitesimal problem. In the 21st century, Sergeyev introduced the Grossone method from the principle of "whole is greater than part", and created another ruler for measuring infinite sets. The development of the infinity theory provides new ideas for solving Hilbert's first problem, and provides a new mathematical foundation for Cosmic Continuum Theory.

Hilbert' s First Problem and the New Progress of Infinity Theory

In the 19th century, Cantor created the infinite cardinal number theory based on the "1-1 correspondence" principle. The continuum hypothesis is proposed under this theoretical framework. In 1900, Hilbert made it the first problem in his famous speech on mathematical problems, which shows the importance of this question. We know that the infinitesimal problem triggered the second mathematical crisis in the 17-18th centuries. The Infinity problem is no less important than the infinitesimal problem. In the 21st century, Sergeyev introduced the Grossone method from the principle of "whole is greater than part", and created another ruler for measuring infinite sets. At the same time, the development of the infinity theory provides new ideas for solving Hilbert's first problem, and provides a new mathematical foundation for Cosmic Continuum Theory.

Mathematical Knowledge and the Origin of Phenomenology

The paper is divided into two parts. In the first one, I set forth a hypothesis to explain the failure of Husserl’s project presented in the Philosophie der Arithmetik based on the principle that the entire mathematical science is grounded in the concept of cardinal number. It is argued that Husserl’s analysis of the nature of the symbols used in the decadal system forces the rejection of this principle. In the second part, I take into account Husserl’s explanation of why, albeit independent of natural numbers, the system is nonetheless correct. It is shown that its justification involves, on the one hand, a new conception of symbols and symbolic thinking, and on the other, the recognition of the question of “the formal” and formalization as pivotal to understand “the mathematical” overall.

ON LOCAL WEAK τ-DENSITY OF TOPOLOGICAL SPACES

A topological space X is locally weakly separable [3] at a point x∈X if x has a weakly separable neighbourhood. A topological space X is locally weakly separable if X is locally weakly separable at every point x∈X. The notion of local weak separability can be generalized for any cardinal τ ≥ℵ0 . A topological space X is locally weakly τ-dense at a point x∈X if τ is the smallest cardinal number such that x has a weak τ-dense neighborhood in X [4]. The local weak density at a point x is denoted as lwd(x). The local weak density of a topological space X is defined in following way: lwd ( X ) = sup{ lwd ( x) : x∈ X } . A topological space X is locally τ-dense at a point x∈X if τ is the smallest cardinal number such that x has a τ-dense neighborhood in X [4]. The local density at a point x is denoted as ld(x). The local density of a topological space X is defined in following way: ld ( X ) = sup{ ld ( x) : x∈ X } . It is known that for any topological space we have ld(X ) ≤ d(X ) . In this paper, we study questions of the local weak τ-density of topological spaces and establish sufficient conditions for the preservation of the property of a local weak τ-density of subsets of topological spaces. It is proved that a subset of a locally τ-dense space is also locally weakly τ-dense if it satisfies at least one of the following conditions: (a) the subset is open in the space; (b) the subset is everywhere dense in space; (c) the subset is canonically closed in space. A proof is given that the sum, intersection, and product of locally weakly τ-dense spaces are also locally weakly τ-dense spaces. And also questions of local τ-density and local weak τ-density are considered in locally compact spaces. It is proved that these two concepts coincide in locally compact spaces.