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

2021 ◽  
Author(s):  
Xijia Wang
Keyword(s):  
Correspondence Principle ◽  
Continuum Hypothesis ◽  
Cardinal Number ◽  
Continuum Theory ◽  
Infinite Cardinal ◽  
Mathematical Foundation ◽  
Mathematical Problems ◽  
New Ideas ◽  
The 19Th Century ◽  
The Continuum

Abstract 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.

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

2021 ◽  
Author(s):  
Xijia Wang
Keyword(s):  
Correspondence Principle ◽  
Continuum Hypothesis ◽  
Cardinal Number ◽  
Continuum Theory ◽  
Infinite Cardinal ◽  
Mathematical Foundation ◽  
Mathematical Problems ◽  
New Ideas ◽  
The 19Th Century ◽  
The Continuum

Abstract 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.

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

2021 ◽  
Author(s):  
Xijia Wang
Keyword(s):  
Correspondence Principle ◽  
Continuum Hypothesis ◽  
Cardinal Number ◽  
Continuum Theory ◽  
Infinite Cardinal ◽  
Mathematical Foundation ◽  
Mathematical Problems ◽  
New Ideas ◽  
The 19Th Century ◽  
The Continuum

Abstract 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.

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

2021 ◽  
Author(s):  
Xijia Wang
Keyword(s):  
Correspondence Principle ◽  
Continuum Hypothesis ◽  
Cardinal Number ◽  
Continuum Theory ◽  
Infinite Cardinal ◽  
Mathematical Foundation ◽  
Mathematical Problems ◽  
New Ideas ◽  
The 19Th Century ◽  
The Continuum

Abstract 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

A complete Lω1ω-sentence characterizing ℵ1

1977 ◽  
Vol 42 (1) ◽  
pp. 59-62 ◽  
Author(s):  
Julia F. Knight
Keyword(s):  
Continuum Hypothesis ◽  
Infinite Cardinal ◽  
Higher Power ◽  
The Continuum

Here an example will be given of a complete Lω1ω-sentence with a model of power ℵ1 but with no model of higher power. The continuum hypothesis is not assumed. The question of whether such an example exists was brought to the author's attention by Professor M. Makkai.An Lω1ω-sentence is said to be complete if its models all satisfy the same Lω1ω-sentences, or, equivalently, if all of the countable L-structures satisfying the sentence are isomorphic. Scott [5] showed that any countable L -structure (where L is countable) must satisfy a complete Lω1ω-sentence. Such a sentence is called a Scott sentence for the structure. An uncountable L-structure need not satisfy any complete Lω1ω -sentence.A complete Lω1ω-sentence σ is said to characterize the infinite cardinal k if σ has a model of power k but not of any higher power. The set of cardinals characterized by complete Lω1ω -sentences will be denoted by CC. By a result of Lopez-Escobar [3], if k ∈ CC, k <⊐ω1.Assuming GCH (so that ⊐α = ℵα, ), Malitz [4] showed that CC = {ℵα: α < ω1}.Without assuming GCH, Baumgartner [1] showed that ⊐α ∈ CC for all α < ω1.Without GCH, it is unknown whether ℵn ∈ CC for n ≥ 2. Now it will be shown that ℵ1, ∈ CC.

scholarly journals An Algorithmic Approach to Solve Continuum Hypothesis

2020 ◽  
pp. 36-49
Author(s):  
Mr. Lam Kai Shun
Keyword(s):  
Set Theory ◽  
Continuum Hypothesis ◽  
Quadratic Equation ◽  
Natural Phenomenon ◽  
Mathematical Notation ◽  
Mathematical System ◽  
The Status ◽  
Mathematical Problems ◽  
Basic Purpose ◽  
The Continuum

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.

On the compactness of some Boolean algebras

1984 ◽  
Vol 49 (1) ◽  
pp. 63-67
Author(s):  
Jacek Cichoń
Keyword(s):  
Boolean Algebra ◽  
Continuum Hypothesis ◽  
Cardinal Number ◽  
Boolean Algebras ◽  
Cardinal Numbers ◽  
Martin’S Axiom ◽  
Reduced Products ◽  
Standard Set ◽  
General Continuum ◽  
The Continuum

We say that the Boolean algebra B is λ-compact, where λ is a cardinal number, if for every family Z ⊆ B∖{0} of power at most λ, if inf Z = 0 then for some finite subfamily Z0 ⊆ Z we have inf Z0 = 0.On the set of all finite subsets of a cardinal number κ, which is denoted [κ]<ω, the sets of the form for any p Є [κ]<ω generate the filter Tκ.This filter is a standard example of a κ-regular filter (see [2]). Because of the importance of κ-regular filters in studying the saturatedness of ultraproducts and reduced products by model-theoretic methods, the question of compactness of the algebra Bκ = P([κ]<ω/Tκ was natural. This question in the most optimistical way was formulated by M. Benda [1, Problem 5c]: is the algebra Bκω-compact for every uncountable κ?In this paper we show that for most of the cardinal numbers which are greater or equal to 2ω the algebra Bκ is not ω-compact. Hence, in view of obtained results, the following question appears: does there exist an uncountable κ such that the algebra Bκ is κ-compact?We use standard set-theoretical notations. CH denotes the Continuum Hypothesis, GCH denotes the General Continuum Hypothesis and MA denotes Martin's Axiom.

scholarly journals Several results on compact metrizable spaces in $$\mathbf {ZF}$$

2021 ◽  
Author(s):  
Kyriakos Keremedis ◽  
Eleftherios Tachtsis ◽  
Eliza Wajch
Keyword(s):  
Continuum Hypothesis ◽  
Compact Spaces ◽  
Power Set ◽  
Compact Metrizable Space ◽  
Permutation Models ◽  
Independence Results ◽  
The Axiom Of Choice ◽  
Metrizable Spaces ◽  
Transfer Theorems ◽  
The Continuum

AbstractIn the absence of the axiom of choice, the set-theoretic status of many natural statements about metrizable compact spaces is investigated. Some of the statements are provable in $$\mathbf {ZF}$$ ZF , some are shown to be independent of $$\mathbf {ZF}$$ ZF . For independence results, distinct models of $$\mathbf {ZF}$$ ZF and permutation models of $$\mathbf {ZFA}$$ ZFA with transfer theorems of Pincus are applied. New symmetric models of $$\mathbf {ZF}$$ ZF are constructed in each of which the power set of $$\mathbb {R}$$ R is well-orderable, the Continuum Hypothesis is satisfied but a denumerable family of non-empty finite sets can fail to have a choice function, and a compact metrizable space need not be embeddable into the Tychonoff cube $$[0, 1]^{\mathbb {R}}$$ [ 0 , 1 ] R .

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