scholarly journals Collatz conjecture(3X+1)There is a hidden theorem ω1: If α Establishment, necessity (3α+ 1) Established

2021 ◽  
Author(s):  
Xie Ling
Keyword(s):  
Mathematical Logic ◽  
Number Theory ◽  
Human Beings ◽  
Computer Data ◽  
Collatz Conjecture

Abstract From a number theory “Collatz conjecture (3X+1)”, Human beings use a large amount of computer data, so far no counterexample has been found. Does mathematical logic support " Collatz conjecture (3X+1)? Collatz conjecture (3X+1) There is a hidden theorem ω1 : If x holds, it must be (3X+1). In reality, human beings will only (3X+1) deduce that x holds.Example: an integer a, and a = 3b +1, b∈N. If b→3x +1 holds. There must be: a→3x +1 is established.In this way, there is no need to deduce (3b + 1).

scholarly journals Collatz conjecture(3X+1)There is a hidden theorem ω1: If α Establishment, necessity (3α+ 1) Established

2021 ◽  
Author(s):  
Xie Ling
Keyword(s):  
Mathematical Logic ◽  
Number Theory ◽  
Human Beings ◽  
Computer Data ◽  
Collatz Conjecture

Abstract From a number theory “Collatz conjecture (3X+1)”, Human beings use a large amount of computer data, so far no counterexample has been found. Does mathematical logic support " Collatz conjecture (3X+1)? Collatz conjecture (3X+1) There is a hidden theorem ω1 : If x holds, it must be (3X+1). In reality, human beings will only (3X+1) deduce that x holds.Example: an integer a, and a = 3b +1, b∈N. If b→3x +1 holds. There must be: a→3x +1 is established.In this way, there is no need to deduce (3b + 1).

scholarly journals Collatz conjecture(3X+1)There is a hidden theorem ω1: If α Establishment, necessity (3α+ 1) Established

2021 ◽  
Author(s):  
Xie Ling
Keyword(s):  
Mathematical Logic ◽  
Number Theory ◽  
Human Beings ◽  
Computer Data ◽  
Collatz Conjecture

Abstract From a number theory “Collatz conjecture (3X+1)”, Human beings use a large amount of computer data, so far no counterexample has been found. Does mathematical logic support " Collatz conjecture (3X+1)? Collatz conjecture (3X+1) There is a hidden theorem ω1 : If x holds, it must be (3X+1). In reality, human beings will only (3X+1) deduce that x holds.Example: an integer a, and a = 3b +1, b∈N. If b→3x +1 holds. There must be: a→3x +1 is established.In this way, there is no need to deduce (3b + 1).

Kolmogorov and mathematical logic

1992 ◽  
Vol 57 (2) ◽  
pp. 385-412 ◽  
Author(s):  
Vladimir A. Uspensky
Keyword(s):  
Twentieth Century ◽  
Mathematical Logic ◽  
Graduate Education ◽  
Set Theory ◽  
Personal Experience ◽  
Foundations Of Mathematics ◽  
Strict Sense ◽  
Human Beings ◽  
The Third ◽  
Intellectual Power

There are human beings whose intellectual power exceeds that of ordinary men. In my life, in my personal experience, there were three such men, and one of them was Andrei Nikolaevich Kolmogorov. I was lucky enough to be his immediate pupil. He invited me to be his pupil at the third year of my being student at the Moscow University. This talk is my tribute, my homage to my great teacher.Andrei Nikolaevich Kolmogorov was born on April 25, 1903. He graduated from Moscow University in 1925, finished his post-graduate education at the same University in 1929, and since then without any interruption worked at Moscow University till his death on October 20, 1987, at the age 84½.Kolmogorov was not only one of the greatest mathematicians of the twentieth century. By the width of his scientific interests and results he reminds one of the titans of the Renaissance. Indeed, he made prominent contributions to various fields from the theory of shooting to the theory of versification, from hydrodynamics to set theory. In this talk I should like to expound his contributions to mathematical logic.Here the term “mathematical logic” is understood in a broad sense. In this sense it, like Gallia in Caesarian times, is divided into three parts:(1) mathematical logic in the strict sense, i.e. the theory of formalized languages including deduction theory,(2) the foundations of mathematics, and(3) the theory of algorithms.

scholarly journals Divisions by Two in Collatz Sequences: A Data Science Approach

2020 ◽  
Vol 21 ◽  
pp. 1-13
Author(s):  
Christian Koch ◽  
Eldar Sultanow ◽  
Sean Cox
Keyword(s):  
Number Theory ◽  
Data Science ◽  
Theory Problem ◽  
Mathematical Methods ◽  
Collatz Conjecture ◽  
Science Approach ◽  
Specific Length

The Collatz conjecture is an unsolved number theory problem. We approach the question by examining the divisions by two that are performed within Collatz sequences. Aside from classical mathematical methods, we use techniques of data science. Based on the analysis of 10,000 sequences we show that the number of divisions by two lies within clear boundaries. Building on the results, we develop and prove an equation to calculate the maximum possible number of divisions by two for any given a Collatz sequence. Whenever this maximum is reached, a sequence leads to the result one, as conjectured by Lothar Collatz. Furthermore, we show how many divisions by two are required for a cycle of a specific length. The findings are valuable for further investigations and could form the basis for a comprehensive proof of the conjecture.

Gödel's Incompleteness Theorems

1992 ◽  
Author(s):  
Raymond M. Smullyan
Keyword(s):  
Mathematical Logic ◽  
Number Theory ◽  
Continuum Theory ◽  
Kurt Gödel ◽  
The World ◽  
Computer Scientists ◽  
Incompleteness Theorems ◽  
The Axiom Of Choice ◽  
Gödel’S Incompleteness Theorems ◽  
The Continuum

Kurt Godel, the greatest logician of our time, startled the world of mathematics in 1931 with his Theorem of Undecidability, which showed that some statements in mathematics are inherently "undecidable." His work on the completeness of logic, the incompleteness of number theory, and the consistency of the axiom of choice and the continuum theory brought him further worldwide fame. In this introductory volume, Raymond Smullyan, himself a well-known logician, guides the reader through the fascinating world of Godel's incompleteness theorems. The level of presentation is suitable for anyone with a basic acquaintance with mathematical logic. As a clear, concise introduction to a difficult but essential subject, the book will appeal to mathematicians, philosophers, and computer scientists.

Mathematics of the 19th Century: Mathematical Logic, Algebra, Number Theory Probability Theory.

1994 ◽  
Vol 101 (4) ◽  
pp. 369
Author(s):  
Karen Hunger Parshall ◽  
A. N. Kolmogorov ◽  
A. P. Yushkevich
Keyword(s):  
Probability Theory ◽  
Mathematical Logic ◽  
Number Theory ◽  
19Th Century ◽  
The 19Th Century
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