https://mklab.gr/ Sat, 12 Sep 2026 07:45:07 +0000 MyBB https://mklab.gr/showthread.php?tid=1861 Sat, 05 Sep 2026 23:56:15 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1861 Author: Natalie Wolchover
Publication date: July 14, 2020
Publisher:Quanta Magazine

The article explains the central idea behind Kurt Gödel’s incompleteness theorems, which overturned the hope that mathematics could be based on a single axiomatic system that was both completely consistent and capable of proving every mathematical truth. Gödel showed that any sufficiently expressive, consistent formal system capable of arithmetic must contain statements that are true but cannot be proved within that system. Moreover, such a system cannot establish its own consistency using only its internal rules. 

The key technical device is Gödel numbering. Gödel assigns numbers to mathematical symbols and then encodes entire formulas as unique integers using prime factorization. For example, a sequence of symbols with codes $a_1,a_2,\ldots,a_n$ can essentially be represented by a number of the form
$2a13a25a3⋯pnan.2^{a_1}3^{a_2}5^{a_3}\cdots p_n^{a_n}$.
Because prime factorization is unique, the original mathematical expression can be reconstructed from its number. Gödel extended this idea to entire proofs, allowing statements about formulas and proofs—metamathematical statements—to be translated into ordinary statements about integers. Mathematics thereby acquires a way of talking about its own syntax and provability. 

Gödel then uses a sophisticated self-reference construction to produce a sentence $G$ that effectively says “$G$ is not provable in this system.” If the system could prove $G$, then $G$ would be false, contradicting consistency. Therefore, assuming the system is consistent, $G$ cannot be proved; but that makes what $G$ says true. Thus there exists a true but unprovable statement, so the system is incomplete. Adding $G$ as a new axiom does not solve the problem permanently: the enlarged system allows another Gödel-type sentence to be constructed. This establishes a permanent gap between mathematical truth and formal provability. 

Key takeaways

• Gödel numbering turns formulas and proofs into integers, letting arithmetic encode statements about mathematics itself.
  
• Gödel constructs a self-referential sentence $G$ asserting its own unprovability.
  
• A sufficiently powerful consistent axiomatic system therefore cannot be both consistent and complete.
  
• Gödel’s second incompleteness theorem shows that such a system cannot, in the relevant formal sense, prove its own consistency. 
  

Central idea: Gödel did not show that mathematics is unreliable; he showed that formal axiomatic methods have intrinsic limits. There will always be mathematical truths lying beyond what any one sufficiently powerful consistent formal system can prove.

ARTICLE / ARTICLE [PDF]]]> Author: Natalie Wolchover
Publication date: July 14, 2020
Publisher:Quanta Magazine

The article explains the central idea behind Kurt Gödel’s incompleteness theorems, which overturned the hope that mathematics could be based on a single axiomatic system that was both completely consistent and capable of proving every mathematical truth. Gödel showed that any sufficiently expressive, consistent formal system capable of arithmetic must contain statements that are true but cannot be proved within that system. Moreover, such a system cannot establish its own consistency using only its internal rules. 

The key technical device is Gödel numbering. Gödel assigns numbers to mathematical symbols and then encodes entire formulas as unique integers using prime factorization. For example, a sequence of symbols with codes $a_1,a_2,\ldots,a_n$ can essentially be represented by a number of the form
$2a13a25a3⋯pnan.2^{a_1}3^{a_2}5^{a_3}\cdots p_n^{a_n}$.
Because prime factorization is unique, the original mathematical expression can be reconstructed from its number. Gödel extended this idea to entire proofs, allowing statements about formulas and proofs—metamathematical statements—to be translated into ordinary statements about integers. Mathematics thereby acquires a way of talking about its own syntax and provability. 

Gödel then uses a sophisticated self-reference construction to produce a sentence $G$ that effectively says “$G$ is not provable in this system.” If the system could prove $G$, then $G$ would be false, contradicting consistency. Therefore, assuming the system is consistent, $G$ cannot be proved; but that makes what $G$ says true. Thus there exists a true but unprovable statement, so the system is incomplete. Adding $G$ as a new axiom does not solve the problem permanently: the enlarged system allows another Gödel-type sentence to be constructed. This establishes a permanent gap between mathematical truth and formal provability. 

Key takeaways

• Gödel numbering turns formulas and proofs into integers, letting arithmetic encode statements about mathematics itself.
  
• Gödel constructs a self-referential sentence $G$ asserting its own unprovability.
  
• A sufficiently powerful consistent axiomatic system therefore cannot be both consistent and complete.
  
• Gödel’s second incompleteness theorem shows that such a system cannot, in the relevant formal sense, prove its own consistency. 
  

Central idea: Gödel did not show that mathematics is unreliable; he showed that formal axiomatic methods have intrinsic limits. There will always be mathematical truths lying beyond what any one sufficiently powerful consistent formal system can prove.

ARTICLE / ARTICLE [PDF]]]> https://mklab.gr/showthread.php?tid=910 Mon, 06 Jul 2026 23:32:47 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=910 List of unsolved problems in mathematics

Summary

The history of mathematics is filled with famous problems that resisted solution for decades or even centuries, and this list highlights many of the most important ones that remain open today. It brings together outstanding questions from virtually every branch of mathematics—including number theory, algebra, geometry, topology, graph theory, analysis, dynamical systems, set theory, and computer science—ranging from deceptively simple puzzles to deep conjectures with far-reaching consequences. Among the best-known are the six remaining Millennium Prize Problems, such as the Riemann hypothesis, P versus NP problem, and the Navier–Stokes existence and smoothness, each carrying a $1 million prize for a correct solution. 

The article also places these problems in historical context by referencing influential collections such as Hilbert's problems, Landau's problems, Smale's problems, and the thousands of questions posed by Paul Erdős. Together, these unsolved problems illustrate that mathematics is a living and evolving discipline, where every major breakthrough not only answers long-standing questions but often opens the door to entirely new areas of discovery. 


ARTICLE]]> List of unsolved problems in mathematics

Summary

The history of mathematics is filled with famous problems that resisted solution for decades or even centuries, and this list highlights many of the most important ones that remain open today. It brings together outstanding questions from virtually every branch of mathematics—including number theory, algebra, geometry, topology, graph theory, analysis, dynamical systems, set theory, and computer science—ranging from deceptively simple puzzles to deep conjectures with far-reaching consequences. Among the best-known are the six remaining Millennium Prize Problems, such as the Riemann hypothesis, P versus NP problem, and the Navier–Stokes existence and smoothness, each carrying a $1 million prize for a correct solution. 

The article also places these problems in historical context by referencing influential collections such as Hilbert's problems, Landau's problems, Smale's problems, and the thousands of questions posed by Paul Erdős. Together, these unsolved problems illustrate that mathematics is a living and evolving discipline, where every major breakthrough not only answers long-standing questions but often opens the door to entirely new areas of discovery. 


ARTICLE]]> https://mklab.gr/showthread.php?tid=615 Mon, 22 Jun 2026 15:09:54 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=615 NIST Digital Library of Mathematical Functions

Summary


The NIST Digital Library of Mathematical Functions (DLMF) is a comprehensive online reference created by the National Institute of Standards and Technology (NIST) that provides authoritative information about mathematical functions, especially special functions used in mathematics, physics, engineering, and scientific computing. It is the modern digital successor to the classic Abramowitz and Stegun Handbook of Mathematical Functions, offering formulas, definitions, identities, numerical methods, references, and interactive tools.
The library covers topics such as gamma functions, Bessel functions, hypergeometric functions, orthogonal polynomials, elliptic functions, zeta functions, and many other advanced areas, making it an essential resource for researchers, students, and anyone working with applied mathematics.


LIBRARY

]]> NIST Digital Library of Mathematical Functions

Summary


The NIST Digital Library of Mathematical Functions (DLMF) is a comprehensive online reference created by the National Institute of Standards and Technology (NIST) that provides authoritative information about mathematical functions, especially special functions used in mathematics, physics, engineering, and scientific computing. It is the modern digital successor to the classic Abramowitz and Stegun Handbook of Mathematical Functions, offering formulas, definitions, identities, numerical methods, references, and interactive tools.
The library covers topics such as gamma functions, Bessel functions, hypergeometric functions, orthogonal polynomials, elliptic functions, zeta functions, and many other advanced areas, making it an essential resource for researchers, students, and anyone working with applied mathematics.


LIBRARY

]]> https://mklab.gr/showthread.php?tid=597 Mon, 22 Jun 2026 13:41:35 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=597 Mathematics Subject Classification

Summary

The Mathematics Subject Classification 2020 (MSC2020) is a standardized system used to organize and categorize mathematical literature by subject area. It is maintained jointly by Mathematical Reviews (MR) and zbMATH and is widely used by journals, publishers, researchers, and databases to classify mathematical papers. 
The system is hierarchical, using codes such as two-digit, three-digit, and five-digit classifications to represent fields ranging from number theory, algebra, geometry, and analysis to applied mathematics, statistics, computer science, and mathematical education. MSC2020 is an updated version of MSC2010 and helps researchers search, index, and identify relevant mathematical work more efficiently. 


Classification]]> Mathematics Subject Classification

Summary

The Mathematics Subject Classification 2020 (MSC2020) is a standardized system used to organize and categorize mathematical literature by subject area. It is maintained jointly by Mathematical Reviews (MR) and zbMATH and is widely used by journals, publishers, researchers, and databases to classify mathematical papers. 
The system is hierarchical, using codes such as two-digit, three-digit, and five-digit classifications to represent fields ranging from number theory, algebra, geometry, and analysis to applied mathematics, statistics, computer science, and mathematical education. MSC2020 is an updated version of MSC2010 and helps researchers search, index, and identify relevant mathematical work more efficiently. 


Classification]]> https://mklab.gr/showthread.php?tid=465 Thu, 18 Jun 2026 06:23:47 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=465
The Wikipedia article on the List of Mathematical Constants presents a broad catalog of numbers that arise naturally throughout mathematics and have fixed, well-defined values. It includes famous constants such as π (the ratio of a circle’s circumference to its diameter), e (the base of natural logarithms), the golden ratio φ, the Euler–Mascheroni constant γ, Apéry’s constant, and many others from fields such as geometry, number theory, analysis, combinatorics, and chaos theory. 

The list also contains entire families of constants, including Bernoulli numbers, Stieltjes constants, and Champernowne constants, along with their definitions, formulas, dates of discovery, and mathematical classifications (e.g., rational, algebraic, irrational, or transcendental). Together, these constants serve as fundamental building blocks that appear repeatedly in mathematical formulas, theorems, and applications across many branches of mathematics. 

ARTICLE]]>
The Wikipedia article on the List of Mathematical Constants presents a broad catalog of numbers that arise naturally throughout mathematics and have fixed, well-defined values. It includes famous constants such as π (the ratio of a circle’s circumference to its diameter), e (the base of natural logarithms), the golden ratio φ, the Euler–Mascheroni constant γ, Apéry’s constant, and many others from fields such as geometry, number theory, analysis, combinatorics, and chaos theory. 

The list also contains entire families of constants, including Bernoulli numbers, Stieltjes constants, and Champernowne constants, along with their definitions, formulas, dates of discovery, and mathematical classifications (e.g., rational, algebraic, irrational, or transcendental). Together, these constants serve as fundamental building blocks that appear repeatedly in mathematical formulas, theorems, and applications across many branches of mathematics. 

ARTICLE]]> https://mklab.gr/showthread.php?tid=464 Thu, 18 Jun 2026 06:21:00 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=464 Summary

The article highlights five fascinating mathematical constants that are far less famous than π but have intriguing stories and applications. The Delian Constant (∛2) arises from the ancient problem of doubling a cube’s volume, while the Sofa Constant comes from the still-unsolved problem of finding the largest sofa that can be moved around a right-angled corner. Kaprekar’s Constant (6174) is a remarkable number that almost any four-digit number reaches after repeatedly rearranging its digits and subtracting. The Lemniscate Constant plays a role for the figure-eight-shaped lemniscate curve similar to π’s role for circles, and Buffon’s Constant emerges from Buffon’s Needle problem, a probability experiment that can be used to estimate π. Together, these constants showcase the surprising variety, beauty, and depth of mathematical ideas beyond the most familiar constants.

ARTICLE]]> Summary

The article highlights five fascinating mathematical constants that are far less famous than π but have intriguing stories and applications. The Delian Constant (∛2) arises from the ancient problem of doubling a cube’s volume, while the Sofa Constant comes from the still-unsolved problem of finding the largest sofa that can be moved around a right-angled corner. Kaprekar’s Constant (6174) is a remarkable number that almost any four-digit number reaches after repeatedly rearranging its digits and subtracting. The Lemniscate Constant plays a role for the figure-eight-shaped lemniscate curve similar to π’s role for circles, and Buffon’s Constant emerges from Buffon’s Needle problem, a probability experiment that can be used to estimate π. Together, these constants showcase the surprising variety, beauty, and depth of mathematical ideas beyond the most familiar constants.

ARTICLE]]> https://mklab.gr/showthread.php?tid=252 Wed, 10 Jun 2026 03:47:59 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=252 Some Fundamental Theorems in Mathematics
by Oliver Knill

Summary  An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of 135 theorems are whether the result can be formulated elegantly, whether it is beautiful or useful and whether it could serve as a guide without leading to panic.

BOOK PAGE]]> Some Fundamental Theorems in Mathematics
by Oliver Knill

Summary  An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of 135 theorems are whether the result can be formulated elegantly, whether it is beautiful or useful and whether it could serve as a guide without leading to panic.

BOOK PAGE]]>