David Meyer<dmm>:
Born #onthisday 118 years ago, Kurt Gödel was an Austrian mathematician and philosopher. Gödel discovered the “Incompleteness Theorem”, which essentially states that there will always be theorems in mathematics that are impossible to prove. Gödel''s discovery of the Incompleteness Theorems effectively drove a stake though the heart of Hilbert's Program [1] (or at least badly damaged it; see Hilbert's Second Problem [2]).
In 1949 Gödel demonstrated the existence of solutions to Einstein's field equations in General Relativity which involve "rotating universes" and featured closed timeline curves which allow for time travel to the past. Gödel's solutions are known as the Gödel metric and are an exact solution of the Einstein field equations [3].
Along with Aristotle, Alfred Tarski and Gottlob Frege, Gödel is considered to be one of the most significant logicians in history and had an immense effect upon scientific and philosophical thinking in the 20th century (and beyond).
Read more about Gödel's life and times here: https://mathshistory.st-andrews.ac.uk/Biographies/Godel.
References
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[1] "Hilbert’s Program", https://plato.stanford.edu/entries/hilbert-program/
[2] "Hilbert's second problem", https://en.wikipedia.org/wiki/Hilbert%27s_second_problem
[3] "THE GODEL SOLUTION TO THE EINSTEIN FIELD EQUATIONS", http://www.math.toronto.edu/~colliand/426/Papers/A_Monin.pdf
#godel #incompletenesstheorems #math #maths #physics
https://mathstodon.xyz/@dmm/112349313059106134
Born #onthisday 118 years ago, Kurt Gödel was an Austrian mathematician and philosopher. Gödel discovered the “Incompleteness Theorem”, which essentially states that there will always be theorems in mathematics that are impossible to prove. Gödel''s discovery of the Incompleteness Theorems effectively drove a stake though the heart of Hilbert's Program [1] (or at least badly damaged it; see Hilbert's Second Problem [2]).
In 1949 Gödel demonstrated the existence of solutions to Einstein's field equations in General Relativity which involve "rotating universes" and featured closed timeline curves which allow for time travel to the past. Gödel's solutions are known as the Gödel metric and are an exact solution of the Einstein field equations [3].
Along with Aristotle, Alfred Tarski and Gottlob Frege, Gödel is considered to be one of the most significant logicians in history and had an immense effect upon scientific and philosophical thinking in the 20th century (and beyond).
Read more about Gödel's life and times here: https://mathshistory.st-andrews.ac.uk/Biographies/Godel.
References
--------------
[1] "Hilbert’s Program", https://plato.stanford.edu/entries/hilbert-program/
[2] "Hilbert's second problem", https://en.wikipedia.org/wiki/Hilbert%27s_second_problem
[3] "THE GODEL SOLUTION TO THE EINSTEIN FIELD EQUATIONS", http://www.math.toronto.edu/~colliand/426/Papers/A_Monin.pdf
#godel #incompletenesstheorems #math #maths #physics
https://mathstodon.xyz/@dmm/112349313059106134
Maths History
Kurt Gödel
Gödel proved fundamental results about axiomatic systems showing in any axiomatic mathematical system there are propositions that cannot be proved or disproved within the axioms of the system.
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#proof #paper #primenumbers
In July 2017, Romeo Meštrović discovered a new, very short proof of the infinitude of primes using a simple "even-odd" argument.
Euclid's theorem on the infinitude of primes has captivated mathematicians for generations since its original proof by Euclid in 300 B.C.
Paul Erdős famously said, "It will be another million years, at least, before we understand the primes," yet this has not deterred mathematicians throughout the centuries from exploring the mysteries of primes and their unique properties.
Many renowned mathematicians from the 18th and 19th centuries, such as Goldbach, Euler, Lebesgue, Kronecker, Hensel, Kummer, Stieltjes, and Hermite, contributed various proofs of the infinitude of primes. Additionally, in the past century, notable mathematicians including I. Schur, K. Hensel, G. Pólya, Erdős, and G. H. Hardy, among others, have offered further compelling proofs, including demonstrations of the infinitude of primes in different arithmetic progressions.
There is no way to determine how many different proofs of the infinitude of primes exist and how many will emerge in the coming years.
Romeo's proof annotated: FermatsLibrary
In July 2017, Romeo Meštrović discovered a new, very short proof of the infinitude of primes using a simple "even-odd" argument.
Euclid's theorem on the infinitude of primes has captivated mathematicians for generations since its original proof by Euclid in 300 B.C.
Paul Erdős famously said, "It will be another million years, at least, before we understand the primes," yet this has not deterred mathematicians throughout the centuries from exploring the mysteries of primes and their unique properties.
Many renowned mathematicians from the 18th and 19th centuries, such as Goldbach, Euler, Lebesgue, Kronecker, Hensel, Kummer, Stieltjes, and Hermite, contributed various proofs of the infinitude of primes. Additionally, in the past century, notable mathematicians including I. Schur, K. Hensel, G. Pólya, Erdős, and G. H. Hardy, among others, have offered further compelling proofs, including demonstrations of the infinitude of primes in different arithmetic progressions.
There is no way to determine how many different proofs of the infinitude of primes exist and how many will emerge in the coming years.
Romeo's proof annotated: FermatsLibrary
Fermat's Library
Fermat's Library | A Very Short Proof of the Infinitude of Primes annotated/explained version.
Fermat's Library is a platform for illuminating academic papers.
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59Amer.Math.Monthly2015-1.pdf
218.7 KB
Romeo Meštrović - A Very Short Proof of the Infinitude of Primes
June 2017. The American Mathematical Monthly 124(6):562
DOI:10.4169/amer.math.monthly.124.6.562
June 2017. The American Mathematical Monthly 124(6):562
DOI:10.4169/amer.math.monthly.124.6.562