INFINITY AND BEYOND
a Movie Review
In the last episode, Marcus du Sautoy looks at some of the great unsolved problems that confronts mathematics in the twentieth century and traveled to the places where great mathematicians lived and the place where they tried to solve this great mathematical problems.Mathematicians like Georg Cantor, who investigated a subject that many of the finest mathematical minds had avoided – infinity. Cantor's greatest contribution to mathematics was Cantor's theorem: Cantor's Theorem: The cardinality of the set of all subsets of any set is strictly greater than the cardinality of the set; i.e., for any set A,cardinality(powerset(A)) > cardinality(A). Cantor's theorem determines the 'infinity of infinities' and he also studied and explained the cardinal and ordinal numbers and their arithmetic.
His contributions results to a fundamentals of lemmas. He discovered the importance of one-to-one correspondence between sets, defined infinite and well-ordered sets, and explained that the "real numbers" are more numerous than the natural numbers.As he throught of a new theory he struggles with it for the rest of his life,about the infinity f smaller fraction and the infinity a larger decimal value that's where Henri Poincaré entered as he was trying to solve one mathematical problem when he accidentally stumbled on chaos theory,that led to a range of ‘smart’ technologies, including machines which control the regularity of heart beats. That later had its fare share of chaos.
Kurt Gödel, an active member of the famous 'Vienna Circle’ of philosophers.Gödel's work was the surprising culmination of a long search for foundations. He proved the incompleteness of axioms for arithmetic (his most famous result), as well as the relative consistency of the axiom of choice and continuum hypothesis with the other axioms of set theory.The basic idea of Gödel's proof, indirect self-reference, is simple but difficult t understand.
Paul Joseph Cohen in this programme, Marcus looks at the startling discoveries of the American mathematician Paul Cohen, Cohen proved the extremely surprising result that both the Continuum Hypothesis and the Axiom of Choice—two of the most basic ideas in mathematics—were actually undecidable using the axioms of set theory. This result meant mathematics could neither prove nor disprove concrete and well known mathematical assertions, caused healthy arguments between philosophers, logicians and mathematicians concerned with the concept of truth.
He also reflects on the contributions of Alexander Grothendieck, who influences in the current mathematical thinking and the mysteries behind the structures of mathematics. Marcus ended his documentation by reviewing all the unsolved problems of mathematics today, that includes Riemann Hypothesis - a conjecture about the distribution of prime numbers – which are the atoms of the mathematical universe with a prize of $1 million and a place in the history books to those who can prove Riemann’s theorem.
Marian <3
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