“Dangerous Knowledge” is a fascinating documentary by the BBC about the story of four masterminds, Goerg Cantor, Ludwig Boltzmann, Kurt Gödel and Alan Turing. What do these four have in common?
Cantor created "set theory". Cantor’s Theorem says that there exist infinitely many infinite sets of different cardinalities. The theory was nonsense and shocking to some of the mathematicians of his time and it created a lot of opposition. Some philosophers saw his theory as a challenge to the uniqueness of “God”, the absolute infinity. Cantor is also known for the Continuum Hypothesis which states that there is no set with cardinality between the cardinality of natural numbers and of real numbers. (We will go over Cantor’s Theorem and the Continuum Hypothesis in detail in class). Cantor could neither prove nor disprove the Continuum Hypothesis. In 1940 and 1960 respectively, Kurt Gödel and Paul Cohen showed that, in fact, one cannot prove or disprove this hypothesis using the set theory axioms. Gödel is best known for his Incompleteness Theorem which states one cannot find a complete and consistent set of axioms for all mathematics. This gave a negative answer to Hilbert’s second problem. If you would like to know more about Cantor, Gödel, Boltzmann and Turing, watch this 90 minute documentary!
Cantor created "set theory". Cantor’s Theorem says that there exist infinitely many infinite sets of different cardinalities. The theory was nonsense and shocking to some of the mathematicians of his time and it created a lot of opposition. Some philosophers saw his theory as a challenge to the uniqueness of “God”, the absolute infinity. Cantor is also known for the Continuum Hypothesis which states that there is no set with cardinality between the cardinality of natural numbers and of real numbers. (We will go over Cantor’s Theorem and the Continuum Hypothesis in detail in class). Cantor could neither prove nor disprove the Continuum Hypothesis. In 1940 and 1960 respectively, Kurt Gödel and Paul Cohen showed that, in fact, one cannot prove or disprove this hypothesis using the set theory axioms. Gödel is best known for his Incompleteness Theorem which states one cannot find a complete and consistent set of axioms for all mathematics. This gave a negative answer to Hilbert’s second problem. If you would like to know more about Cantor, Gödel, Boltzmann and Turing, watch this 90 minute documentary!
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