On The Continuum Hypothesis
While studying the continuum hypothesis, Cohen is quoted as saying in 1985 that he "had the feeling that people thought the problem was hopeless, since there was no new way of constructing models of set theory. Indeed, they thought you had to be slightly crazy even to think about the problem."
"A point of view which the author feels may eventually come to be accepted is that CH is obviously false. The main reason one accepts the axiom of infinity is probably that we feel it absurd to think that the process of adding only one set at a time can exhaust the entire universe. Similarly with the higher axioms of infinity. Now is the cardinality of the set of countable ordinals, and this is merely a special and the simplest way of generating a higher cardinal. The set is, in contrast, generated by a totally new and more powerful principle, namely the power set axiom. It is unreasonable to expect that any description of a larger cardinal which attempts to build up that cardinal from ideas deriving from the replacement axiom can ever reach .
Thus is greater than, where, etc. This point of view regards as an incredibly rich set given to us by one bold new axiom, which can never be approached by any piecemeal process of construction. Perhaps later generations will see the problem more clearly and express themselves more eloquently."
An "enduring and powerful product" of Cohen's work on the Continuum Hypothesis, and one that has been used by "countless mathematicians" is known as "forcing", and it is used to construct mathematical models to test a given hypothesis for truth or falsehood.
Shortly before his death, Cohen gave a lecture describing his solution to problem of the Continuum Hypothesis at the Gödel centennial conference, in Vienna in 2006. A video of this lecture is now available online.
Read more about this topic: Paul Cohen (mathematician)
Famous quotes containing the words continuum and/or hypothesis:
“The further jazz moves away from the stark blue continuum and the collective realities of Afro-American and American life, the more it moves into academic concert-hall lifelessness, which can be replicated by any middle class showing off its music lessons.”
—Imamu Amiri Baraka (b. 1934)
“The hypothesis I wish to advance is that ... the language of morality is in ... grave disorder.... What we possess, if this is true, are the fragments of a conceptual scheme, parts of which now lack those contexts from which their significance derived. We possess indeed simulacra of morality, we continue to use many of the key expressions. But we have—very largely if not entirely—lost our comprehension, both theoretical and practical, of morality.”
—Alasdair Chalmers MacIntyre (b. 1929)