Difference between revisions of "Continuum hypothesis"

Latest revision as of 00:18, March 31, 2019

The Continuum hypothesis is a conjecture by Georg Cantor which states that there is no set with cardinality greater than all the natural numbers but less than the cardinality of the real numbers (Continuum). Stated another way, this hypothesis holds that there is no set which as a cardinality between that of the natural numbers and that of the real numbers.

The Generalized Continuum hypothesis states that there is no set whose cardinality lies between the cardinality of a given set and the cardinality of the set of all subsets of that given set.

In standard mathematical nomenclature, the cardinality of a set is denoted by the Hebrew letter $\aleph$ .

Cantor died without knowing the answer to his conjecture of the Continuum hypothesis. Kurt Godel and Paul Cohen have since shown that the Continuum hypothesis is undecidable in Zermelo-Fraenkel Set Theory.

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-The '''Continuum hypothesis''' is a conjecture by [[Georg Cantor]] which states that the cardinality of the real numbers (Continuum) equals that of <math>\mathcal{P}(\mathbb{N})</math>, the power set of the [[natural numbers]]. The cardinality of this set is denoted with the Hebrew letter <math>\aleph</math>.  [[Kurt Godel]] and [[Paul Cohen]] has since shown that the Continuum hypothesis is undecidable in [[Zermelo-Fraenkel Set Theory]].
+The '''Continuum hypothesis''' is a conjecture by [[Georg Cantor]] which states that there is no set with [[cardinality]] greater than all the [[natural number]]s but less than the cardinality of the [[real number]]s (Continuum).  Stated another way, this hypothesis holds that there is no set which as a cardinality between that of the natural numbers and that of the real numbers.
-In layman's terms, Dr. Amir Aczel, author of "The Mystery of the Aleph: Mathematics, the Kabbalah and the Search for Infinity," (Four Walls Eight Windows: 2000) explained this concept on National Public Radio:<ref>AUTHOR AMIR ACZEL DISCUSSES CONCEPTS OF INFINITY National Public Radio (NPR) January 19, 2001</ref>
+The ''Generalized'' Continuum hypothesis states that there is no set whose cardinality lies between the cardinality of a given set and the cardinality of the set of all subsets of that given set.
-:This is called the continuum hypothesis. And it has many equivalents in mathematics, by the way. There's something called the axiom of choice and many other equivalents of this continuum hypothesis. And what the continuum hypothesis says, very simply, is that there is one level of infinity, which is that of the numbers, it's called the whole numbers...1, 2, 3, 4, 5. And yet there is a higher level of infinity for the continuum, meaning numbers on a stretch of numbers, the real line where you can have irrational numbers and rational numbers and so on. So what Cantor said is that--and he died not knowing the answer...was that one--the numbers are, he called them aleph zero, their level of infinity. The continuum has a higher level of infinity called aleph one. Now Kurt Godel proved in the '30s that there is no solution to--he really didn't finish the proof; he went one way, and Paul Cohen of Stanford University finished that proof in 1963, that within our system of mathematics, it is not possible for us to know what the answer is. It's not that there is no answer, there is an answer, but we can't know what the answer is.
+In standard mathematical nomenclature, the cardinality of a set is denoted by the Hebrew letter <math>\aleph</math>.
-== References ==
+Cantor died without knowing the answer to his conjecture of the Continuum hypothesis. [[Kurt Godel]] and Paul Cohen have since shown that the Continuum hypothesis is [[undecidable]] in [[Zermelo-Fraenkel]] Set Theory.
-<references/>
+[[Category:Set Theory]]
+[[category:infinity]]
-[[category:set theory]]