In the foundations of mathematics, Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction. The same paradox had been discovered a year before by Ernst Zermelo but he did not publish the idea, which remained known only to Hilbert, Husserl and other members of the University of Göttingen. According to naive set theory, any definable collection is a set. Let R be the set of all sets that are not members of themselves. If R qualifies as a member of itself, it would contradi...

A synagogue, also spelled synagog (from Greek: transliterated synagogē, meaning "assembly"; beyt knesset, meaning "house of assembly"; beyt t'fila, meaning "house of prayer"; shul; esnoga; kal) is a Jewish house of prayer. When broken down, the word could also mean "learning together" (from the Greek συν syn, together, and αγωγή agogé, learning or training). Synagogues have a large hall for prayer (the main sanctuary), and can also have smaller rooms for study and sometimes a social hall and offices. Some have a separate room for Torah study, called the beit midrash(Sfard) "beis ...

In number theory, Hillel Furstenberg's proof of the infinitude of primes is a celebrated topological proof that the integers contain infinitely many prime numbers. When examined closely, the proof is less a statement about topology than a statement about certain properties of arithmetic sequences. Unlike Euclid's classical proof, Furstenberg's proof is a proof by contradiction. The proof was published in 1955 in the American Mathematical Monthly while Furstenberg was still an undergraduate student at Yeshiva University. Furstenberg's proof Define a topology on the integers Z, called the...

Beal's conjecture is a conjecture in number theory proposed by Andrew Beal in 1993. While investigating generalizations of Fermat's last theorem in 1993, Beal formulated the following conjecture: :If :: A^x +B^y = C^z, :where A, B, C, x, y, and z are positive integers with x, y, z > 2, then A, B, and C have a common prime factor. For a proof or counterexample published in a refereed journal, Beal initially offered a prize of US $5,000 in 1997, rising to $50,000 over ten years, but has since raised it to US $1,000,000. Examples To illustrate, the solution 33 + 63 = 35 has bases with...

The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. As of May 2013, six of the problems remain unsolved. A correct solution to any of the problems results in a US$1,000,000 prize (sometimes called a Millennium Prize) being awarded by the institute. The Poincaré conjecture, the only Millennium Prize Problem to be solved so far, was solved by Grigori Perelman, but he declined the award in 2010. P versus NP The question is whether, for all problems for which an algorithm can verify a given solution quickly (that is,...