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Freiling's Axiom Of Symmetry
Freiling's axiom of symmetry (\texttt) is a set-theoretic axiom proposed by Chris Freiling. It is based on intuition of Stuart Davidson but the mathematics behind it goes back to Wacław Sierpiński. Let A\subseteq\mathcal( ,1^ denote the set of all functions from ,1/math> to countable subsets of ,1/math>. (In other words, A = \left(\big 0,1big]^\right)^ where is the collection of subsets of X of cardinality at most \kappa .) The axiom \texttt then states: :For every f\in A, there exist x,y\in , 1/math> such that x\not\in f(y) and y\not\in f(x). A theorem of Sierpiński says that under the assumptions of ZFC set theory, \texttt is equivalent to the negation of the continuum hypothesis (CH). Sierpiński's theorem answered a question of Hugo Steinhaus and was proved long before the independence of CH had been established by Kurt Gödel and Paul Cohen. Freiling claims that probabilistic intuition strongly supports this proposition while others disagree. There ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathematics – is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of ''naive set theory''. After the discovery of Paradoxes of set theory, paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox), various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied. Set the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chris Freiling
Christopher Francis Freiling is a mathematician responsible for Freiling's axiom of symmetry in set theory. He has also made significant contributions to coding theory, in the process establishing connections between that field and matroid theory. Freiling obtained his Ph.D. in 1981 from the University of California, Los Angeles under the supervision of Donald A. Martin. He is a member of the faculty of the Department of Mathematics at California State University, San Bernardino California State University, San Bernardino (Cal State San Bernardino or CSUSB) is a public research university in San Bernardino, California. Founded in 1965, it is part of the California State University system. The main campus sits on in th .... , CSUSB Mathematics Department, retrieved 2015-04-11. Selected pu ...
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Wacław Sierpiński
Wacław Franciszek Sierpiński (; 14 March 1882 – 21 October 1969) was a Polish mathematician. He was known for contributions to set theory (research on the axiom of choice and the continuum hypothesis), number theory, theory of functions, and topology. He published over 700 papers and 50 books. Three well-known fractals are named after him (the Sierpiński triangle, the Sierpiński carpet, and the Sierpiński curve), as are Sierpiński numbers and the associated Sierpiński problem. Early life and education Sierpiński was born in 1882 in Warsaw, Congress Poland, to a doctor father Konstanty and mother Ludwika (''née'' Łapińska). His abilities in mathematics were evident from childhood. He enrolled in the Department of Mathematics and Physics at the University of Warsaw in 1899 and graduated five years later. In 1903, while still at the University of Warsaw, the Department of Mathematics and Physics offered a prize for the best essay from a student on Vorono ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
