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Encyclopedia > Cantor's theorem

In Zermelo-Fränkel set theory, Cantor's theorem states that the power set (set of all subsets) of any set A has a strictly greater cardinality than that of A. Cantor's theorem is obvious for finite sets, but surprisingly it holds true for infinite sets as well. In particular, the power set of a countably infinite set is un-countably infinite. To illustrate the validity of Cantor's theorem for infinite sets, just test an infinite set in the proof below. Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. The relationship of one set being a subset of another is called inclusion. ... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ... In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ... In mathematics, a countable set is a set with the same cardinality (i. ...

Contents


The proof

Let f be any function from A into the power set of A. It must be shown that f is necessarily not surjective. To do that, it is enough to exhibit a subset of A that is not in the image of f. That subset is In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ... A surjective function. ...

B=left{,xin A : xnotin f(x),right}.

To show that B is not in the image of f, suppose that B is in the image of f. Then for some yA, we have f(y) = B. Now consider whether yB or yB. If yB, then yf(y), but that implies, by definition of B, that yB. On the other hand, if yB, then yf(y) and therefore yB. Either way, we get a contradiction.


Because of the double occurrence of x in the expression "xf(x)", this is a diagonal argument. A variety of diagonal arguments are used in mathematics. ...


A detailed explanation of the proof when X is countably infinite

To get a handle on the proof, let's examine it for the specific case when X is countably infinite. Without loss of generality, we may take X = N = {1, 2, 3,...}, the set of natural numbers. In mathematics the term countable set is used to describe the size of a set, e. ... Without loss of generality or simply WLOG is a frequently used expression in mathematics. ... In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory and computer science. ...


Suppose that N is bijective with its power set P(N). Let us see a sample of what P(N) looks like: In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ... In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ...


P(mathbb{N})={varnothing,{1, 2}, {1, 2, 3}, {4}, {1, 5}, {3, 4, 6}, {2, 4, 6,...},...}


P(N) contains infinite subsets of N, e.g. the set of all even numbers {2, 4, 6,...}, as well as the empty set. In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ...


Now that we have a handle on what the elements of P(N) look like, let us attempt to pair off each element of N with each element of P(N) to show that these infinite sets are bijective. In other words, we will attempt to pair off each element of N with an element from the infinite set P(N), so that no element from either infinite set remains unpaired. Such an attempt to pair elements would look like this: In mathematics, an element (also called a member) is an object contained in a set (or more generally a class). ...


Xbegin{Bmatrix} 1 & Longleftrightarrow & {4, 5} 2 & Longleftrightarrow & {1, 2, 3}  3 & Longleftrightarrow & {4, 5, 6}  4 & Longleftrightarrow & {1, 3, 5}  vdots & vdots & vdots end{Bmatrix}P(mathbb{N})


Some natural numbers are paired with subsets that do not contain them. For instance, in our example the number 1 is paired with the subset {4, 5}. Other natural numbers are paired with subsets that do contain them. For instance, the number 2 is paired with the subset {1, 2, 3}. A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. The relationship of one set being a subset of another is called inclusion. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. The relationship of one set being a subset of another is called inclusion. ...


Using this idea, let us build a special set of natural numbers. This set will provide the contradiction we seek. Let D be the set of all natural numbers which are paired with subsets that do not contain them. By definition, our power set P(N) must contain this set D as an element. Therefore, D must be paired off with some natural number. However, this causes a problem -- which natural number will be paired with D? It cannot be a member of D, since D was specially constructed to contain only those natural numbers which are *not* paired with subsets containing them. On the other hand, if the natural number paired with D is not contained in D then . . . well, then it must be contained in D, again by the definition of D! Reductio ad absurdum (Latin for reduction to the absurd, traceable back to the Greek ἡ εις το αδυνατον απαγωγη, reduction to the impossible, often used by Aristotle) is a type of logical argument where we assume a claim for the sake of argument, arrive at an absurd result, and then... In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ...


This is a contradiction because the natural number cannot be both inside and outside of D at the same time. Therefore, there is no natural number which can be paired with D, and we have contradicted our original supposition, that there is a bijection between N and P(N). A bijective function. ...


Through this proof by contradiction we have proven that the cardinality of N and P(N) cannot be equal. We also know that the cardinality of P(N) cannot be less than the cardinality of N because P(N) contains all singletons, by definition, and these singletons form a "copy" of N inside of P(N). Therefore, only one possibility remains, and that is the cardinality of P(N) is strictly greater than the cardinality of N, and this proves Cantor's theorem. Reductio ad absurdum (Latin for reduction to the absurd, traceable back to the Greek ἡ εις το αδυνατον απαγωγη, reduction to the impossible, often used by Aristotle) is a type of logical argument where we assume a claim for the sake of argument, arrive at an absurd result, and then... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ... In mathematics, a singleton is a set with exactly one element. ... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ...


History

Cantor gave essentially this proof in a paper published in 1891 Über eine elementare Frage der Mannigfaltigkeitslehre, where the diagonal argument for the uncountability of the reals also first appears (he had earlier proved the uncountability of the reals by other methods). The version of this argument he gave in that paper was phrased in terms of indicator functions on a set rather than subsets of a set. He showed that if f is a function defined on X whose values are 2-valued functions on X, then the 2-valued function G(x) = 1 − f(x)(x) is not in the range of f. Cantors diagonal argument is a proof devised by Georg Cantor to demonstrate that the real numbers are not countably infinite. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... Contrary to what most mathematicians believe, Georg Cantors first proof that the set of all real numbers is uncountable was not his famous diagonal argument, and did not mention decimal expansions or any other numeral system. ...


Russell has a very similar proof in Principles of Mathematics (1903, section 348), where he shows that that there are more propositional functions than objects. "For suppose a correlation of all objects and some propositional functions to have been affected, and let phi-x be the correlate of x. Then "not-phi-x(x)," i.e. "phi-x does not hold of x" is a propositional function not contained in this correlation; for it is true or false of x according as phi-x is false or true of x, and therefore it differs from phi-x for every value of x." He attributes the idea behind the proof to Cantor. In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ...


Ernst Zermelo has a theorem (which he calls "Cantor's Theorem") that is identical to the form above in the paper that became the foundation of modern set theory ("Untersuchungen über die Grundlagen der Mengenlehre I"), published in 1908. See Zermelo set theory. Ernst Friedrich Ferdinand Zermelo (July 27, 1871 – May 21, 1953) was a German mathematician and philosopher. ... Zermelo set theory, as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern set theory. ...


For one consequence of Cantor's theorem, see beth numbers. In mathematics, the Hebrew letter (aleph) with various subscripts represents various infinite cardinal numbers (see aleph number). ...


See also


 
 

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