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Encyclopedia > Cantor's diagonal argument

Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers. Such sets are now known as uncountable sets, and the size of infinite sets is now treated by the theory of cardinal numbers which Cantor began. Year 1891 (MDCCCXCI) was a common year starting on Thursday (link will display the full calendar) of the Gregorian calendar (or a common year starting on Saturday of the 12-day slower Julian calendar). ... Georg Cantor Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845, St. ... In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. ... Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... A bijective function. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In mathematics, an uncountable or nondenumerable set is a set which is not countable. ... Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set. ...


The diagonal argument was not Cantor's first proof of the uncountability of the real numbers; it was actually published three years after his first proof, which appears in 1874. However, it demonstrates a powerful and general technique, which has since been reused many times in a wide range of proofs, also known as diagonal arguments by analogy with the argument used in this proof. The most famous examples are perhaps Russell's paradox, Gödel's first incompleteness theorem, and Turing's answer to the Entscheidungsproblem. The term diagonalization is used in two different senses in mathematics: The process of finding a diagonal matrix similar to a given square matrix or representing a given linear map. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... 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. ... Analogy is both the cognitive process of transferring information from a particular subject (the analogue or source) to another particular subject (the target), and a linguistic expression corresponding to such a process. ... Part of the foundation of mathematics, Russells paradox (also known as Russells antinomy), discovered by Bertrand Russell in 1901, showed that the naive set theory of Frege leads to a contradiction. ... Gödels incompleteness theorem - Wikipedia /**/ @import /skins-1. ... The Entscheidungsproblem (German for decision problem) is the challenge in symbolic logic to find a general algorithm which decides for given first-order statements whether they are universally valid or not. ...

Contents

An uncountable set

Cantor's original proof considers infinite sequences of elements of the form (x1, x2, x3, ...) where each element xi is either 0 or 1.


Consider any infinite listing of some of these sequences. We might have for instance:

s1 = (0, 0, 0, 0, 0, 0, 0, ...)
s2 = (1, 1, 1, 1, 1, 1, 1, ...)
s3 = (0, 1, 0, 1, 0, 1, 0, ...)
s4 = (1, 0, 1, 0, 1, 0, 1, ...)
s5 = (1, 1, 0, 1, 0, 1, 1, ...)
s6 = (0, 0, 1, 1, 0, 1, 1, ...)
s7 = (1, 0, 0, 0, 1, 0, 0, ...)
...

And in general we shall write

sn = (sn,1, sn,2, sn,3, sn,4, ...)

that is to say, sn,m is the mth element of the nth sequence on the list.


It is possible to build a sequence of elements s0 in such a way that its first element is different from the first element of the first sequence in the list, its second element is different from the second element of the second sequence in the list, and, in general, its nth element is different from the nth element of the nth sequence in the list. That is to say, s0,m will be 0 if sm,m is 1, and s0,m will be 1 if sm,m is 0. For instance:

s1 = (0, 0, 0, 0, 0, 0, 0, ...)
s2 = (1, 1, 1, 1, 1, 1, 1, ...)
s3 = (0, 1, 0, 1, 0, 1, 0, ...)
s4 = (1, 0, 1, 0, 1, 0, 1, ...)
s5 = (1, 1, 0, 1, 0, 1, 1, ...)
s6 = (0, 0, 1, 1, 0, 1, 1, ...)
s7 = (1, 0, 0, 0, 1, 0, 0, ...)
...
s0 = (1, 0, 1, 1, 1, 0, 1, ...)

(The elements s1,1, s2,2, s3,3, and so on, are here highlighted, showing the origin of the name "diagonal argument".)



And yet it may be seen that this new sequence s0 is distinct from all the sequences in the list. This follows from the fact that if it were identical to, say, the 10th sequence in the list, then we would have s0,10 = s10,10. In general, if it appeared as the nth sequence on the list, we would have s0,n = sn,n, which, due to the construction of s0, is impossible.


From this it follows that the set T, consisting of all infinite sequences of zeros and ones, cannot be put into a list s1, s2, s3, ... Otherwise, it would be possible by the above process to construct a sequence s0 which would both be in T (because it is a sequence of 0's and 1's which is by the definition of T in T) and at the same time not in T (because we can deliberately construct it not to be in the list). T, containing all such sequences, must contain s0, which is just such a sequence. But since s0 does not appear anywhere on the list, T cannot contain s0.


Therefore T cannot be placed in one-to-one correspondence with the natural numbers. In other words, it is uncountable.


The interpretation of Cantor's result will depend upon one's view of mathematics, and more specifically on how one thinks of mathematical functions. In the context of classical mathematics, functions need not be computable, and hence the diagonal argument establishes that, there are more infinite sequences of ones and zeros than there are natural numbers. To those constructivists who countenance only computable functions, Cantor's proof (merely) shows that there is no recursively enumerable set of indices (for example, Gödel numbers) for the programs computing them. Classical mathematics, as a term of art in mathematical logic, refers generally to mathematics constructed and proved on the basis of classical logic and ZFC set theory, i. ... Computable functions (or Turing-computable functions) are the basic objects of study in computability theory. ... In the philosophy of mathematics, constructivism asserts that it is necessary to find (or construct) a mathematical object to prove that it exists. ... Computable functions (or Turing-computable functions) are the basic objects of study in computability theory. ... In computability theory, often less suggestively called recursion theory, a countable set S is called recursively enumerable, computably enumerable, semi-decidable or provable if There is an algorithm that, when given an input — typically an integer or a tuple of integers or a sequence of characters — eventually halts... This article or section may be confusing or unclear for some readers, and should be edited to rectify this. ...


Real numbers

The uncountability of the real numbers was already established by Cantor's first uncountability proof, but it also follows from this result. It can be shown that the set T can be placed into one-to-one correspondence with the real numbers, that is, it has the cardinality of the continuum. As T is uncountable, it follows that the real numbers must also be uncountable. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... 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. ... In mathematics, the cardinality of the continuum is the cardinal number of the set of real numbers R (sometimes called the continuum). ...


General sets

A generalized form of the diagonal argument was used by Cantor to prove Cantor's theorem: for every set S the power set of S, i.e., the set of all subsets of S (here written as P(S)), is larger than S itself. This proof proceeds as follows: In Zermelo-Fränkel set theory, Cantors theorem states that the power set (set of all subsets) of any set A has a strictly greater cardinality than that of A. Cantors theorem is obvious for finite sets, but surprisingly it holds true for infinite sets as well. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... 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 is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ...


Let f be any one-to-one function from S to P(S). It suffices to prove f cannot be surjective. That means that some member of P(S), i.e., some subset of S, is not in the image of f. That set is In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ... In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ...

T={,sin S: snotin f(s),}.

If T is in the image of f, then for some t in S we have T = f(t). Either t is in T or not. If t is in T, then t is in f(t), but, by definition of T, that implies t is not in T. On the other hand, if t is not in T, then t is not in f(t), and by definition of T, that implies t is in T. Either way, we have a contradiction.


Note the similarity between the construction of T and the set in Russell's paradox. Its result can be used to show that the notion of the set of all sets is an inconsistent notion in normal set theory; if S would be the set of all sets then P(S) would at the same time be bigger than S and a subset of S. Part of the foundation of mathematics, Russells paradox (also known as Russells antinomy), discovered by Bertrand Russell in 1901, showed that the naive set theory of Frege leads to a contradiction. ... In set theory, referring to the set of all sets typically leads to a paradox. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...


The above proof fails for W. V. Quine's "New Foundations" set theory, which has a different version of the axiom of comprehension in which {s in S: snotin f(s),} cannot in general be shown to exist. The set {s in S: snotin f({s}),} (where P1(S) is the set of one-element subsets of S and f is supposed to be a bijection from P1(S) to P(S)) can be shown to exist in New Foundations, so the theorem one is able to prove is that |P1(S)| < |P(S)|: if f({r}) were equal to the set above (which must be true for some r in S if f is a map onto P(S)), then r in f({r}) would imply and be implied by its own negation. W. V. Quine Willard Van Orman Quine (June 25, 1908 - December 25, 2000) was one of the most influential American philosophers and logicians of the 20th century. ... In mathematical logic, New Foundations (NF) is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types of Principia Mathematica. ... In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of specification, or axiom schema of separation, or axiom schema of restricted comprehension, is a schema of axioms in Zermelo-Fraenkel set theory. ... In mathematical logic, New Foundations (NF) is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types of Principia Mathematica. ...


For a more concrete account of this proof that is possibly easier to understand see Cantor's theorem. In Zermelo-Fränkel set theory, Cantors theorem states that the power set (set of all subsets) of any set A has a strictly greater cardinality than that of A. Cantors theorem is obvious for finite sets, but surprisingly it holds true for infinite sets as well. ...


Analogues of the diagonal argument are widely used in mathematics to prove the existence or nonexistence of certain objects. For example, the conventional proof of the unsolvability of the halting problem is essentially a diagonal argument. In computability theory the halting problem is a decision problem which can be informally stated as follows: Given a description of a program and a finite input, decide whether the program finishes running or will run forever, given that input. ...


The diagonal argument shows that the set of real numbers is "bigger" than the set of integers. Therefore, we can ask if there is a set whose cardinality is "between" that of the integers and that of the reals. This question leads to the famous continuum hypothesis. Similarly, the question of whether there exists a set whose cardinality is between |S| and |P(S)| for some infinite S leads to the generalized continuum hypothesis. 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 and injections, and another which uses cardinal numbers. ... In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ... In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ...


External links

  • Original German text of the 1891 proof, with English translation

  Results from FactBites:
 
Cantors Diagonal argument - Wikipedia (681 words)
A logical argument devised by Georg Cantor to demonstrate that the real numbers are not countably infinite.
The diagonal argument is an example of reductio ad absurdum because it proves a certain proposition (the interval (0,1) is not countably infinite) by showing that the assumption of its negation leads to a contradiction.
A generalized form of the diagonal argument was used by Cantor to show that for every set S the power set of S, i.e., the set of all subsets of S (here written as P(S)), is larger than S itself.
  More results at FactBites »

 
 

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