Cantor's diagonal argument - Wikipedia. An illustration of Cantor's diagonal argument (in base 2) for the existence of uncountable sets. The sequence at the bottom cannot occur anywhere in the enumeration of sequences above.
Nevertheless, infinite sets of different cardinalities exist, as Cantor's diagonal argument shows. In set theory, Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1. Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one- to- one correspondence with the infinite set of natural numbers. Diagonalization arguments are often also the source of contradictions like Russell's paradox. He begins with a constructive proof of the following theorem: If s. In the example, this yields: s.
By construction, s differs from each sn, since their nth digits differ (highlighted in the example). Hence, s cannot occur in the enumeration. Based on this theorem, Cantor then uses a proof by contradiction to show that: The set T is uncountable. He assumes for contradiction that T was countable.
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Then all its elements could be written as an enumeration s. Applying the previous theorem to this enumeration would produce a sequence s not belonging to the enumeration. However, s was an element of T and should therefore be in the enumeration. This contradicts the original assumption, so T must be uncountable. Interpretation. To constructivists, the argument shows no more than that there is no bijection between the natural numbers and T. It does not rule out the possibility that the latter are subcountable. In the context of classical mathematics, this is impossible, and the diagonal argument establishes that, although both sets are infinite, there are actually more infinite sequences of ones and zeros than there are natural numbers.
Real numbers. To see this, we will build a one- to- one correspondence between the set T of infinite binary strings and a subset of R (the set of real numbers). Since T is uncountable, this subset of R must be uncountable. Hence R is uncountable. To build this one- to- one correspondence (or bijection), observe that the string t = 0. This suggests defining the function f(t) = 0. T. Unfortunately, f(1. So this function is not a bijection since two strings correspond to one number.
The idea is to remove the . From (0, 1), remove the numbers having two binary expansions. Put these numbers in a sequence: a = (1/2, 1/4, 3/4, 1/8, 3/8, 5/8, 7/8, .
From T, remove the strings appearing after the binary point in the binary expansions of 0, 1, and the numbers in sequence a. Put these eventually- constant strings in a sequence: b = (0. A bijection g(t) from T to (0, 1) is defined by: If t is the nth string in sequence b, let g(t) be the nth number in sequence a; otherwise, let g(t) = 0. To build a bijection from T to R: start with the tangent function tan(x), which provides a bijection from (. Next observe that the linear functionh(x) = . The composite function tan(h(x)) = tan(.
Compose this function with g(t) to obtain tan(h(g(t))) = tan(. This means that T and R have the same cardinality. The example mapping f happens to correspond to the example enumeration s in the above picture. 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; that is, the set of all subsets of S (here written as P(S)), has a larger cardinality than S itself. This proof proceeds as follows: Let f be any function from S to P(S). It suffices to prove f cannot be surjective.
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That means that some member T of P(S), i. As a candidate consider the set: T = .
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If s is in T, then by definition of T, s is not in f(s), so T is not equal to f(s). On the other hand, if s is not in T, then by definition of T, s is in f(s), so again T is not equal to f(s); cf. For a more complete account of this proof, see Cantor's theorem. Consequences. If S were the set of all sets then P(S) would at the same time be bigger than S and a subset of S. Russell's Paradox has shown us that naive set theory, based on an unrestricted comprehension scheme, is contradictory. Note that there is a similarity between the construction of T and the set in Russell's paradox. Therefore, depending on how we modify the axiom scheme of comprehension in order to avoid Russell's paradox, arguments such as the non- existence of a set of all sets may or may not remain valid.
The diagonal argument shows that the set of real numbers is . Therefore, we can ask if there is a set whose cardinality is . This question leads to the famous continuum hypothesis. Similarly, the question of whether there exists a set whose cardinality is between .
For example, the conventional proof of the unsolvability of the halting problem is essentially a diagonal argument. Also, diagonalization was originally used to show the existence of arbitrarily hard complexity classes and played a key role in early attempts to prove P does not equal NP. Version for Quine's New Foundations. In NF, the naive axiom scheme of comprehension is modified to avoid the paradoxes by introducing a kind of . In this axiom scheme. On the other hand, we might try to create a modified diagonal argument by noticing that.
In which case, if P1(S) is the set of one- element subsets of S and f is a proposed bijection from P1(S) to P(S), one is able to use proof by contradiction to prove that . We would conclude that if r is not in f(. Jahresbericht der Deutschen Mathematiker- Vereinigung 1.
From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics, Volume 2. Oxford University Press. Universality and the Liar: An Essay on Truth and the Diagonal Argument.
Cambridge University Press. ISBN 9. 78- 0- 5. Principles of Mathematical Analysis (3rd ed.). New York: Mc. Graw- Hill. The Real Numbers and Real Analysis. ISBN 9. 78- 0- 3.
Stanford encyclopedia of philosophy. Principles of mathematics. Universality and the Liar: An Essay on Truth and the Diagonal Argument. Cambridge University Press. ISBN 9. 78- 0- 5.