word looked up :

Cantor's diagonal argument

Cantor's diagonal argument is a proof devised by Georg Cantor to demonstrate that the real numbers are not countably infinite. (It is also called the diagonalization argument or the diagonal slash argument.) It does this by showing that the interval (0,1), that is, the set of real numbers larger than 0 and smaller than 1, isn't countably infinite. The diagonal argument is a proof by contradiction which proceeds as follows:

(1) Assume that the interval (0,1) is countably infinite.
(2) We may then enumerate the numbers in this interval as a sequence, { r₁, r₂, r₃, ... }
(3) We shall now construct a real number x between 0 and 1 by considering the n^th digit after the decimal point of the decimal expansion[?] of r_n. Assume, for example, that the decimal expansions of the beginning of the sequence are as follows.

       r₁ = 0 . 0 1 0 5 1 1 0 ... 
       r₂ = 0 . 4 1 3 2 0 4 3 ...
       r₃ = 0 . 8 2 4 5 0 2 6 ... 
       r₄ = 0 . 2 3 3 0 1 2 6 ...
       r₅ = 0 . 4 1 0 7 2 4 6 ... 
       r₆ = 0 . 9 9 3 7 8 3 8 ...
       r₇ = 0 . 0 1 0 5 1 3 0 ... 
       ...

The digits we will consider are indicated in bold. From these digits we define the digits of x as follows.

- if the n^th digit of r_n is 0 then the n^th digit of x is 1
- if the n^th digit of r_n isn't 0 then the n^th digit of x is 0

For the example above this will result in the following decimal expansion.

        x = 0 . 1 0 0 1 0 0 1 ...

The number x is clearly a real number between 0 and 1.

(4) However, it differs in the n^th decimal place from r_n, so x isn't in the set { r₁, r₂, r₃, ... }.
(5) This set is therefore not an enumeration of all the reals in the interval (0,1).
(6) This contradicts with (2), so the assumption (1) that the interval (0,1) is countably infinite must be false.

Note: Strictly speaking, this argument only shows that the number of decimal expansions of real numbers between 0 and 1 isn't countably infinite. But since there are expansions such as 0.01999... and 0.02000... that represent the same real number, this doesn't immediately imply that the corresponding set of real numbers is also not countably infinite. This can be remedied by disallowing the decimal expansions that end with an infinite series of 9's. In that case it can be shown that for every real number there is a unique corresponding decimal expansion. It is easy to see that the proof then still works because the number x contains only 1's and 0's in its decimal expansion.

The diagonal argument is an example of reductio ad absurdum because it proves a certain proposition (the interval (0,1) isn't 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. This proof proceeds as follows:

(1) Assume that P(S) isn't larger than S.
(2) Then there is a function f from S to P(S) that is surjective, i.e., for every V in P(S) there is an s in S such that f(s) = V.
(3) We shall now construct a set T that is in P(S) as follows.

T = { s in S | s isn't in f(s) }

(4) The set T isn't in the range of f, i.e., there is no s in S such that f(s) = T. This can be shown as follows. It will either hold that s is in f(s) or not.
- If s is in f(s) then it follows by definition of T that s isn't in T, so T isn't equal to f(s).
- If s isn't in f(s) then it follows by definition of T that s is in T, so also then T isn't equal to f(s).

Since in both cases f(s) isn't equal to T it follows that f(s) is never equal to T.

(5) The function f is therefore not surjective.
(6) This contradicts with (2), so the assumption (1) that P(S) isn't larger than S must be false.

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.

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 halting problem is essentially a diagonal argument.

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 s, leads to the Generalized continuum hypothesis.

Princess of Orange, he denounced the man in unmeasured terms. "He is the for a year before, Maurice was the enemy of the preacher.html">preacher. On the following Sunday, July 23, Maurice went in solemn state.html">state to the accompanied by his cousin, the famous Count William Lewis of Nassau, the Contra-Remonstrants, and by all the chief officers of his household one. As the martial stadholder at the head of his brilliant cavalcade palace--where the ancient sovereign.html">sovereign Dirks and Florences of Holland had so Kneuterdyk and so through the Voorhout, an immense crowd thronged around soldier were marching to siege or battle-field where fresher glories than four.html">four thousand persons were present at the service or crowded around aisles; while the Great Church was left comparatively empty, a few called the Prince's Church, and a great revolution was beginning even stadholders and their military attendants. He knew that he was now to captain-general of the state, to renounce his dreams of religious unequal struggle in which he might utterly succumb. But his iron nature he is said to have vowed vengeance against the immediate instruments by feloniously seized. He meant to strike a blow which should startle the country, and teach men to beware how they trifled with the sovereign him their chief functionary. He resolved--so ran the tale of the preacher Trigland, who told it to seized at midnight from their beds four men whom he considered the on the square in the midst of the city, to have their heads cut off at then to summon all the citizens at dawn of day, by ringing of bells and .

On wordlookup.net

All is still licensed under the GNU FDL.
It uses material from the wikipedia.

navig stuff