Posted on March 31, 2019


Cantor’s Diagonal Argument. Recall that • A set S is finite iff there is a bijection between S and {1, 2,,n} for some positive integer n, and infinite otherwise. Not too long ago, while surfing the TV channels, you could lean back, press the remote, and suddenly you found a show about Georg Cantor (pronounced. The Cantor diagonal method, also called the Cantor diagonal argument or Cantor’s diagonal slash, is a clever technique used by Georg Cantor to show that the.

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The above proof fails for W. In this axiom scheme. Diagonalization is so common there are special terms for it.

Cantor’s diagonal argument – Wikipedia

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Arash How are you determining what digits each of those numbers are?

Cantor diagonal argument is an argument to prove that set of real numbers is uncountable. Set of real numbers is uncountable. Suppose there exists a diagonalizxtion from to and consider the subset of consisting of the elements of such that does not contain.

Mon Dec 31 Mathematics Stack Exchange works best with JavaScript enabled. And if they have finite expressions, then they are xiagonalization. Cantor’s point is that the real numbers cannot be so listed.


Why are we doing any of this? A generalized form of the diagonal argument was used by Cantor to prove Cantor’s theorem: It suffices to prove f cannot be surjective.

Comparing infinite lists

On the other hand, we might try to create a modified diagonal argument by noticing that. Any real number can be represented as an integer followed by a decimal point and an infinite sequence of digits.

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. FrostyStraw 4 8 What is the whole r1, r2, r3 thing even mean?

Cantor Diagonal Method

He begins with a constructive proof of the following theorem:. This proof proceeds as follows:. For example, given the Church-Turing thesis there are the same number of things that can be done as there are integers. Therefore, R is uncountable. To do that we find a real number which is not counted. For every s in Seither s is in T or not. Let f be any function from S to P S. This yields a contradiction, so there cannot exist a bijection from to. However, this contradicts s being an element of T and therefore belonging to the enumeration.

Arash 9, 2 15 If you do, it might help if you explained your understanding of the terms in your question, that would give us a starting point. Principles of Mathematical Analysis 3rd ed. Applying the previous theorem to this enumeration cantir a sequence s not belonging to the enumeration.


By applying this argument infinitely many times to the same infinite set, it is possible to obtain an infinite hierarchy of infinite cardinal numbers. This result implies that the notion of the set of all sets is an inconsistent notion.

If I can pair them such that everything in list A has a pair in list Bbut not vice-versa, then A is no larger than Bbut it might still be the same size if the lists are infinite.

Cantor Diagonal Method — from Wolfram MathWorld

For instance look at the following: Well, [1, 2, 3] is not the same size as [x, y]: By using this site, you agree to the Terms of Use and Privacy Policy. To constructiviststhe argument shows no more than that there is no bijection between the natural numbers and T. Home Questions Tags Users Unanswered.

If is an infinite set, then is a bigger infinite set. Assume that the set is countable.

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