Proof of uncountability of cantor set pdf

Repeating this argument for each level of removal, it can. Although the cantor set is defined in a general manner, the. The cantor set is the intersection of closed sets, so f is closed. Axiomatic set theory introduced to control these paradoxes. At first, i was excited to see a variant proof as it did not use the diagonal argument explicitly. Interesting properties we have already showed that the cantor set is nowhere dense. Titleabstractpreliminariesconstruction and formulaproperties and proofs abstract the cantor set is a famous set. Construction of the cantor set an uncountable set having measure zero. Here cantor s original theorem and proof 1, 2 are sketched briefly, using his own symbols. Pdf it is shown that the pillars of transfinite set theory, namely the uncountability proofs, do not hold. Cantor s theorem and its proof are closely related to two paradoxes of set theory. The set of limit points of p constituted the second derived set p2, etc.

In this section, ill concentrate on examples of countably in. Georg cantor s first uncountability proof demonstrates that the set of all real number s is uncountable. In its construction we remove the same number of points as the number left behind to form the cantor set, which leads us to this result. We refer to readers the book 11 for detailed proofs. That is, is there an enormously large set that still somehow contrives to take up no space. For any set s let 2s denote the set of subsets of s. This is an immediate consequence of the previous result. Cantors set is the set c left after this procedure of deleting the open middle third.

This is really a generalization of cantors proof, given above. Minkowski and hausdorff dimensions definitions and examples. It is shown that the pillars of transfinite set theory, namely the uncountability proofs, do not hold. Aug 06, 2015 the remaining cases are shown in file. Im personally not aware of a proof that doesnt use it. However, as time passed, i began to see that the proof was just the old one veiled under new terminology. Cantor s first set theory article contains georg cantor s first theorems of transfinite set theory, which studies infinite sets and their properties. Well bypass a formal proof of the uncountability of the cantor set. In 1879, cantor published a new uncountability proof that modifies his 1874 proof. For instance, part b is the famous \pigeonhole or \dirichlets box. Cantor s diagonal proof of the uncountability of the continuum.

Let s all sets of type 2 set of all sets not containing themselves as elements s. Discussion about cantor s first proof using the nextintervalfunction, potential and actual infinity. The cantor set acquires a very important role in many branches of mathematics, mainly in set theory and fractal theory 9, 10. Pdf the cantor set is an interesting example of an uncountable set of measure zero and has many.

Then for each a 2a we have an associated subset of a, s a. The nature of infinity and beyond cantors paradise medium. We will show that the cantor ternary set has uncountable many elements by contradiction. The cantor set is uncountable trinity college dublin.

The diagonal argument was not cantors first proof of the uncountability of the real numbers, which appeared in 1874. We show that no function from an arbitrary set s to its power set. Discussion about cantors first proof using the nextintervalfunction, potential and actual infinity. Details on a proof of the uncountability of the cantor set. Surprisingly, a small twist on cantors line of reasoning shows that the golden ratio is irrational, as we shall demonstrate. Uncountability of the cantor set so one of my profs gave us a few voluntary exercises over the semester break and one of exercises are various proofs about the cantor set. This is an elegant diagonal argument based on decimal representation of real numbers. The canonical proof that the cantor set is uncountable does not use cantor s diagonal argument directly.

Proof that the set of real numbers is uncountable aka there is no bijective function from n to r. One of these theorems is his revolutionary discovery that the set of all real numbers is uncountably, rather than countably, infinite. What are the ways of proving that the cantor set is. This theorem is proved using cantors first uncountability proof, which differs from the more familiar proof. Pdf on mar 17, 2016, peng kuan and others published on cantors first proof of uncountability find, read and cite all the research you need on researchgate. To get a contradiction, suppose that the set of real numbers can. Diagonalization, countability and uncountability duration. Cantors first proof of the uncountability of the real numbers after long, hard work including several failures 5, p.

We will show that in fact cantors set has amuch larger cardinality i. Here cantors original theorem and proof 1, 2 are sketched briefly, using his own symbols. Cantors proof of this result 2, 2 makes use of nested intervals, but today a proof based on another ingenious idea of cantor is more popular, namely the diagonal method,which he introduced in 1891 to prove the uncountability of 2 n 3. We use a method of proof known as cantors diagonal argument. Cantors diagonal proof a reformatted version of this article can be found here. Continued discussion of the cantor set construction. The result was given, almost as an aside, in a paper 1 whose most prominent result was the countability of the algebraic numbers. Dec 17, 2018 cantors most fruitful use of countability as a concept occurred in the third result of his 1874 paper when he demonstrated the uncountability of the real numbers the first set shown to lack. Give a proof of fact 4 which is rigorous enough for your taste. Generalizations and properties of the ternary cantor set and. The earliest appearance in print that i know is on page 43 of the theory of sets of points by w. As far as my intention in this proof, i used the specific example only to have a quick, easy way of seeing that the cantor set was infinite in case we didnt happen to know that. Pdf on cantors first proof of uncountability kuan peng. The exact date of birth can be ascertained as december 7.

An informal proof suggests that the formal proof can be carried out if one would want to do so. Ive managed to proof all of them but im a bit unsure about my proof of uncountability. All text books contain informal proofs and its good that they do. We will show, as in the last theorem that this is not surjective by constructing a subset of a which is not hit by f. Cantor s original proof of uncountability of the reals did not explicitly mention diagonalization. E if p lies partially or completely in the interval. We use a method of proof known as cantor s diagonal argument. That is, cantors set has the same cardinality as the interval 0,1. He first defines the topological notion of a point set p being everywhere dense in an interval. An informal proof presents people the basic ideas and reasoning of a proof. Cantors original proof of uncountability of the reals did not explicitly mention diagonalization.

S contradiction but every element must either be in s or not in s. Cantors first set theory article contains georg cantors first theorems of transfinite set theory, which studies infinite sets and their properties. I have no problem believing that cantor himself realized that a diagonal proof of the uncountability of r was possible but i have not even found an allusion to this in his collected works. A set was said to be of the first species so long as pn was empty for some finite value of n.

Perhaps the most interesting property is that it is also uncountable. Given a point set p, cantor defined the set of all its limit points as p. So, till now i believe that any proof of the uncountability of the reals must use cantor s diagonal argument. Lecture 22 cantor and cantorlike sets contd waterloo fractal. Set of real numbers is uncountable proof by cantors. For any set x, let px denote the power set of x, i. Cantors set is uncountable we already know that cantors set is in nite. This is the object mentioned earlier whose existence is of interest to mathematicians.

Cantor s paradox is the name given to a contradiction following from cantor s theorem together with the assumption that there is a set containing all sets, the universal set. In order to distinguish this paradox from the next one discussed below, it is important. Now we give cantors another proof of the uncountability of. The power set of a set is always of greater cardinality than the set itself. I find it especially confusing that the rational numbers are considered to be countable, but the real numbers are not.

Im trying to understand the significance of cantors diagonal proof. Sep 14, 2016 proof that the set of real numbers is uncountable aka there is no bijective function from n to r. Interestingly, cantors paper starts with a proof that the set of algebraic numbers is count. More than a decade and a half before the diagonalization argument appeared cantor published a different proof of the uncountability of r. This theorem is proved using cantor s first uncountability proof, which differs from the more familiar proof.

An exploration of the cantor set rosehulman scholar. Cantors diagonalization proof of the uncountability of the. Let s be a set with m elements and t a set with n elements. For any set, there is no function mapping its members onto all its subsets. Any infinite subset of a countable set is countable we wont prove this. Cantor s first proof of the uncountability of the real numbers after long, hard work including several failures 5, p. This will be a proof by contradiction using similar.