Countability of cantor set pdf

The set of limit points of p constituted the second derived set p2, etc. Talk about what went wrong with 2 c on the midterm. Set of real numbers is uncountable proof by cantors. Probability via measure theory measurable spaces borel probability spaces constructions of probability spaces coin toss space, markov chains, etc.

In words, a set is countable if it has the same cardinality as some subset of the natural numbers. 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. Sep 14, 2016 proof that the set of real numbers is uncountable aka there is no bijective function from n to r. We show that no function from an arbitrary set s to its power set. The power set of a set is always of greater cardinality than the set itself. The set of real numbers is uncountable see cantor s first uncountability proof, and so is the set of all infinite sequences of natural numbers. Although the cantor set is defined in a general manner, the. The set of real numbers is uncountable see cantors first uncountability proof, and so is the set of all infinite sequences of natural numbers. Sep 11, 2017 this short video presents an application of cantor s diagonalization argument in showing that the real numbers are unaccountably infinite. Given a point set p, cantor defined the set of all its limit points as p.

This is a good example of a result that seems fairly obvious and therefore hard to prove properly. The cantor set is a meager set or a set of first category as a subset of 0,1 although not as a subset of itself, since it is a baire space. We say that f is a bijection if every element a 2a has a. This is really a generalization of cantors proof, given above.

Mathematicians are interested in the cantor set because it is a good example of an object whose existence is not at first clear. Cantor is known as the inventor of set theory, and some of his discoveries have important and. Cantor took this notion much further, showing that infinite sets come in an infinite number of sizes. There is a measurable set, a subset of the cantor set, that is not a borel set. We will show that in fact cantors set has amuch larger cardinality i.

In fact, its impossible to construct a bijection between n and the interval 0. So a countable set could contain an infinity of elements. The cantor set thus demonstrates that notions of size in terms of cardinality, measure, and baire category need not coincide. The german mathematician, georg cantor 18451918, challenged the foundations of mathematics, and in the process made enemies of several prominent mathematicians of his time. Cantors diagonal argument to show the reals are uncountable. A set a is called uncountable if, and only if, it is not countable. A set e is said to be countable if it can be put in onetoone correspondence with a subset of n 1,2,3. Proof that the set of real numbers is uncountable aka there is no bijective function from n to r. Cantors diagonal argument to show the reals form a uncountable set. R is the disjoint union of the set of real algebraic numbers, which is countable, and the set of real transcendental numbers.

This short video presents an application of cantors diagonalization argument in showing that the real numbers are unaccountably infinite. The set of points that remain after all of these intervals are removed is not an interval, however, it is uncountably infinite. In mathematics, a countable set is a set with the same cardinality number of elements as some subset of the set of natural numbers. This paper organizes and extends some of the ideas found in the papers listed in the references.

Alphabet a,b,c consider the set of all possibly infinite strings over a. I there is a listing, l, that contains all subsets of n. The cantor set is an interesting subset of 0, 1, which we will encounter several times in this course. Moreover, this is essentially the only way in which ambiguity can. More mathematically, e is countable if there exists a surjection f. This is really a generalization of cantor s proof, given above. Some technical details edit the proofs of the statements in the above section rely upon the existence of functions with certain properties. If an element x belongs to a set x then we denote this fact by writing x. Cantors set is uncountable we already know that cantors set is in nite. In fact, q is countable, even though q seems substantially bigger than z.

A set was said to be of the first species so long as pn was empty for some finite value of n. The integers are obviously discrete, in that theres a big gap between successive pairs of integers. The cantor ternary set is created by repeatedly deleting the open middle thirds of a set of line segments. Cantor on the set of rational numbers from interv al 0,2 it is p ossible to make the sequences that determine progression of elements.

A set is finite when its cardinality is a natural number. In mathematics objects have been traditionally introduced only with reluctance, but a more arithmetical rather than geometrical approach to the continuum. Cantors first set theory article contains georg cantors first theorems of transfinite set theory, which studies infinite sets and their properties. The cantor set is uncountable trinity college dublin. Set theory from cantor to cohen akihiro kanamori set theory is an autonomous and sophisticated. Set theory from cantor to cohen 3 the expansion of the function concept which shifted the emphasis from the con tinuum taken as a whole to its extensional construal as a collection of objects. The cantor set is uncountable february, 2009 every x 20. The general problem is to characterize topological properties of the. Cantor is known as the inventor of set theory, and some of his discoveries have important and surprising implications for calculus. This theorem is proved using cantor s first uncountability proof, which differs from the more familiar proof. The proof of the following theorem is known as cantors famous diagonalization argument. For any set s let 2s denote the set of subsets of s.

There are infinitely many uncountable sets, but the above examples are some of the most commonly encountered sets. Countability and cantor sets countable and uncountable sets the concept of countability will be important in this course and we shall revise it here. For each element x 2a input, f must specify one element fx2b output. If the latter set were countable, r would be countable. Introduction two finite sets have the same cardinality if there exists a onetoone correspondence or bijection between them. The cantor set acquires a very important role in many branches of mathematics, mainly in set theory and fractal theory 9, 10. Elements of set theory the real number system countability the cantor set the vitali paradox chapter b. If you arent familiar with the idea of countability, then you may be interested in the article all about infinity.

Thus to prove that a set is finite we have to discover a bijection between the set 0,1,2,n1 to the set. Interesting properties we have already showed that the cantor set is nowhere dense. In practise we will often just say \countable when we really mean \countably in nite. Georg cantor proved this astonishing fact in 1895 by showing that the the set of real numbers is not countable.