Countability of cantor set pdf

The set of points that remain after all of these intervals are removed is not an interval, however, it is uncountably infinite. A set a is called uncountable if, and only if, it is not countable. A set x is infinite if and only if there is an injection f from n the set of all natural numbers to x. The set of real transcendental numbers is uncountable. Thus to prove that a set is finite we have to discover a bijection between the set 0,1,2,n1 to the set. 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. Probability via measure theory measurable spaces borel probability spaces constructions of probability spaces coin toss space, markov chains, etc. 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. In fact, its impossible to construct a bijection between n and the interval 0. Cantors diagonal argument to show the reals form a uncountable set. In mathematics objects have been traditionally introduced only with reluctance, but a more arithmetical rather than geometrical approach to the continuum. Moreover, this is essentially the only way in which ambiguity can.

Although the cantor set is defined in a general manner, the. We take it as obvious that n has n elements, and also that the empty set. The set of limit points of p constituted the second derived set p2, etc. One of these theorems is his revolutionary discovery that the set of all real numbers is uncountably, rather than countably, infinite. When in that situation, you should always go back to first principles that is the definitions of finite and infinite. The cantor set acquires a very important role in many branches of mathematics, mainly in set theory and fractal theory 9, 10. This is really a generalization of cantors proof, given above. This paper organizes and extends some of the ideas found in the papers listed in the references. A set e is said to be countable if it can be put in onetoone correspondence with a subset of n 1,2,3.

In mathematics, a countable set is a set with the same cardinality number of elements as some subset of the set of natural numbers. 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. Proof that the set of real numbers is uncountable aka there is no bijective function from n to r. Perhaps the most interesting property is that it is also uncountable. We will show that in fact cantors set has amuch larger cardinality i. If you arent familiar with the idea of countability, then you may be interested in the article all about infinity. A set is finite when its cardinality is a natural number. The cantor set is uncountable trinity college dublin.

A set was said to be of the first species so long as pn was empty for some finite value of n. A set ais said to be countably in nite if jaj jnj, and simply countable if jaj jnj. For each element x 2a input, f must specify one element fx2b output. This is really a generalization of cantor s proof, given above. The power set of a set is always of greater cardinality than the set itself. The general problem is to characterize topological properties of the. Alphabet a,b,c consider the set of all possibly infinite strings over a. 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 set is uncountable we already know that cantors set is in nite.

So a countable set could contain an infinity of elements. Mathematicians are interested in the cantor set because it is a good example of an object whose existence is not at first clear. Countability and cantor sets countable and uncountable sets the concept of countability will be important in this course and we shall revise it here. Introduction two finite sets have the same cardinality if there exists a onetoone correspondence or bijection between them. In fact, q is countable, even though q seems substantially bigger than z. If an element x belongs to a set x then we denote this fact by writing x. Sep 11, 2017 this short video presents an application of cantor s diagonalization argument in showing that the real numbers are unaccountably infinite. If the latter set were countable, r would be countable. Some technical details edit the proofs of the statements in the above section rely upon the existence of functions with certain properties. Cantor is known as the inventor of set theory, and some of his discoveries have important and surprising implications for calculus. The cantor ternary set is created by repeatedly deleting the open middle thirds of a set of line segments. For any set s let 2s denote the set of subsets of s. In words, a set is countable if it has the same cardinality as some subset of the natural numbers.

There is a measurable set, a subset of the cantor set, that is not a borel set. The set of real numbers is uncountable see cantors first uncountability proof, and so is the set of all infinite sequences of natural numbers. We say that f is a bijection if every element a 2a has a. This is a good example of a result that seems fairly obvious and therefore hard to prove properly. That is, it is impossible to construct a bijection between n and r. Interesting properties we have already showed that the cantor set is nowhere dense. There are infinitely many uncountable sets, but the above examples are some of the most commonly encountered sets. Given a point set p, cantor defined the set of all its limit points as p.

We show that no function from an arbitrary set s to its power set. The proof of the following theorem is known as cantors famous diagonalization argument. Set of real numbers is uncountable proof by cantors. Set theory from cantor to cohen akihiro kanamori set theory is an autonomous and sophisticated. Cantor is known as the inventor of set theory, and some of his discoveries have important and. In practise we will often just say \countable when we really mean \countably in nite. Elements of set theory the real number system countability the cantor set the vitali paradox chapter b.

This short video presents an application of cantors diagonalization argument in showing that the real numbers are unaccountably infinite. Cantor took this notion much further, showing that infinite sets come in an infinite number of sizes. The integers are obviously discrete, in that theres a big gap between successive pairs of integers. More mathematically, e is countable if there exists a surjection f. 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. Cantors diagonal argument to show the reals are uncountable. R is the disjoint union of the set of real algebraic numbers, which is countable, and the set of real transcendental numbers.

The cantor set is uncountable february, 2009 every x 20. Sep 14, 2016 proof that the set of real numbers is uncountable aka there is no bijective function from n to r. The cantor set is an interesting subset of 0, 1, which we will encounter several times in this course. Georg cantor proved this astonishing fact in 1895 by showing that the the set of real numbers is not countable. 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. Cantors first set theory article contains georg cantors first theorems of transfinite set theory, which studies infinite sets and their properties. This theorem is proved using cantor s first uncountability proof, which differs from the more familiar proof. Cantor s first set theory article contains georg cantor s first theorems of transfinite set theory, which studies infinite sets and their properties. Let sand tbe sets, suppose tis countable, and let f. Talk about what went wrong with 2 c on the midterm. I there is a listing, l, that contains all subsets of n.