The cantor set is the intersection of closed sets, so f is closed. The cantor set is uncountable february, 2009 every x 20. Cantors uncountability theorem there are uncountably many in. The set is made by starting with a line segment and repeatedly removing the middle third. So, till now i believe that any proof of the uncountability of the reals must use cantors diagonal argument. Moreover, this is essentially the only way in which ambiguity can. This article carefully analyzes this proof to clarify its logical reasoning please read the article at analyze of the proof of cantors theorem. Discussion about cantors first proof using the nextintervalfunction, potential and actual infinity. Well bypass a formal proof of the uncountability of the cantor set. Cantor as we have seen, the naive use of classes, in particular the connection betweenconceptandextension,ledtocontradiction. This is the object mentioned earlier whose existence is of interest to mathematicians. Counting the empty set as a subset, a set with members has a total of subsets, so. I of all elements in the sequence defines a new set called the cantor set.
The uncountability of the real numbers was already established by cantor s first uncountability proof, but it also follows from the above result. Cantor point out the uncountability of the reals altogether. 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. In elementary set theory, cantors theorem is a fundamental result which states that, for any set, the set of all subsets of the power set of, denoted by has a strictly greater cardinality than itself. The cantor set is a subset of real numbers with certain properties that are interesting to mathematicians. The nature of infinity and beyond cantors paradise. Set of real numbers is uncountable proof by cantors. In set theory, cantors diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by georg cantor as a mathematical proof that there are infinite sets which cannot be put into onetoone correspondence with the infinite set of natural numbers 20 such sets are now known as uncountable sets, and the size of. Ive managed to proof all of them but im a bit unsure about my proof of uncountability. Sep 27, 2016 georg cantor tried hard to find such set. However, it is shown that uncountability is not proven, then the cardinality of real numbers is questionable. However, as time passed, i began to see that the proof was just the old one veiled under new terminology. The power set of a set is always of greater cardinality than the set itself.
The set of limit points of p constituted the second derived set p2, etc. Another wa y to arrive at the uncountability of the unit interv al is th rough power sets. Continuity and uncountability mathematics science forums. Georg graduated from high school in 1860 with an outstanding report, which mentioned in par. The cantor set is the infinite set of points left over. Additional topics may be discussed according to student interest. However, it is shown that uncountability is not proven, then. One of these theorems is his revolutionary discovery that the set of all real numbers is uncountably, rather than countably, infinite. These properties relate to topology, measurement, geometry, as well as set theory. Uncountability article about uncountability by the free. Perhaps the most interesting property is that it is also uncountable. There exist transcendental numbers numbers that are not the solutions of polynomial equations because the real numbers are not countable. Why does the cantor set have zero lebesgue measure. 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.
We call sets that can be put into a bijection with the natural numbers countably in. Cantors first set theory article contains georg cantors first theorems of transfinite set theory, which studies infinite sets and their properties. The power set, denoted p x, of a set x is the set of all subsets of x. The cantor set is more infinite than the set of natural numbers 1, 2, 3. This is really a generalization of cantor s proof, given above. The stated purpose of the paper where cantor published the diagonal argument is to prove the existence of uncountable infinities, avoiding the theory of. 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. Now we give cantors another proof of the uncountability of. In cantors original proof of the uncountability of the reals not the diagonalization argument, he constructs, given any countable sequence of real numbers, a real number not in the sequence. Since the set of real numbers is not columnable, and the set of algebraic numbers is permissible. Cantor took this notion much further, showing that infinite sets come in an infinite number of sizes. To show that the cantor set is uncountable, we need to construct a.
We will show that the cantor ternary set has uncountable many elements by contradiction. The cantor ternary set is created by repeatedly deleting the open middle thirds of a set of line segments. Cantor and continuity akihiro kanamori may 1, 2018 georg cantor 18451919, with his seminal work on sets and number, brought forth a new eld of inquiry, set theory, and ushered in a way of proceeding in mathematics, one at base in nitary, topological, and combinatorial. Cantor set uncountability mathematics stack exchange. This is really a generalization of cantors proof, given above. Suppose that the cardinality of x is equal to the cardinality of px. When informed of this, students might naturally ask so, can. A set is considered uncountable when the number of elements cannot be written as some subset of the. Cantor s first set theory article contains georg cantor s first theorems of transfinite set theory, which studies infinite sets and their properties. Analysis of the proof of cantors theorem cantors theorem states that the power set of. Henry john stephen smith discovered it in 1875, and cantor first described it in 1883. Given a point set p, cantor defined the set of all its limit points as p. This theorem is proved using cantors first uncountability proof, which differs from the more familiar proof.
This presentation is suggestive of cantors natural caution in overstepping mathematical sense at the time. We call sets that can be put into a bijection with the natural numbers countably. Sep 27, 2016 analysis of the proof of cantors theorem cantors theorem states that the power set of. I there is a listing, l, that contains all subsets of n. As to whether the set of real numbers can be listed, cantor gave a letter to dedekind in 1873, but soon he got the answer. Life father, georg waldemar cantor, born in denmark, successful merchant, and stock broker in st petersburg. This presentation is suggestive of cantor s natural caution in overstepping mathematical sense at the time. Notes prepared by stanley burris march, 2001 set theory. Discussion about cantor s first proof using the nextintervalfunction, potential and actual infinity. It has the same cardinality as the set of real 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. This is an elegant diagonal argument based on decimal representation of real numbers. We understand what cantor called equivalent to mean the sets are the same size, or have the same cardinality. Defining the sets, ck, of intervals in this way creates a sequence of sets.
Sep 23, 2012 the power set, denoted p x, of a set x is the set of all subsets of x. Pdf on cantors first proof of uncountability kuan peng. Fat cantor sets last month, i wrote about the cantor set, a mathematical space that is an interesting mix of small and large. For any set x, let px denote the power set of x, i. Cantor set is created by starting with the closed interval 0,1. To put it simply, aristotle had admitted infinity as a passive uncountability in terms of the quality of matter to be submissible to infinite division.
For any set s let 2s denote the set of subsets of s. One of these theorems is cantor s revolutionary discovery that the set of all real numbers is uncountably, rather than countably, infinite. A set was said to be of the first species so long as pn was empty for some finite value of n. Lecture 22 cantor and cantorlike sets contd waterloo fractal. In papers of 1873 and 1874, georg cantor outlined the basics of in. The set of algebraic numbers solutions of polynomial equations is countable because the polynomials are countable and every polynomial has finitely many solutions. Although the cantor set is defined in a general manner, the. At first, i was excited to see a variant proof as it did not use the diagonal argument explicitly. We will show that in fact cantors set has amuch larger cardinality i. There is a measurable set, a subset of the cantor set, that is not a borel set. However, one may show the cantor set is uncountable the same way one shows any continuum is uncountable. Proof that the set of real numbers is uncountable aka there is no bijective function from n to r. Generalizations and properties of the ternary cantor set and. An exploration of the cantor set rosehulman scholar.
When represented geometrically, the set is a fractal, it has a hausdorff dimension which is not an integer. The reader is referred to the discussion in pages 112115 of the book encounters by d. For finite sets, cantors theorem can be seen to be true by simple enumeration of the number of subsets. So, till now i believe that any proof of the uncountability of the reals must use cantor s diagonal argument. We show that no function from an arbitrary set s to its power set. That being said, we shall show that c is uncountable by a somewhat indirect method. That is, is there an enormously large set that still somehow contrives to take up no space. We recall cantors diagonalization proof of his eponymous theorem. The nature of infinity and beyond cantors paradise medium. Its in the lowest third, and in the highest third of that, and in the lowest third of that, and in the highest third of that, and so on, alternating. For any set, there is no function mapping its members onto all its subsets. 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. Georg cantor published his first set theory article in 1874, and it contains the first theorems of transfinite set theory, which studies infinite sets and their properties.
A set that is not countable is said to be uncountable. Interesting properties we have already showed that the cantor set is nowhere dense. Pdf the cantor set is an interesting example of an uncountable set of measure zero and has many interesting properties and consequences in the fields. The cantor set acquires a very important role in many branches of mathematics, mainly in set theory and fractal theory 9, 10. The cantor set is uncountable trinity college dublin. Prior to cantors time, 1 was mainly a metaphor used by theologians not a precisely understood mathematical concept a source of paradoxes, disagreement, and confusion 2. Aug 06, 2015 the remaining cases are shown in file. By this means cantor provided a new proof of joseph liouvilles result 1844,1851 that there are transcendental numbers real nonalgebraic numbers and only afterward did cantor point out the uncountability of the reals altogether. Now we give cantor s another proof of the uncountability of. To prove this, an injection will be constructed from the set t of infinite binary strings to the set r of real numbers. Cantors set is uncountable we already know that cantors set is in nite. Uncountability of the cantor set mathematics stack exchange. The earliest appearance in print that i know is on page 43 of the theory of sets of points by w. In 1856, because of fathers poor health, family moved to germany.
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