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The Magical World of Infinities

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The Magical World of Infinities The Magical World of Infinities Figure 12 Figure 13 Part II Old monuments, newer buildings, grills on However due to copyright restrictions we cannot Features windows, balcony railings, fences, and so on reproduce photographs of Escher’s work here. It too provide for interesting examples to view via is recommended that the reader peruse his art symmetry. The collage in Figure 12 provides www.mcescher.com. We examples of this and can be analysed in a manner on the official website similar to the earlier collage. puzzle inspired by Escher to enjoy at leisure and to discoverhowever itsleave secrets the reader via symmetry; with an imagesee Figure of a floor13. It would be a grave lacuna not to mention the artwork of M C Escher in an article on symmetry. Shashidhar Jagadeeshan Bibliography Introduction In the previous article we encountered the strange world of 1. M. A. Armstrong, Groups and Symmetry, Springer Verlag, 1988. infinities, where a lot of our intuitive sense of how infinite sets 2. David W. Farmer, Groups and Symmetry: A Guide to Discovering Mathematics, American Mathematical Society, 1995. should behave started breaking down. We saw for example that 3. Joseph A. Gallian, Contemporary Abstract Algebra, 7th edition, Brooks/Cole Cengage Learning, 2010. infinite sets can have subsets which have as many elements asthe original set; we also saw our intuition about length breaking 4. Kristopher Tapp, Symmetry: A Mathematical Exploration, Springer, 2012. down. No matter what the lengths of the two lines are, they 5. Herman Weyl, Symmetry, Princeton University Press, 1952. ended up having the same number of points. Moreover, all the examples of infinite sets we encountered ended up having the same number of elements. You might naturally assume that there is only one kind of infinity – which is what perhaps you had GEETHA VENKATARAMAN is a Professor of Mathematics at Ambedkar University Delhi. Her area of research assumed right from the beginning? is in finite group theory. She has coauthored a research monograph, Enumeration of finite groups, published by Cambridge University Press, UK. She is also interested in issues related to math education and women in In the last section of Part I of this article (AtRiA, March 2016), mathematics. She completed her MA and DPhil from the University of Oxford. She taught at St. Stephen’s College, we had hinted at the possibility of there being different kinds of University of Delhi from 1993 to 2010. Geetha has served on several curriculum development boards at the school level, undergraduate level and postgraduate level. She was Dean, School of Undergraduate Studies at Ambedkar infinities. If there are, can we mathematically prove they exist? University Delhi during 2011-2013. She is currently Dean, Assessment, Evaluation and Student Progression at How many different infinities are there really? In this article we Ambedkar University Delhi. will answer these questions. Recall that the cardinality of a set counts the number of elements it contains. We denote the cardinality of a set X by X . In some | | cases we have special symbols denoting the cardinality of sets. Keywords: Infinity, set, cardinality, countable, natural number, real number, unit interval, continuum, one-one correspondence, injective function 1617 AtAt Right Right AnglesAngles | Vol. 5,5, No.No. 2, 2, July July 2016 2016 Vol. 5, No. 2, July 2016 | At Right Angles 17 1 For example, 0 represents the cardinality of the Schroeder-Bernstein Theorem (sometimes correspondence between the set (0, 1) and . We we have found an element b such that ℵ N set N of natural numbers, and c (called the Cantor’s name is also added) states that if we have do so by assuming the contrary; that is, we assume b (0, 1) and b / a , a , a ,...,a ,... continuum) represents the cardinality of the set R an injective function f : X Y and another that there does exist a 1-1 correspondence between 1 2 3 n → ∈ ∈{ } of real numbers. If an infinite set has cardinality injective function g : Y X (note that f and g these two sets and keep arguing logically, step by → But this is a contradiction—because we had 0, then we say that this set is countable. 0 and c need not be inverses of each other), then X = Y . step, until something goes wrong! The only reason ℵ ℵ | | | | assumed that every element in the set (0, 1) is are examples of cardinal numbers. for something to go wrong could then be that we We are now ready to compare the set R of real accounted for the list a1, a2, a3,...,an,... ! made an erroneous assumption in the beginning. { } At this stage it would be good to introduce some numbers with the set N of natural numbers and to Where did we go wrong? If you go back and check ideas and techniques that we use to compare two show that 0 < c. If there is a 1-1 correspondence between the set all the steps in our argument, you will find that ℵ N infinite sets, since we will be using them quite of natural numbers and the set (0, 1), we can the mistake was in assuming that there is a often. In the previous article we introduced the ( , ) one-to-one correspondence between (0, 1) and N. Are real numbers countable? assign a natural number to each element in 0 1 . idea of 1-1 correspondence between two sets X and ( , ) In fact, what Cantor managed to show was that no Let us start by comparing the set of all real Let us denote the number in 0 1 associated with Y, and said that X = Y if and only if we can find 1 as a , the number associated with 2 as a and so matter how clever you are, you cannot come up | | | | numbers between 0 and 1 (we denote this set by 1 2 a 1-1 correspondence between X and Y. In a 1-1 on, allowing us to enumerate the elements of the with a 1-1 correspondence between the above two (0, 1)) and the set N of natural numbers. correspondence we have a function which set (0, 1), using natural numbers thus: sets, because the moment you do, and you associates every element of X with a unique element We remind readers of the fact that the real numbers enumerate the elements of (0, 1) using the natural (0, 1)= a , a , a , ..., a , ... numbers, the diagonalization process guarantees of Y, and by inverting this association, every have decimal representations. Furthermore, by { 1 2 3 n } element of Y is associated with a unique element of inserting a string of zeros, we can make it an that you will always come up with an element in 1 Let us further denote each element in the above X. A slightly weaker notion than 1-1 infinite decimal representation. For example, = (0, 1) which is not in the list that you had made! 4 list as a decimal expansion, and let us do it in a correspondence is the idea of an injective function 0.25000 ... (with infinitely many trailing zeros) This establishes the fact that the set of real 1 manner in which a clear pattern emerges. numbers contains more elements than the set of (often referred to as a ‘1-1 function’ as opposed to and 7 = ‘1-1 correspondence’, but we will use the term 0.14285714285714285714285714285714 .... natural numbers, and therefore that 0 < (0, 1) a1 = 0.a1,1 a1,2 a1,3 ... a1,n ... ℵ | | injective function to avoid confusion). An injective The question is, are these representations unique? or in other words (0, 1) is not countable. = . ... ... function f : X Y is a function that satisfies the You might have come across the curious fact that a2 0 a2,1 a2,2 a2,3 a2,n → What about the real numbers, are they countable? property that if f(a)=f(b), then a = b. Notice 0.99999 = 1 (this is really fascinating, if you = . ... ... ··· a3 0 a3,1 a3,2 a3,3 a3,n In Part I of this article we showed that there is a that in an injective function we cannot be sure that have not already done so, see if you can prove it for 1-1 correspondence between the set ( 1, 1) and . − every element in the set Y has a partner in X; yourself). So it appears that we have two possible . R. We can use a similar argument to show that however, if an element in the set Y does have a decimal representations for some real numbers. It there is a 1-1 correspondence between (0, 1) and an = 0.an,1 an,2 an,3 ... an,n ... partner in X, then that partner is unique. Figure 1 turns out that if we can take care of the case of R and, in fact, between any open interval in the illustrates an injective function. repeating nines, we can then have unique decimal . set of real numbers and R. I hope this amazing representations for all real numbers. So, if we fact has not slipped by the reader, that the set of decide that we will choose to represent numbers Now here is where Cantor’s brilliance can be seen real numbers R and any open interval contained . ... ... like 0 2999 by 0 3000 , then every member again. He defines a new element in R have the same cardinality; namely, the of our set has a unique decimal representation. continuum c. Clearly, since (0, 1) is not = . ... ... Here is Cantor’s proof that there are more real b 0 b1 b2 b3 bn countable, and R has the same cardinality as numbers between 0 and 1 than there are natural (0, 1), R is not countable and 0 < c.
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