The Different Sizes of Infinity

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1 The Different Sizes of Infinity New York City College of Technology Cesar J. Rodriguez November 11, 2010

2 A Thought to Ponder At... Does Infinity Come in Varying Sizes? 2

3 Points of Marked Interest(s) General Analysis of Properties/Characteristics of Various Sizes of Infinities. Cardinality (Size) of Sets. Injections/Surjections/Bijections Between Sets Contributions and Achievements of Famous Mathematician Georg Cantor on the Topic of Infinity. 3

Cardinal Rules of Cardinality Simple Example 1: The set Sample = {1, 3, 5, 7} contains four elements. Definition: The Cardinality of a set is a measure of a size of the given set.

4 Cardinal Rules of Cardinality Simple Example 1: The set Sample = {1, 3, 5, 7} contains four elements. Definition: The Cardinality of a set is a measure of a size of the given set. Pose question to audience: How do you convince me both Example sets have the same cardinality without using the notion of the number 4? Methods for Determining Cardinality: Comparing sets using Injections/Surjections/Bijections. Simple Example 2: The set Circles contains four elements. 4

5 Relational Ties Between Injection/Surjection/Bijection Injections/Surjections/Bijections are all Functions. Injection (1-1) Every element in the first set is connected with exactly one element in the second set but not every element of the second set needs to have something connected to it. Surjection (onto) Every element in the first set is connected with one element of the second set and every element of the second set has to have something different connected to it. Bijection (1-1 and onto) Every element in one set is in connection with one and only one other element in the second set. Bijection is an Injection and Surjection. 5

6 Drawing the Connection Question: Is it possible to have a Surjection in the first graph? Answer: No because set EX1 is smaller than set EX2. Conclusion: If there is an Injection from set EX1 to set EX2, then the size of set EX1 is less than or equal to the size of set EX2. If there is an Injection from set EX1 to set EX2, but it is not possible to have a Surjection, then the size of set EX1 is strictly less than the size of set EX2. Question: Is it possible to have an Injection in graph 2? Answer: No because set EX1 is larger than set EX2. Conclusion: If there is a Surjection from set EX1 to set EX2, then the size of set EX1 is greater than or equal to the size of set EX2. If there is a Surjection from set EX1 to set EX2, but no Injection, then EX1 is strictly greater than EX2. Question: What can we conclude in graph 3? Answer: The two sets are the same size. 6

7 Set of Circumstances Situation 1: EX1 = EX2 The two sets EX1 and EX2 share the same Cardinality if there exists a Bijection between them. Situation 2: EX1 EX2 The set EX1 has a smaller or equal Cardinality than set EX2 if there exists an Injection from EX1 into EX2. Situation 3: EX1 < EX2 The set EX1 has a smaller Cardinality than set EX2 if there is an Injection from EX1 to EX2 but no Bijection. 7

8 In Cantor We Trust Georg Ferdinand Ludwig Philipp Cantor was a Russian Mathematician known for his contributions to the field of Set Theory and Real Analysis. Interesting Fact: During periods of depression, Cantor turned away from Mathematics and shifted his attention to philosophy and the literary works of Shakespeare. Contributions/Discoveries: Defining the notion of Cardinality for infinite sets. Proving the set of Rational Numbers is Countable (Bijection with Natural Numbers). Proving the set of Real Numbers is larger than the set of Natural Numbers as well as proving for every infinite set there exists a larger infinite set. Proving a Bijection of points on the interval [0, 1] and points in p-dimensional space. Discovering Cantor s Paradox (there is no largest infinite set) 8

9 Setting the Bar on Sets We define that two sets have the same Cardinality (size) if there is a Bijection between them.. 0 א The Cardinality of Natural Numbers is denoted as A set X with a Cardinality less than א 0 is a Finite Set א) 0 is the smallest Infinite set). A set X with the same Cardinality as א 0 is classified as Countable Infinite set (you can count through it). A set X with a Cardinality greater than א 0 is said to be Uncountable 9

10 See The Connection Between Sets 10

Hooking Up With Cantor s Pairing FunctionQuestion: Does the set of Rational Numbers Problem: How do you order the Rational Numbers? Solution: Graph.

11 Hooking Up With Cantor s Pairing FunctionQuestion: Does the set of Rational Numbers Problem: How do you order the Rational Numbers? Solution: Graph. have the same Cardinality as the set of Natural Numbers? Answer: Yes. N= {0, 1, 2, } is the set of Natural Numbers. Q = {1/3, 5/17, } is the set of Rational Numbers. Proving there is a Bijection between these sets through graphical means. The number 0 in the set of Natural Numbers would be connected with 1/1 in the set of Real Numbers; 1 to 2/1 ; 2 to 1/2 and so on (following the direction of the arrows in the graph). Conclusion: The Set of Natural Numbers forms a Bijection with the Set of Rational Numbers and hence, it is Countable. Every element in N is connected to one and only one element in Q. 11

12 Who s s Bigger: א 0 or R? Scenario: b11 = 1 if a11 1; otherwise b11 = 2 b22 = 1 if a22 1; otherwise b22 = 2 b33 = 1 if a33 1; otherwise b33 = 2... bnn = 1 if ann 1; otherwise bnn = 2 12 Suppose there is a Bijection between the set of Natural Numbers and the set of [0,1], then we have a situation as in the image where every number indicated on the left-hand side of the equal sign would have be connected with one and only one value on the right-hand side of the equal sign. There must be a number in the interval [0,1] which is not on the list: 0. b 11 b 22 b 33 b 44 b nn Therefore, the number 0. b 11 b 22 b 33 b 44 b nn found in [0,1] would not have a number within N to be paired up with, hence, [0,1] contains more elements than N this would lead us to conclude that the interval [0,1] and consequently, the set of Real Numbers is, in fact, larger than the set of Natural Numbers. Different Sizes of Infinites

13 13 How Big is it Exactly? Recall, N (the set of Natural Numbers), is the smallest, countable infinite set. Now, where exactly then would the set of Real Numbers fall in comparison with א 0? The Continuum Hypothesis (proposed by Georg Cantor) suggest the following: R = א 1, the next largest infinity after א 0 Due to an endless list of scenarios, where there would be a vast list of assumptions, Cantor was unable to effectively prove the set of Real Numbers was directly, the next largest set of countable infinities after the set of Natural Numbers ( א 0 ). It was not, without-a-doubt, confirmed whether R could not be equal to, let s say, א 2 or א 3 or א 100 or etc. Mathematicians were not able to answer Cantor s question for nearly 100 years. In 1963, they proved that the Axioms of Set Theory used by Cantor are not sufficient to decide the question one way or another.

14 Same Rules Don t t Apply The characterizations of what it means for sets to be,,, or = are all equivalent for finite sets, but no longer for infinite sets. Different descriptions of size when extended to infinite sets, will break different rules held for finite sets. Regarding Infinite Sets: They may be the same size but have more or less elements. If you were to delete an element from an Infinite Set, would the set be considered smaller in size even though the set is still referred to as Infinite? When the rule is applied to a Finite Set, it is easily interpreted as being smaller in size. 14

Hilbert s s Paradox of the Grand Hotel Scenario: Consider a hypothetical hotel with countable infinitely many rooms, all of which are occupied that is to say every room contains a guest.

15 Hilbert s s Paradox of the Grand Hotel Scenario: Consider a hypothetical hotel with countable infinitely many rooms, all of which are occupied that is to say every room contains a guest. One might be tempted to think that the hotel would not be able to accommodate any newly arriving guests, as would be the case with a finite number of rooms. Case 1: Suppose a new guest arrives and wishes to be accommodated in the hotel. Because the hotel has infinitely many rooms, we can move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3 and so on, and fit the newcomer into room 1. Case 2: It is also possible to accommodate a countably infinite number of new guests: just move the person occupying room 1 to room 2, the guest occupying room 2 to room 4, and in general room n to room 2n, and all the odd-numbered rooms will be free for the new guests. In a finite hotel with more than one room, the number of odd-numbered rooms is obviously smaller than the total number of rooms. However, in Hilbert s Grand Hotel, the quantity of odd-numbered rooms is as many as the total quantity of rooms. The properties of infinite "collections of things" are quite different from those of finite "collections of things". 15

To Infinity and Beyond

To Infinity and Beyond University of Waterloo How do we count things? Suppose we have two bags filled with candy. In one bag we have blue candy and in the other bag we have red candy. How can we determine which bag has more