 The Continuum Hypothesis << Back to Meta Next Paper >> NOTE: "?" denotes greek or hebrew mathematical symbol that is unable to be displayed.         How can we (or can we at all) justify claims about the concept of infinity, so that we may incorporate it in mathematical theory? Firstly, the concept of infinity poses several paradoxes which need to be resolved in order to form a concrete understanding of what it is (or what is it not) to be infinite, if we wish to claim any particulars at all. The continuum hypothesis, symbolically represented as 2?0 = ?1 roughly translates to mean that the infinite realm of rational numbers is "one level of infinity" smaller than the infinite realm of real numbers which includes both rational and irrational numbers. A further claim of the continuum hypothesis is that the infinite realm of real numbers can be paralleled to the infinity of the infinitely small, because both are infinites by division. Likewise it correlates the infinite realm of rational numbers with the infinity of the infinitely big, since both are infinities by addition. My paper will propose a comprehensive understanding of the motivation for, and context of the continuum hypothesis which will be necessary to address the creditability of the claims the continuum hypothesis makes concerning comparable infinities.         The motivation for the continuum hypothesis began an attempt to resolve the paradoxes posed by the infinitely big. Consider the statement "there are as many even natural numbers as there are natural numbers." Intuitively one would disagree with this statement based on the knowledge that an even natural number occurs after every odd natural number, making the set of all natural numbers double the size of the set of even natural numbers. But we also know that the set of natural numbers extends on infinitely. So if we were to draw two sets of numbers, no matter how long we went on listing numbers there would always be a one to one correspondence between the sets (making them equal to each other). This deduction indicates the paradox posed by the infinitely large, since it is contradictory to say that something that is boundless is larger than something else that is boundless.         Avoiding this paradox the infinitely large poses was the motive Cantor used to develop his criteria for sets. He postulated that sets could be compared in a correlative sense, or in a subset sense. In the correlative sense, one asks if the members of a set can be paired to members of another set, while the subset sense asks if the members of a set belong to another set. With this criterion we can now state that the set of even numbers is a subset of all numbers, thus smaller than the set of natural numbers in the subset sense. Cantor had developed a methodology of referring to a set with an infinite number of members as a whole, and promoted the idea that the concept of infinity was set-theoretic.         Cantor's set theory did not come without problems. One of the problems that Cantor encountered can be illustrated with Russell's paradox which states "Let R be the set of all sets which do not belong to themselves" returning the contradiction that "R belongs to itself if and only if R does not belong to itself." This seems to show that addressing the paradoxes of the infinitely big with set theory, causes the paradoxes of the one and the many to arise.         To address the paradoxes of the one and the many, Cantor further developed diagonal arguments. These diagonal arguments expand on Cantor's correlation criterion, adding that any set A is bigger than set B when all members of B can be paired with some members of A, but not with all of them. A result (of Cantor's diagonal arguments) showed that there are more real numbers between 0 and 1 than there are natural numbers, which became the fundamental motivation to develop the continuum hypothesis. The proof for this claim is illustrated by creating two sets of numbers where you pair natural numbers with decimal expansions of real numbers between 0 and 1. You can then always construct a real between 0 and 1 that is not listed in the table by applying the following conditions: "Create a new decimal by following the table diagonally and for each decimal place where X occurs write Y, and write X where every number that is not X occurs. " Doing so will always construct a real that is not listed in the table. But the initial table constructed contains all natural numbers, so there are members of the set of real numbers that cannot be paired within the set of natural numbers. This makes the set of real numbers larger in proportion to the set of natural numbers, even though both sets are infinite. Cantor further expanded his diagonal arguments to show that there are more sets of natural numbers than there are natural numbers. To compensate he applies the concept of a powerset, in which the powerset of A is the set of all subsets of A. The concequence to the statement that the powerset of A is larger than the initial set A is that now there is no limit to how large an infinite set can be. This leads to the non-trivial question Cantor posed that the continuum hypothesis addresses that says "The set of natural numbers is infinite, but its powerset (the set of real numbers) is larger. So how much larger is the set of real numbers than the set of natural numbers?"         There are two possibilities to answer "Cantor's unanswered question." One is that the set of real numbers is the "next infinite size up" from the set of natural numbers, the other is that there are intermediate sizes of infinity between the set of real numbers and the set of natural numbers. The continuum hypothesis is the claim that the answer to Cantor's question is that the set of real numbers is the next size of infinity up from the set of natural numbers.         Motivated to talk about levels of infinity, Cantor developed a hierarchy of sets. Levels in the hierarchy are measured in the Ordinal numbers, which specifically measure the "length" of a well ordered set in correlation sense. The ordinals are also in itself, well ordered. The first ordinals on the hierarchy are the sets of natural numbers. The ordinal that succeeds the set of natural numbers is Omega (?) and so on up to Kappa (? = ??) which is the first cardinal that can only be named by the ordinal it is (due to it's massive "size"). To avoid the paradox posed by the fact that there is no limit to how big an infinite set can be, we can assume that there is a limit to how small a set can be. So by definition, cardinal numbers allows us to measure the "size" of sets, and are the smallest ordinals of a given set size. The finite cardinals include the natural numbers, and measure the size of finite sets. The first infinite cardinal number is the set of all natural numbers (?0), making the second cardinal number, the set of all real numbers (?1).         Now that there is a system for referring to infinite sets (the ordinals and cardinals) we can now look at several transfinite arithmetic equations, specifically ones that are involved in the continuum hypothesis. Suppose that Kappa is a cardinal number, logically it would be less than its powerset (2?) which follows Cantor's claim that the powerset of any given set is larger than the initial set. Therefore the set of natural numbers is less than the powerset of natural numbers (?0 < 2?0). Further, we can now claim that the set of real numbers is larger than the set of natural numbers by one cardinal number since cardinals measure the "size" of sets and the set of natural numbers is less than its powerset, and its powerset is equal to the set of real numbers (2?0 = ?1).         A "continuum" has two basic properties: that between any two points there is another point (denseness), and that there are no gaps between points. The set of rational numbers is dense, being that between any two rationals there is another rational number, but there are gaps where irrational numbers occur. The set of real numbers includes both irrational and rational numbers, which is represented as ?1 and measures the "size" of a continuum since it measures how close together points on a line are. This illustrates that the set of the infinitely divisible (small) has members that cannot be paired with members of the infinitely addable (big) since there are gaps between natural numbers where irrational numbers (accountable by division) occur. Although technically we can neither prove or disprove the continuum hypothesis within any formal set theory (ZF).         In conclusion I don't believe that the argument in favor of the continuum hypothesis is a strong one. Firstly CH cannot justifiably identify the set of real numbers with the infinitely divisible. The infinite by division only states that there exists another point between any two points on a line. This only identifies with the property of denseness for which we only need the rational numbers, and not the real numbers. And the size of the set of rational numbers is ?0 and not 2?0. Secondly, the continuum hypothesis cannot justifiably identify the set of rational numbers with the infinitely addable, since we can extend addition to ordinals beyond omega ?. Although we now have a formal theory of sets, which enable us to talk and refer to objects and instances in the infinite realm, it is unconvincing to claim that the continuum hypothesis is true. << Back to Meta Next Paper >>