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chapters1-3 - Chapter 1 Introduction 1.1 Mathematical...

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Chapter 1 Introduction 1.1 Mathematical Paradoxes Until recently, till the end of the 19th century, mathematical theories used to be built in an intuitive or axiomatic way. In other words, they were based either intuitive ideas concerning basic notions of the theory - ideas taken from the reality- or on the properties of these notions expressed in systems of axioms. The historical development of mathematics has shown that it is not sufficient to base theories on an intuitive understanding of their notions only. This fact became especially obvious in set theory. The basic concept of this theory, set , is certainly taken from reality, for there we come across many examples of various sets, all of which are finite. But in mathematics it is also necessary to consider infinite sets, such as the set of all integers, the set of all rational numbers, the set of all segments, the set of all triangles. By a set, we mean intuitively, any collection of objects- for example, the set of all even integers or the set of all students in a class. The objects that make up a set are called its members (elements). Sets may themselves be members of sets for example, the set of all sets of integers has sets as its members. Most sets are not members of themselves; the set of all students, for example, is not a member of itself, because the set of all students is not a student. However, there may be sets that do belong to themselves - for example, the set of all sets. However, a simple reasoning indicates that it is necessary to impose some limitations on the concept of a set. Russell, 1902 Consider the set A of all those sets X such that X is not a member of X. Clearly, by definition, A is a member of A if and only if A is not a member of A. So, if A is a member of A, the A is also not a member of A; and if A is not a member of A, then A is a member of A. 1
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2 CHAPTER 1. INTRODUCTION In any case, A is a member of A and A is not a member of A. This paradox arises because the concept of set was not precisely enough defined, and was too liberally interpreted. Russell noted the self-reference present in his paradox (and other paradoxes, two of them stated below) and suggested that every object must have a definite non-negative integer as its type . Then an expression x is a member of the set y is meaningful if and only if the type of y is one greater than the type of x . So, according to the theory of types, it is meaningless to say that a set belongs to itself, there can not be such set A, as stated in the Russell paradox. The paradoxes concerning the notion of a set are called logical paradoxes (anti- nomies) . Two of the most known (besides the Russell’s) logical paradoxes are Cantor and Burali-Forti antinomies. Both were stated at the end of 19th cen- tury. The Cantor paradox involves the theory of cardinal numbers , Burali-Forti paradox is the analogue to Cantor’s in the theory of ordinal numbers . They will make real sense only to those already familiar with both of the theories, but we
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