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Unformatted text preview: Introduction: Mathematical Paradoxes Intuitive approach. Until recently, till the end of the 19th century, mathematical theories used to be built in an intuitive or axiomatic way. Intuitive approach is not sufficient. The his- torical development of mathematics has shown that it is not sufficient to base theories on an intuitive understanding of their notions only. Example: By a set, we mean intuitively, any collection of objects- for example, the set of all even integers or the set of all stu- dents 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. 1 Example: Most sets are not members of them- selves; 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 them- selves - for example, the set of all sets. Russell Paradox, 1902: Consider the set A of all those sets X such that X is not a member of X. Clearly, 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. In any case, A is a member of A and A is not a member of A. 2 Russell solution: every object must have a definite non-negative integer as its type . 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 . Theory of types says that it is meaningless to say that a set belongs to itself, there can not be such set A, as stated in the Russell paradox. Development: by Whitehead and Russell in years 1910 - 1913. It is successful, but difficult in practice and has certain other drawbacks as well. 3 The paradoxes concerning the no- tion of a set are called LOGICAL PARADOXES or ANTINOMIES General solution to Logical Paradoxes : a mod- ern development of Axiomatic Set The- ory as one of the most important fields of Modern Mathematics or more specifi- cally Mathematical Logic, or Foundations of Mathematics. Zermello, 1908-first paradoxes free axiomatic set theory. 4 Two of the other most known logical para- doxes are Cantor and Burali-Forti antinomies. They were stated at the end of 19th century. Cantor paradox involves the theory of cardi- nal numbers . Burali-Forti paradox is the analogue to Can- tors in the theory of ordinal numbers . 5 Cardinal number: cardX = cardY or X and Y are equinumerous if and only if there is one-to-one correspondence that maps X and Y ). cardX cardY means that X is equinumerous with a subset of Y . cardX < cardY means that cardX cardY and cardX 6 = cardY ....
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This note was uploaded on 02/12/2011 for the course CSE 541 taught by Professor Bachmair,l during the Spring '08 term at SUNY Stony Brook.

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1slide - Introduction: Mathematical Paradoxes Intuitive...

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