Mathematicians have been using sets since the very beginning of the subject. For example, Greek mathematicians defined a circle as the set of  points at a fixed distance r from a fixed point P. However, the concepts of 'infinite set' & 'finite set'  eluded mathematicians and philosophers over the centuries. For example, Hindu minds conceived of infinite in their scriptural text Ishavasy-opanishad as follows: "The Whole is there. The Whole is here. From the hole imanates the Whole. Taking away the Whole from the Whole , what remains is still a Whole". Phythagoras(~ 585-500 B.C.), a Greek mathematicians, associated good and evil with the limited and the unlimited, respectively. Aristotle(384-322 B.C.) said, "The infinite is imperfect, unfinished and therefore, unthinkable; it is formless and confused." The Roman Emperor and philosopher Marcus Aqarchus(121-180 A.D.) said infinity is a fathomless gulf, into which all things vanish". English philosopher Thomas Hobbes(1588-1679) said, "When we say anything is infinite, we signify only that we are not able to conceive the ends and bounds of the thing named".   

        The working mathematician, as well as the street, is seldom concerned with the unusal question: what is a number? But the attempt to answer this question precisely has motivated much of the work by mathematicians and philosophers in the foundations of mathematics during the past hundred years. Characterization of the integers, rational numbers and real numbers has been a central problem for the classical researches of Weierstrass, Dedekind, Kronecker, Frege, Peano, Russell, Whitehead, Brouwer, and others. The researches of Georg Cantor around 1870 in the theory of infinite series and related topics of analysis gave a new direction for the development of the set theory. Cantor, who is usually considered the founder of the set theory as a mathematical discipline, was led by usually considered the founder of set theory as a mathematical discipline, was led by his work into a consideration of infinite sets or classes of arbitrary character.   

        However, Cantor's results were not immediately accepted by his contemporaries. Also, it was discovered that his definition of a set leads to a contradictions and logical paradoxes. The most well known among these was given in 1918 by Bertrand Russell(1872-1970), now known as Russell's paradox.   

        In effort to resolve these paradoxes, the first reaction of mathematicians was to 'axiomatize' Cantor's intuitive set theory. Axiomatization means the following: starting with a set of unambiguous statements called axioms, whose truth is assumed, one is able to deduce all the remaining propositions of the theory from these axioms using axioms of logical inference. Russell and Alfred North Whitehead(1861-1974) in 1903 proposed an axiomatic theory of sets in their three-volume work called Principia Mathematicians found it awkward to use. An axiomatic set theory which is workable and is fully logistic was given in 1908 by Ernst Zermello(1871-1953). This wa improved in 1921 by Abraham A. fraenkel (1891-1965) and T. Skolem (1887-1963) and is now known as 'Zermello-frankel(ZF)-axiomatic theory of sets.

 

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