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Set theory is one of the branches of mathematical logic which deals with the study of sets. According to Georg Cantor, a set is a collection of definite, distinguishable objects of perception or thought conceived as a whole. Set refers to a combination of various objects.These objects include; members or elements of the set. In set theory, sets are usually represented by letters A and B while elements are represented by small letters x and y. Set method is usually applied in cases whereby objects or elements have a similarity to mathematics. George Cantor’s theory, therefore, is a fundamental system for mathematics. Set theory is necessary when it comes to studying of sets that are infinite(Kuratowski, K.,2014). Set theory is therefore important since it is a foundation of mathematical analysis, object algebra, and discrete mathematics.
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DEVELOPMENT OF SET THEORY
Georg Cantor is the proponent of set theory. Georg Cantor contributed greatly to the development of set theory.(Johnson, P.E, 1972). It is through Cantor’s work that set theory was put into mathematical basis. Georg Cantor’s early contribution was his work on number theory. He published some articles on number theory between 1867 and 1871.A major development took place in 1872. In 1872, Georg Cantor traveled to Switzerland and met Richard Dedekind who also made a great contribution to Cantor’s set theory.(Dedekind,1888). Dedekind believed in the logical way of thinking and this significantly impacted on Cantor’s ideas on set theory.
Georg Cantor then moved from theory to papers on trigonometric series. These articles consisted of Georg Cantor’s first ideas and various concepts on set theory.(Miller, A. W. 2017). It also included results of irrational numbers. In 1874, Georg Cantor further published another article in Crelle’s journal. The article that was published contributed further to the development of set theory.
By 1878, set theory was the center of controversy. In Cantor’s next publication, he introduced the idea of equivalence of sets. Georg Cantor believed that two sets are equivalent if the sets can be put in a 1-1 correspondence. Between 1879 and 1884, Georg Cantor published another article on set theory. Cantor’s work appeared in Mathematische Annalen. Kronecker strongly disagreed with Cantor’s ideas that were presented in his set theory. Kronecker only accepted mathematical objects that could only be constructed finitely from the intuitively given set of natural numbers.Georg Cantor based his argument on infinite sets. These two opposing ideas brought conflict between the two scholars.
In 1883, Georg Cantor published his work on set theory. His work was a discussion on well-ordered sets. According to Cantor, ordinal numbers are introduced as the types of well-arranged sets. Multiplication and summation of transfinite numbers are also defined in Cantor’s work.( Chen, Y. (2010).
Georg Cantor believed that mathematics is a free field and only concepts can be introduced in the mathematical area if the thoughts do not lead to any contradictory ideas in the field. Other scholars who gave their opinions on the concepts of infinity include; Aristotle, Descartes, Berkeley, Leibniz, and Bolznam.
In 1885, Cantor continued to develop his set theory. Between 1895 and 1897, Cantor published his final double treatise on set theory. Cantor proved that if A and B set with A equal to a subset of B, and B similar to a subset of A, then A and B are equivalent. This was proofed by Felix Bernstein and E. Schroder. In 1886, Cantor discovered paradox. In 1897, the first international congress for mathematics was held in Zurich. In this contest, Cantor’s work was ranked among the best and praised by many including Hurwitz and Hadamard. In 1899, Cantor discovered another paradox which arises from the set of all sets. By 1900, set theory was considered as one of the branches of mathematics. In 1908, Carmelo was the first to attempt an axiom at is the icon of set theory. Other mathematicians who also attempted to axiomatize set theory include; Fraenkel, Von Neumann, and Bernays. These people played an important role in the development of set theory.
In conclusion, set theory is very important in the study of mathematics and statistics. Set theory enables mathematicians to study infinity. This means that infinite sets can be studied using set theory. It is therefore evident that set theory is the mathematical theory of the actual as opposed to the potential. Through the help of set theory, mathematicians and statisticians are in a position to familiarize themselves with all mathematical notions and arguments.
- Karwowski, W., & Mital, A. (Eds.). (2014). Applications of fuzzy set theory in human factors (Vol. 6). Elsevier.
- Miller, A. W. (2017). Descriptive set theory and forcing (Vol. 4). Cambridge University Press.
- Kunen, K., & Vaughan, J. (Eds.). (2014). Handbook of set-theoretic topology. Elsevier.
- Kuratowski, K. (2014). Introduction to set theory and topology(Vol. 101). Elsevier.
- Jech, T. (2013). Set theory. Springer Science & Business Media.
- Chen, Y. (2010). Introduction to probability theory. The lecture notes on information theory. Duisburg-Essen University.
- Kechris, A. (2012). Classical descriptive set theory (Vol. 156). Springer Science & Business Media.
- Johson, P.E (1972). A history of set theory. London: Academic Press
- Pollard, S. (1990) Philosophical introduction to set theory: Notre Dame Press
- Cohen, P.J. (1966).Set theory and the Continuum Hypothesis. New York: W.A. Benjamin, Inc
- Dauben,J.W, Cantor,G(1979).Mathematics and philosophy of the infinite.Cambridge