The Concept “Infinity”
March 23, 2011 Leave a comment
Does one concentrate purely on the mathematical aspects of the topic or does one consider the philosophical and even religious aspects?
The most famous among those is the problem of Infinity.The concept ” Infinity ” cause the great difficulties for many thinkers over centuries. If u was a professional thinker,u might thought that at least . Because it is correspond not only to the reason of human thought ,but also even religious aspects.
Expressed in one website.The interaction of philosophy and mathematics is seldom revealed so clearly as in the study of the infinite among the ancient Greeks. The dialectical puzzles of the fifth-century Eleatics, sharpened by Plato and Aristotle in the fourth century, are complemented by the invention of precise methods of limits, as applied by Eudoxus in the fourth century and Euclid and Archimedes in the third.
And then, the people including mathematicians began to think about the time , about the infinity and the world they lived in.Would the world go on forever or not ? Or had the world always existed ? Whatever happened to say, the foundation of those questions is the concept ” Infinity ” .
In Modern time,the mathematician who tried to solve that was Cantor,[ Georg Ferdinand Ludwig Phillip Cantor ].But he was not the first mathematician to formulise the concept of the infinite .Prior to Cantor ,Richard Dedekind made the first giant step by deciding how to recognize the infinite ,rather than construct it, thereby avoiding objections such as the following made by Gauss;
” I protest against the use of infinite magnitude as something completed, which in mathematics is never permissible.Infinity is merely a facon de parler,the real meaning being a limit which certain ratios approach indefinitely near,while others are permitted to increase without restriction.”
Cantor asked two questions: First,can infinity be recognized without making reference to the natural numbers? Second,are there different degrees of infinity?
For his first question ,he defined a set as being infinite if it could be put into a one-to-one correspondence with a proper subset of itself.We must consider the mapping that 0_1,1_2,2_3,3_4,….etc,.That is any set that satisfies Dedekind’s definition of the infinite automatically satisfies Cantor’s definition.His solution for second question built on this.He defined two sets as being equinumerous if they could be put into a one-to-one correspondence with each other.The positive integers are equinumerous with the negative integers by the mapping n_/_n ( for a positive integer n ).By a similar mapping,the positive real numbers are equinumerous with the negative real numbers .That ‘s the point which should be considered right now.