Mathematicians have always faced problems as they expand their knowledge of their field. Most problems can be solved.However, some seem to be no solution can be challenging and even mathematics, which is why they always cause problems such as mathematics. This is known as a paradox, a statement which seems to contradict themselves or appear illogical, but it still could be true. An example is to say, "I always lie." If you lie, you tell the truth, but if you tell the truth, you lied. Zeno paradox with infinite, of Cantor and Russell with set theory, and the twin paradox in relativity physics has created problems and arguments for mathematicians, as well as forcing them to think about the subject of mathematics in a different way than before.Zeno, the Greek philosopher who lived in the fifth century BC, created several paradoxes to demonstrate the idea of space and time apart, and that by dividing them one comes to many contradictions. Two of several paradoxes that presented examples of such contradictions.
The first says that the tortoise and Achilles sprinter will race, and that the tortoise will be given a head start. Zeno tells Achilles that if you want to beat the tortoise, he must first catch up with it, but to do that he first must cover himpunanengah distance between them. Then, Zeno says that Achilles himpunanelah not make himpunanengah of the original distance between himself and the tortoise, the tortoise will have moved forward, creating a new gap between the two. Achilles then have to close this gap himpunanengah from new before catching turtles. However, once he closes this gap himpunanengah of the new, the tortoise would be moving again and create a new gap again. This means that Achilles will continue to cover the distance himpunanengah gap, only to discover that he had to cover the distance himpunanengah new loopholes. Zeno concluded that during the tortoise has a head start, Achilles can never catch him because he will always include a limited distance in an infinite sequence of time intervals.
The second paradox studying an arrow in flight. Zeno said that if you start to break down a small flight time to and gradual, then you can check the arrow at a certain moment, and then the arrow will move. He went on to say that if the time is composed of instants, the arrow never moves because at a certain instant the arrow is at a point in space but not in motion (Katz 57).
Zeno paradox creates a problem for mathematicians because they are researching the idea of infinity and infinitesimals in confined spaces. Aristotle was the first person who tried to deny this statement, claiming that in the example of Achilles, "a finite object can not come in contact with things quantitatively infinite," which means split-infinite time will not affect runners. In the matter of the arrow Aristotle says that time is not composed of indivisible instants, the assumption Zeno, and that even though the arrow may not move at some point, the motion is not defined at instants but for a certain period (Katz 56-7). However, because the infinite has no real value and no mathematically real, there is always a lot of controversy around it.
Zeno's paradoxes caused mathematicians to think carefully about the concept of infinity and infinitesimals and not make assumptions about them. In a lecture about Pythagoras and the Pythagoreans by Dr. Shirley we learn that the infinitesimals create problems for the Greeks. Pythagoreans encountered the first major crisis in mathematics when they find the square root of 2 when working with triangles. They think all right-angled triangle will have a limited length, and were surprised when they find a 45-45-90 triangle, which has a square root of 2 as the length of the hypotenuse.
Zeno infinite research is essential to mathematics as it helps lead the major developments in the calculus. Find the limit function approach as close to infinity, and in Dr. Shirley lectures on the calculus we learn it is the boundary that solved the crisis both in the mathematics of how to interpret an "extra" dx the derivative problem. Furthermore, in the 1600's Leibniz became disturbed by his use of infinitesimals in differentiation, and decided to justify their use. Although for Leibniz it never really mattered whether or not there are infinitesimals, he found that if a certain ratio is true when the quantity is limited, then the same ratio would apply when dealing with limits and values of the infinite. Manipulation technique becomes very useful for Johann and Jakob Bernoulli who received infinitesimals as mathematical entities and use them to make important discoveries in calculus and its applications (Katz 530-1).
The paradox created by Cantor in the second half of the 19th century includes the concept of the cardinality of the set theory and its relationship with (Katz 734). Cardinality basically explains how many numbers in one set, because the finite set it is as simple as counting, but an infinite set can not have cardinality that can be represented by a whole number. He found that if members of an infinite set can be put into one-to-one correspondence with each other, leaving no additional digits in the set either, then the two sets have the same cardinality. One-to-one correspondence means that for himpunaniap members in one set, there is a corresponding member in the second set. For example, in an e-mail with my professor, Dr. Shirley noted that the set of positive integers and the set of perfect squares are both limited and have a relationship n2 n for each member of the set, which means they have one-to-one correspondence. Cantor proved that the set of real numbers has cardinality larger than the set of integers, the paradox is that the infinite set of real numbers is "bigger" than the infinite set of integers. In general, Cantor's paradox begins by stating that the set of all sets (call it set B) is the set of his own powers, where the power set is the set of all subsets of a set A.Power set is always greater than the set associated with them (Weisstein, "Power set" 1). Paradoxically concluded that given a set B, the cardinality of the set B must be greater than himself. To understand the paradox, we must consider the Cantor theorem, which states that the cardinality of the set is lower than the cardinality of all subsets companies (Weisstein, "Cantori Theorem 1). The paradox is that if a set B is the set of all sets, then the cardinality of a subset of B will greater than B the set, but the cardinality of the set B must be the same as the set B and the same subset of B (Weisstein, Cantor's paradox 1).
Russell's paradox, discovered in the early 20th century, providing even more general view of the paradox discovered by Cantor's set theory. It states that R is the set of all sets which are not members of themselves, which means that all sets in R does not contain themselves as elements. The question then becomes, does R contain itself as element? If we assume that R does not contain itself, then by definition R can not contain itself and vice versa. The problem is most often given as the barber paradox.Suppose that in a small town there's only one barber is defined as a person who shaves all those who do not shave themselves.Then the question is "who shaves the barber?" If the barber does not shave himself, then he is not by definition. If the barber does not shave himself, then by definition he does (Russell Paradox 3).
Cantor and Russell's paradox is very important to the field of set theory because they are caused mathematicians to examine their assumptions previously made. This paradox suggests that set theory at the time (many designed by Cantor) has many inconsistencies because a lot of it is purely intuitive and not based on any type of axioms or proof. Mathematicians are forced to formulate a way to create a more consistent set theory and to provide a clear limitation. In the 1900s Ernst Zermelo set of seven axioms that provide clear rules for set theory (Katz 809-11). One of them, the axiom of separation (or regularity) and the Russell paradox Cantori avoided by restricting ourselves to swallow the set ("Russell's Paradox" 1). This paradox is very important for the development of set theory because they expressed the need for rules, as in algebra or geometry.
Although the paradox is annoying and confusing by nature, they remain important for mathematics in identifying problems and inconsistencies in mathematics throughout history. Moreover, by challenging the thinking time, paradoxically can cause more brilliant discoveries even in mathematics. Clearly, the paradox has been important for mathematics, and discipline may not be where it is today without them.
Sources:http://tiger.towson.edu/ ~ gstiff1/paradoxpaper.htm
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