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there is no end and we say that we have an infinite set. But is it true that the subsets of this sets are less numerous than the entire set? Consider all the even numbers 2,4,6,8.... and all the natural numbers 1,2,3,4,.... We may think that there are more natural numbers than even natural numbers since the even numbers are a subset. But is this true? Let us count the number of even natural numbers. We have to find a mapping from the natural numbers to the even natural numbers. For each natural number m there is a even natural number 2m. Therefore there are as many even number as there are even and odd numbers! Consider the 2 concentric circles in the Figure above. Has the larger outer circle more points than the smaller circle? We think that the answer is probably yes! But is this true? We have to count the number of points in both circles. If we can find a mapping that assigns for each point on the larger circle a point in the smaller circle then both circles have the same number of points. Every point B of the larger circle can be connected with a line going through the center. This line will pass through a point A of the smaller circle. We could consider this as the mapping that we wanted. Every point of the larger circle can be mapped to a point on the smaller circle. How is this possible? Should the larger circle not contain more points than the smaller circle? The point A can be very close to the point C. So we have to recognize that our common logic does not work for the infinite large sets. Aristotle considering the Paradox due to the infinite small and large considered two types of infinities. The so-called “actual” infinite and the “potential” infinite. If we have to travel a distance then we do not travel over a actual infinite set which is impossible but we travel over a potential infinite set in that this set could be divided actually for ever in smaller and smaller pieces. It was actually considered that the for real objects the actual infinite is not possible, that an object could not have something which exceeds all limits. An actual infinity is regarded as a completed totality. A potential infinity is more like a finite but indefinitely long, unending series of events. According to that the diagonal of a square cannot be described as a ratio of natural numbers as there is no such ratio for the square root of the number 2. In this way actually the ancient Greeks discovered that there is no mapping of the natural to the real numbers that are more infinite than the infinite natural numbers 1,2,3,... etc.
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