A set of line segments, known is as Cantor’s set is still mind-boggling. Take a line of unit length. Then chop out the middle one-third. Then chop out the middle one-third of the two remaining line segments, and then chop the middle one-third of those remainders and so on. With each step, the total length of the line (adding all the bits together) is two-thirds of the previous length, continuing in this way, the total length diminishes with each step until we are left with almost a point or entirely empty space. Cantor proved that even when an infinite number of chops have been performed, and the line has shrunk to zero total length, it still contains more points than the discrete row of dots.
The fact that the length of an infinite row of discrete points is zero and the length of Cantor's points is zero, does not mean this contains the same number of points. The extra-ordinary conclusion is that a line of zero length can contain as many points as the stars in the sky.
It is helpful to understand the relation between infinity and zero. In a sense, zero is infinitely small, the opposite limit to infinity. Problems occur if zero and infinity is multiplied and such operations are to be handled with care. In mathematics, one divided by infinity is zero and one divided by zero equals' infinity.
Enough has been said to show that the properties of infinity sets or collections are frequently counter- intuitive and common sense reasoning may well lead to nonsense. Nevertheless, by discovering these elaborate properties, mathematicians can use infinity without fear, so long as they stick carefully to the rules, however strange these rules may seem. The question is – is infinity relevant, or are these observations just an imagination by mathematicians for their own amusement? The answer is that infinity does crop up repeatedly when we attempt to develop theories to model the physical world.
Understanding the existence of infinity, in mathematics, is the key to understanding what it means for concepts like ‘Infinity’ or “Imaginary numbers” to exist, something that puzzles many people when the first encounter these infinite objects. Mathematical objects do not exist in the same sense that physical object exists; instead mathematical objects are abstracts concepts, often isolated from a real world situation.
When we ask whether a mathematical object exists, we must have in mind an appropriate context, a particular, precisely different collection of concepts. Then we ask, “Among these concepts, is there one which matches the object we are looking for?” if so, we say that the object exists; if not, it does not exist. So the existence of infinity depends on the context, we are referring in mathematics. · In the context of any number system infinity does not exist. Explanation: One-Way to think, what would be infinity minus 1? It could not be a finite number, since no finite number plus 1 equal's infinity. We see the rules of arithmetic are violated; since if they hold, one could subtract infinity from both sides to conclude that it gives rise to -1 = 0, which is not true.
Therefore, there is no number system, which agrees with the rules of arithmetic and in which infinity exists. In other words, infinity does not exist in the context of number system. · In the context of topological space, infinity does exist.
Topological space is a set of objects. Here the concept of infinity refers to some sequences of real numbers that converge. Think about drawing the real numbers on an interval of finite length. Here, on this line, represent larger numbers closer. For instance, you could mark number 1 at a distance 1 unit from the end, 2 at a distance of 1/2; from the end, 3 at a distance of 1/3 and so on. Now consider the end points in the interval and call them “infinity”. Now we can say that sequences of numbers converge to infinity if and only if the dots where the numbers are represented converge to the right-hand end point of the interval. We can say in this context, infinity does exist. 3. In the context of sets, the infinity exists:
In the context of measuring the sizes of sets in which “infinity” concepts do exist but there are more than one of them; since not all infinite sets have the same size. So there does not exist any one single “infinity” concept; instead, there exists a whole collection of things called “infinite cardinal numbers”.