To infinityand beyond Buzz Lightyear Please write 400500 wor

To infinity.....and beyond!\" (Buzz Lightyear) Please write 400-500 words about how Real Analysis has influenced your thinking about infinity.

Solution

Zeno’s paradoxes had first alerted philosophers to the concept of infinity in 450 B.C.E. when he argued that a fast runner such as Achilles has an infinite number of places to reach during the pursuit of a slower runner. Since then, there has been a struggle to understand how to use the notion of infinity in a coherent manner. Philosophers wanted to know whether there is more than one coherent concept of infinity, which entities and properties are infinitely large, infinitely small, infinitely divisible, and infinitely numerous, and what arguments could justify the answers one way or the other. The term “the infinite” refers to whatever it is that the word “infinity” correctly applies to. For example, the infinite integers exist just in case there is infinity of integers. We also speak of infinite quantities, but what does it mean to say a quantity is infinite. A clear concept of infinite number is needed in order to have a clear concept of infinite quantity. Looking back over the last 2,500 years of use of the term “infinite,” three distinct senses stand out: actually infinite, potentially infinite, and transcendentally infinite. The concept of potential infinity treats infinity as an unbounded or non-terminating process developing over time. The concept of actual infinity treats the infinite as timeless and complete. Transcendental infinity is the least precise of the three concepts and is more commonly used in discussions of metaphysics and theology to suggest transcendence of human understanding or human capability. To give some examples, the set of integers is actually infinite. Another example is a continuous line that has an actual infinity of points. A single point on this line has no next point. For mathematical purposes, the actual infinite has turned out to be the most useful of the three concepts. Calculus is the area of mathematics that is thought of as a technique for treating a continuous change as being composed of an infinite number of infinitesimal changes.

The new technical treatment of infinity originated with Dedekind in 1888. It provided a definition of \"infinite set\" rather than simply “infinite.” Dedekind says an infinite set is a set that is not finite. The notion of a finite set can be defined in various ways. We might define it numerically as a set having n members, where n is some non-negative integer. When creating set theory, mathematicians did not begin with the belief that there would be so many points between any two points in the continuum nor with the belief that for any infinite cardinal there is a larger cardinal. These were surprising consequences discovered by Cantor. The main issue here is whether we can coherently think about infinity to the extent of being said to have the concept. If we understand negation and have the concept of finite, then the concept of infinite is merely the concept of not-finite.