Den onsdag 4 oktober 2017 kl. 23:00:06 UTC+2 skrev Doctor Allan: > does infinity exists and if so how can i prove it by math ? > > Infinity is a concept based on observation, in the same way as there is a concept of "stone"--imagine "stone"; you know what this concept means, because you can find examples of it, but no single example is the actual gestalt. Just like the concept of "one", "infinity" is not a standalone object, but a descriptive concept. > > What does it mean? Well, there are many meanings, even in mathematics, as another poster has recounted. A simple one is a proof of the infinity of the number of prime numbers. 2,3,5,7.... Let's see what happens if we assume the number of primes is finite. Let's see what we get when we multiply them all together and add one. > > x=2*3*5*7*...*lastprime+1. > > x is clearly larger than all the primes. Is x a prime? Let's try factoring it! x/2 leaves remainder 1, x/3 leaves remainder 1... in fact, x/any-prime leaves a remainder of 1, so x must be prime! But this contradicts the idea that we have only a finite number of primes, as x was created by multiplying by all of them. Therefore, the number of primes is infinite. > > Now you may want to see an example of infinity in the real world. Draw a line segment and also draw next to it (not connected to it) a circle. Put your pencil now over one end of the line segment and trace the entire path, stopping when you get to the end. As you can tell, the maximum path length is exactly what you think--finite. Now try this with a circle. Notice that you can smoothly move your pencil round and round and round the circle without ever coming to an end. Like a treadmill, the maximum path length on a circuit is infinite. This is useful, for track, car, horse, dog races and also for gyms, that they can provide a track that will allow for more than any finite distance. This, my friend, is an example of infinity you can point at!
There is a long list of "infinities (with no claim to exhaustiveness): infinity of the one-point compactification of N, infinity of the one-point compactification of R, infinity of the two-point compactification of R, infinity of the one-point compactification of C, infinities of the projective extension of the plane, infinity of Lebesgue-type integration theory, infinities of the non-standard extension of R, infinities of the theory of ordinal numbers, infinities of the theory of cardinal numbers, infinity adjoined to normed spaces, whose neighborhoods are complements of relatively compact sets, infinity adjoined to normed spaces, whose neighborhoods are complements of bounded sets, infinity around absolute G-delta non-compact metric spaces, infinity in the theory of convex optimization, etc.;
each of these has a clear definition and a set of well-defined rules for handling it. In mathematics, the textbook definitions applies. And definitions can vary from textbook to textbook and from field to field. For example, "dimension" has several meanings in mathematics. The same can be said about different notions of infinity, as above.