Different Sized Infinities

[ccb_v2 0]

I am going to show that the irrational numbers are uncountably infinite. That is that they are a finite set that cannot be mapped to the natural numbers, {1, 2, 3, 4, ...}. I also am going to show that the rational numbers are indeed countable.

 

Both proofs are in reality pretty simple and can be understood by most people that have been educated to the point of knowing algebra. Of course, if you need any clarification, feel free to ask, but to find the proofs just follow the links to my blog.

 

The proof that the rationals are countable.

 

The proof that the irrationals are uncountable.

 

I hope that you enjoy them.

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[Sarah81 0]

We went over this idea, and the proofs of course, in an honors-level math class that I took last semester. Once I wrapped my brain around the basic concepts, the whole thing made perfect sense. Fortunately, I'm taking a very basic math class for my other math requirement so that I can spend more study time doing something that I enjoy - literature *grins*

[mitchellmckain 0]

To follow up on this topic, the irrational numbers are not only uncountable but this only represents a higher order of infinity that can also be exceed by and even higher order of infinity. Just as there is a one to one correspondence between the rational numbers and the natural numbers, thus representing the lowest order of infinity C1. But there is a one to one correspondence between the points of an infinite plane and the points in a line segment from zero to one, representing a higher order of infinity C2. An even higher order of infinity is represented by the number of possible functions on a real line segment: C3.

These things were first proved by Ludwig Cantor, but they were not well received at the time. The following quote from webite Cantor relates how Cantor, was himself was surprised by some of these results. The website continues to describe how other mathematicians continued to reject his discoveries through most of his life.

Cantor pressed forward, exchanging letters throughout with Dedekind. The next question he asked himself, in January 1874, was whether the unit square could be mapped into a line of unit length with a 1-1 correspondence of points on each. In a letter to Dedekind dated 5 January 1874 he wrote [1]:-
Can a surface (say a square that includes the boundary) be uniquely referred to a line (say a straight line segment that includes the end points) so that for every point on the surface there is a corresponding point of the line and, conversely, for every point of the line there is a corresponding point of the surface? I think that answering this question would be no easy job, despite the fact that the answer seems so clearly to be "no" that proof appears almost unnecessary.

Cantor continued to correspond with Dedekind, sharing his ideas and seeking Dedekind's opinions, and he wrote to Dedekind in 1877 proving that there was a 1-1 correspondence of points on the interval [0, 1] and points in p-dimensional space. Cantor was surprised at his own discovery and wrote:-

I see it, but I don't believe it!