An infinite deal of nothing...or something? - Week 11

This week we have been discussing uncomputable functions. Specifically in lecture today we began to discuss the problem of cardinality. In turn, this came down to the question of how do you count infinite sets? In counting, we have come up with two key methods to ensure that we are not over or under-counting. These methods work for finite sets but it becomes more problematic when dealing with infinite sets.

The two rules of counting are as follows:

1) You have one-to-one matching

2) You map only once something from your domain to your range

For infinite sets it is much harder to apply these principles. When we are trying to compare the size of two infinite sets we can determine if one is actually larger than the other. If they are the same size, we can match them up in such a way that they have a one to one ratio. However, some infinite sets are not countable. The definition for being countable is having a one to one match up with the natural numbers. While the set of irrational numbers are countable, the set of real numbers are not.  Due to this fact, we can say that infinity of the real numbers is larger than the irrational numbers.

I find these concepts to be incredibly interesting but it is taking some time to wrap my head around the details. I think it is a good way to finish up this class as it is opening the doors to interesting future topics without overwhelming us with an overload of information. I look forward to hearing more about infinite loops…I just hope that my future code is not responsible for them!