Nature of Mathematics: Infinity

The other day in class when we had the discussion of what zero divided by zero is: zero, one, infinity... etc. At some point during this discussion, I believe professor Golden briefly mentioned something about how there are some mathematicians who don't think infinity should be included in math. For some reason or another this really intrigued me, so I decided to research more on my own about infinity and the problems along with it. While investigating, I came across an article from Quanta magazine, which I wanted to share with you through this blog. Just a disclaimer, I'm definitely not an expert on this subject and some of still confuses me, however I'd like to share my thoughts and knowledge before and after reading this article and the subsequent research I did on the topics presented by it. I'll start by briefly summarizing the article and then move on toward my views on the debate.

Today in mathematics, the "ZFC" or Zermelo-Fraenkel set theory with the axiom of choice is the standard form of axiomatic set theory and the most common foundation in mathematics (Princeton.edu). From my understanding its basically the most accepted concept of fundamental laws of mathematics and set theory. However, there are some holes in it, namely infinity. One of these problems that has brought up much debate is the "continuum hypothesis" (Wolchover).

The continuum hypothesis states: "There is no set whose cardinality is strictly between that of the integers and the real numbers." (Wikipedia). The problem is that with the "ZFC" there is no way to prove if this hypothesis is true or false. In 1940, Kurt Godel showed that the continuum hypothesis could not be disproved, whereas in 1960, Paul Cohen showed that it couldn't be proven either, both assuming the ZFC axioms to be true (Wikipedia). This is where to two differing theories presented come in to play. First there is the forcing axioms which say the continuum hypothesis is false and then there is the inner model axiom, "V=ultimate L," which says the continuum hypothesis is true (Wolchover).

The V=ultimate L model, from my understanding, says that you must construct some set L that is equal to V, the universe of sets. In this constructed L, you must start with the empty set and use it to build bigger and bigger sets until encompassing all of them. In this manner, the resulting universe of sets would prove the continuum hypothesis to be true (Wolchover). Hugh Woodin, a mathematician at the university of California, Berkeley, found that such an L could, in theory, be constructed. However, such an L has not actually been made yet (Wolchover).

On the other hand, the forcing axiom looks not to create a new model that encompasses everything, but rather expand and simply fill some of the holes of the existing model. Stevo Tordorcevic, one of the leading activists for this method, likens it to the idea of complex numbers (Wolchover), which I thought made it really easy to understand. Just like how complex numbers fill gaps in our understanding of the number system by adding on to it, the forcing axioms work in the same way.

Before reading this article, I honestly had no idea that this problem with infinity existed, or any problem with it for that matter. That's why I thought it was so interesting when it was mentioned in class. After doing some research, I found out that are actually quite a few problems with infinity. For instance, infinity has two distinguishable types, actual infinity and potential infinity. The difference between the two is also quite intriguing but for fear of digressing I'll leave it at that.

So on to my thoughts about the article. After learning about both sides of the argument, I think I'm more inclined to agree with the forcing axioms. The sole reason for this is because the forcing axioms have concrete results that deal with the problems we have now. Natalie Wolchover, the author of this article, puts it in to words much better than I could by saying, "while V=ultimate L is busy building a castle of unimaginable infinities, forcing axioms fill some problematic potholes in everyday mathematics." I've always enjoyed the more applied part of mathematics so perhaps that is why the forcing axioms appeal more to me. However, the idea that one day a set could be found that encompasses and fixes everything is also fairly appealing. As of right now, its only a matter of opinions on which one you like more until anything can be further proven, so I'd like to ask which side you would chose ?