The Problem with Infinity (and Cantor's Theorem)

In this post I discuss my thoughts on infinity and math which acts upon it. I cover what infinity is and why it doesn't make sense to give it mathematical properties the way Cantor does.

Infinity is not a number, it's a concept. Most people agree with this easily and fail to understand the significance of this. What is a concept? Well, some examples of concepts are remorse and happiness. You can't point to them and it's difficult to argue their existence, but everyone has felt them before and knows that they exist. Then there are concepts like democracy and bigfoot. The existence of these two can be argued at length. Infinity is somewhere between happiness and bigfoot. It can't be pointed to, it can't be held, it can't even be felt like happiness can. Why is this important?

One reason this is important is because concepts are something for which everyone has a different idea of their exact meaning. For example, no two people in the world have the exact same understanding of what "happiness" means. So when talking about infinity, it's impossible for two people to have the same understanding of it. This makes it difficult to prove anything about it mathematically.

Additionally, when infinity is treated as a number, it results in inconsistency. Why is this important? Inconsistent things in life, logic, and mathematics are cause for concern and disbelief. Inconsistency in life is often called hipocrisy, like when a politician's motto is "family first" and he's later found to have multiple mistresses. In logic, inconsistency is called a fallacy. In math, it's called a contradiction.

Treating infinity as a number is a contradiction because it's not a number, it's a concept. Infinity + 1 = Infinity, and Infinity + 2 = infinity as well. Unfortunately, this means that

infinity + 1 = infinity + 2

Okay, so cross out the infinities and we're left with 1 = 2, a contradiction.

So why is it okay to say that "1 < infinity?" Well it's not. Giving infinity mathematical properties is like asking "What is Marxism divided by 2?"

Apparently, the mathematical community as a whole (for the most part), has accepted Cantor's theorem as something mathematical that can be proven (there is some debate here). In fact Cantor's theorem, and any other theorem that involves infinity, is not math at all. Ideas dealing with infinity are philosophy, not math, since what is being discussed is a concept.

Related links.
http://scienceblogs.com, Infinity is not a number.
http://en.wikibooks.org, Infinity is not a number.