Yesterday I studied the cardinality of infinite sets and, I must say, it was the first time I have ever distrusted math. That is, it was the first time I have not bought - hook, line and sinker - something about math that was proven to me by a mathematics professor.
This is not to say that I think it is untrue. I am saying that the truth of the cardinality of infinite sets is so at odds with my intuition that I am having trouble believing it. And I could use a bit more explanation, I think, to clear up a few things.
- The distinction between the rules governing finite and infinite sets. For example, we can say that the odd numbers are a proper subset of the integers. We know this absolutely, correct? Yet, we can then prove using diagonalization that the set containing all odds and the set containing all integers have the same cardinality, which is Aleph-null.
How is that we can allow such discontinuity in our understanding of these sets? How can both of those two statements be simultaneously true? How, if A is a proper subset of B, can |A|=|B|? I know this doesn't hold when we limit the conversation to finite sets. However, I don't understand why we can't make the same assertion about these two infinite sets - or at least not definitively say they both have cardinality Aleph-null.
- Proving that the reals don't form a bijective map to Aleph-null. This was proven by trying to map the reals onto the naturals. More specifically, the professor considered only the continuum and showed that the continuum can't map to the naturals and therefore neither can the reals.
The professor did this by showing that any list of natural numbers could never contain all the reals because we can always create at least one new number not contained in the list.
HOWEVER, it felt like a bit of flashy wording to me. If we can do this and then say "look, we didn't account for this number" and furthermore not be allowed to say "okay, but I still have this natural number to which I can map that real number" in this instance, why do the same arguments not apply to other instances? That is, why can't I do the exact same thing when dealing with the mapping between odds/integers?
I know this isn't the typical question for stackexchange, but I really want to nail this down. Don't get me wrong, I know the information. I can apply the information. However, I don't necessarily believe the information and that bothers me a bit.