We started talking about infinity today. Usually when I think of infinity, I just think of the general, “not stopping idea”, and leave it at that. What else is there to say about infinity? It goes on forever, and that’s it.
Yeah, I was really wrong.
We began by defining cardinality, which I also hadn’t really thought in depth about. Cardinality, apparently, can only be used when comparing sets; it doesn’t make sense to use cardinality to describe one set. That’s like saying “I’m taller than.” Taller than what? It’s an incomplete thought.
We defined cardinality in terms of an equivalence relation (I’m pretty sure Casey is obsessed with them at this point, just saying), by saying that:
A~B if |A| = |B|, which means that there is a bijective function from A to B.
We tested the three properties of ~, so this really is an equivalence relation.
- Reflexive: f(n) = n
- Symmetric: inverse property (If f:A→B exists, then f:B→A exists)
- Transitive: composition property (If f:A→B, and g:B→C, then g∘f: A→B→C)
We all agreed that with finite sets, their cardinality is equal if they have the same amount of elements.
Then Casey brought up infinite sets, and that’s when things got weird. We can prove that sets that are intuitively not equal, are equal. Like |ℤ| = |2ℤ|. It’s true, because there’s a bijective function between these two sets that maps from ℤ to 2ℤ: f(n) = 2n.
We also settled on the concept of uncountable, which I still can’t believe. Apparently, the natural numbers are countable, even though it is an infinite set.
But get this. The real numbers, they are uncountable. This infinite set is so big, that even the set of real numbers between (0,1) is uncountable.
There are different kinds of infinity.