So...mind blown...
I am overwhelmingly fascinated by the fact that we can define the cardinality of an uncountable set. And that we can do so simply! I already knew that we could call it "c," but the defining of 2^A being 2^aleph-null...it just seems so simple. Well, I suppose that nasty looking aleph null symbol doesn't make it look too simple, but it is!
The main problem I have with these proofs is that they seem to all be proving the theorem. (They are.) I just don't understand how I'm going to figure out how to do these problems without the infinite hints of the example. For instance. In the proof of Theorem 10.19, I don't think I would've even known how to define the sets without the help of the example. To put it simply, I am must more scared of this test than I was of the previous two tests.
What exactly is the benefit of knowing that the cardinality is less than another cardinality? I see great benefit in knowing that if (in the language of LaTex) A\leq B and B \leq A, then A=B. But what if the only thing you can do is prove A \leq B? Then we can assume that A is not equal to B, but that doesn't tell us that it's denumerable, does it? Questions, questions...
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