I have had some difficulty with proofs involving showing a function is bijective. I still need to get to a point where I easily comprehend and can adequately construct similar proofs. I would expect them to keep appearing in this section, not to mention the final exam. I fail to understand the logic of the proof of Theorem 10.14. Aside from the fact that I get tripped up when I see different characters, some with sub-characters, I still don't understand how you set things equal to each other and then that shows an injection. Then you set other things equal to each other and it is onto. I really need to focus on the fundamentals of these types of proofs. At least I understand that it is a bijection if it is injective and onto...at least.
The Continuum Hypothesis cannot be proven or disproven. That is interesting. I would think that there are other hypotheses and conjectures that are impossible to prove or disprove, but I'm not that familiar with mathematics to name one. In a field that I sometimes see as very black and white, I am always surprised when I see that mathematical ideas cannot be proven or disproven. I guess there are many mysteries. I wonder what the opinion is for mathematicians - if it can't be proven, then is it false or, rather, if it can't be disproven, then is it true?
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