Since we discussed in class a little bit of 10.3 already, before I read the section, it made a little more sense. However, the proofs as seen in class and in the text seem very difficult. In fact, the proof of Theorem 10.8 "The open interval (0, 1) of real numbers is uncountable" was shown in the book and in class, but I still don't really get it. What does it mean when b (sub) i is separated into a sort of piecewise function of 4 if a (sub) ii equals 5 and 5 if a (sub) ii does not equal 5? I don't see the connection between doing this and showing that it is not onto. I recall that it may have something to do with the game.
Speaking of the game, although it wasn't mentioned in the reading, it was discussed in class and has applications, I believe, to the proofs of denumerable and uncountable sets. I kind of enjoyed the game, although it fast became easy and only Player 2 has a winning strategy. It reminds me of tic-tac-toe and other such games in that once you know some tactics, you can never lose. I thought that it might be fun to try the game from the homework out with my son who is in kindergarten. He might think that it is fun with the X's and O's and it might teach him to think a bit more logically. No doubt he'll eventually never want to be Player 1. Anyway, I thought that the game was interesting and maybe I can use it to understand the logic and proofs in this section better.
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