1. I LOVE INDUCTION. I have always loved mathematical induction, but these proofs were just awesome. So simple and yet mind blowing. Who knew it could be so easy to prove something for an "every element" set. I loved it. Probably the hardest thing for me so far was trying to think of what sets have a least element and which don't. It seems easy, but there were a couple that I guessed incorrectly. So my analysis of it is that if there isn't a set end point, there can't be a least number. So open ended sets (that either end with infinity, negative infinity or they approach a value but aren't equal to it) have no least value. I take it that because of this, we will mostly be working with the natural numbers.
2. So this well-ordered set business was pretty fun. It reminded me of the test! When you make a subset of the natural numbers, one of those sets is going to have to be open ended and go to infinity. So it can't be well-ordered. Hooray for seeing an way that a test question could be expanded to have a part b!
My only question is this: Why do we need to know that the implication works for P(1)? I get that we can assume the hypothesis (direct proof) and so assuming for P(k) is ok, but I am confused about why we need to know that P(1) works. It's nice to know, I just don't see the value.
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