October 6, 2000: I took advantage of one of those teachable moments today. The class came in discussing/complaining about their TE class the previous day and how she had asked them why 2+3=3+2. Their reply was that it was an axiom, but that wasn't what she wanted. Apparently, judging by my class, she wanted them to explain why that was the choice of an axiom. The difficulty is that the language was messed up. I told them that it was actually an interesting question because an axiom system for a ring and numbers only came about after recognizing their properties. Thus one needed some kind of explanation for why. At this point, the discussion wandered off into what infinity is. We then ended up discussing countability and why the natural numbers have the same cardinality as the integers. Many of them didn't know the meaning of cardinality, so we defined that concept too. Then we discussed Hilbert's infinite hotel. I put off the discussion of the uncountability of the real numbers until later. I then took the tail end of today's lecture to discuss how you find the inverse of an element of the ring Q[2^{1/3}].
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