Going off topic in class: I think it worked--- this time

Recently I went off topic in a class. I think it was okay but I want YOUR thoughts.

1. On Monday I
   1. defined Primitive Recursive functions
   2. showed them that addition, mult, exp are all primitive recursive,
   3. talked a little bit about TOWER and WOWER, both primitive recursive,
   4. constructed by diag a computable function that was not prim rec,
   5. noted that this function is not natural
   6. noted that there are natural computable functions that are not prime rec (NOTE that I noted this),
   7. made the point that I can do the same diag argument for ANY reasonable system of (total) computable functions.
2. On Wednesday I was all set to start Turing Machines when a student named Amber said You promised to show us a natural example of a computable function that is not primitive recursive! I didn't recall exactly promising that but I very well might have and if a student cares about these things you want to nurture that curiosity.
3. I told the class that I would toss out today's lesson plan and talk about this, but I would be informal and some of what I said might not be quite right, but would get the idea across.
4. I asked one of the students who had his laptop up to stop surfing porn and look up Ackermann's function for me, so I could get the definition right.
5. I talked about the Union-Find Data Structure that has amortized O(n INVERSE-ACK(n)) running time for n operations. I noted that this would not be an impressive use of the Ackermann's function if Amber could find an O(n) structure. I had the class vote on if Amber could do that. This was about fifty-fifty. I then told them that Amber could not, not because she is not a fine programmer, but because there is a lower bound proof that nobody can do better. That is, the lower bounds has been proven.
6. I then talked about Goodstein Sequences which lead to computable functions that are not primitive recursive.

I went off topic. I am one lecture behind. Was it worth it?

1. This class is not a prereq to anything else so I do not HAVE TO COVER everything.
2. If I don't cover the topics on the syllabus then the guy who made up the syllabus, Gasarch, might get mad at me. This does not worry me.
3. I doubt I will end up cutting anything important. I usually spend the last few lectures on misc topics (e.g., sparse sets cannot be NPC unless P=NP) which I can skip.
4. They were interested in this. Not just Amber, but the whole class. That makes it worthwhile.

Misc:

1. I told the linguistics major in the class that Ackermann's function has applications in linguistics. He believed me until I told him YOU"VE BEEN PUNK'D!!
2. Ackerman's function gets 2,870,000 hits on Google. Ackermann's function gets 6,740,000 hits on Google. Does that make Ackermann correct? What if the wrong spelling got more hits?