Minor typo (I think) in your lecture John:

>But you can still try to give me a "best approximation" to the nonexistent natural number a with 2a=4.

Think that should be 2a=**3**?

If not, it's a "thinko" on my part (kudos to [Patrick O'Neill](https://forum.azimuthproject.org/discussion/comment/16151/#Comment_16151) for the "thinko" concept!).