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Then I guess I better get out of the teaching game entirely! (NOT!)
My freshman calculus course didn't finish the definition of the real
numbers until the second semester. This axiomatization was
interesting, but ultimately convinced me to stick with the 'reality'
of physics as opposed to the self-consistent mindgames of math.
Biology teachers can't agree on a complete/consistent definition of
'life' and WE will not be able to come with a (non-trivial)
definition of energy. Kurt Godel may have offered some insight into
this difficulty?