> So long as it is not self-contradictory I can make it an axiom of a 
> mathematical
> basis.  It may not be very interesting mathematics to postulate:
> Axiom 1: There is a purple cow momentarily appearing to Anna and then 
> vanishing.

I fear this is not an "axiom of a mathematical basis".  :)

The problem with improperly-founded axioms is the same problem
encountered with the naive set theory of Frege.  You can't ever be
certain that a set of axioms isn't self-contradictory.  In fact,
Frege's unstated Axiom of Unrestricted Comprehension, which roughly
states "for any property P, there exists a set containing all and only
the things that satisfy that property", is self-contradictory by


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