> 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.

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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
itself.
Anna
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