A. Wolf wrote:
>> 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.  

I can if there's no rule of inference.  Perhaps that's crux.  You are requiring 
that a "mathematical structure" be a set of axioms *plus* the usual rules of 
inference for "and", "or", "every", "any",...and maybe the axiom of choice too.

>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.

Well not  entirely by itself - one still needs the rules of inference to get to 
Russell's paradox.

But then what is the justification for limiting "universes" to those which 
the usual rule of inference?  And remember that because of Godelian 
incompleteness an infinite number of axioms can be added even to those 
without running into contradictions.


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