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A positive answer was given in the paper "Sets, Classes, and Categories" by F.A.Muller (British Journal for the Philosophy of Science 52 (2001) 539-573)

also available online where an extension (named ARC) of Ackermann set theory was suggested, and the author stated in the abstract of the paper that:

ARC "founds Cantorian set-theory as well as category-theory and therefore can pass as a founding theory of the whole of mathematics".

Can someone motivate a negative answer?

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4 
 What is the question? – David Roberts Apr 2 2012 at 4:42
7 
 The other answer is negative. – Andrej Bauer Apr 2 2012 at 6:26
2 
 @Andrej: I always thought you didn't believe in the law of excluded middle. – Emil Jeřábek Apr 2 2012 at 11:12
2 
 @Emil: Even intuitionistically, the only truth value distinct from true is false. – Andreas Blass Apr 2 2012 at 12:59
2 
 The last question, 'what is your favourite...', is not appropriate for MO. And I'm leaning towards closing this question. – David Roberts Apr 2 2012 at 16:09
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closed as subjective and argumentative by Emil Jeřábek, Mike Shulman, Neil Strickland, Andy Putman, David Roberts Apr 2 2012 at 20:13

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