[FOM] Arithmetical compatibility of higher axioms

[joeshipman@aol.com]

It is a well-supported empirical observation that no seriously proposed 
axioms extending ZF conflict with any others regarding statements of 
number theory, although they certainly contradict each other when 
talking about higher types of sets.

How much of this observation extends to statements of second-order 
arithmetic? Is there any reasonably simple statement S of second-order 
arithmetic for which it can be plausibly argued that there exist two 
mathematicians A and B, such that A believes that S is true because it 
is implied by her favorite axiom extending ZF, while B believes S is 
false because its negation is implied by his favorite axiom extending 
ZF? (For the record, I do not regard as "reasonably simple" the 
statement of second-order arithmetic coding up "a non-constructible 
real number exists").

-- JS