[FOM] Arithmetical compatibility of higher axioms

[Harvey Friedman]

>
> On Aug 13, 2009, at 12:59 AM, joeshipman at aol.com wrote:
>
> 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?

It is a theorem of mine that "there is a PCA well ordering of the  
reals" iff "there is a real such that every real is constructible from  
it". This is incompatible with, say, every PCA set is measurable, or  
every uncountable PCA set has a perfect subset.

Harvey Friedman