[FOM] Arithmetical compatibility of higher axioms

[Dmytro Taranovsky]

Joe Shipman 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?

There are no known natural plausible statements A and B expressible in second
order arithmetic such that ZFC+A+B is known to be inconsistent but both ZFC+A
and ZFC+B are known/believed to be consistent.  (Note: I am not sure how to
come up with a counterexample even if A and B are not required to be natural.)
This is because:
1. The only known way to come up with such statements is by forcing.
2. The theory of second order arithmetic is generically absolute between all
generic extensions of V that satisfy projective determinacy (under reasonable
axioms, this includes all generic extensions of V).
3. After a detailed study of the relevant theories, one concludes that the
negation of projective determinacy is implausible.

This lack of bifurcation extends substantially beyond second order arithmetic
(for example, it extends beyond L(R)), and it is not known how far up one can
go.  My view is that there is exactly one natural completion of ZFC (that is
non-restrictive and consistent).

> 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?

I would say that such an example is:
(*) Every uncountable coanalytic set has a perfect subset.

I am not sure if you consider (*) reasonably simple (its explicit 
representation
as a Pi^1_4 statement is not very simple), but this is a very natural 
statement
-- more natural than the simple claim that there is a non-constructible real.

The difference from what I wrote above is that existence of a 
mathematician who
believes a statement is not sufficient to make the statement plausible.  There
are different degrees of implausibility. V=L, while implausible, is more
plausible than, for example, inconsistency of ZFC + an inaccessible 
cardinal. It is plausible that at least one mathematician (out of 
thousands) believes in
V=L, and thus rejects (*).

Here is why a mathematician may believe in V=L.  Constructible universe is to
set theory what a predicative universe is to analysis. The definition of
L_{alpha+1} is based on quantification over L_alpha (which was previously
constructed) and does not rely on unbounded quantification over arbitrary sets
or even arbitrary real numbers.  In other words, the construction of L is
almost predicative.  A mathematician who rejects general impredicative
definitions but accepts the notion of an arbitrary ordinal -- and the ability
to iterate a construction any ordinal (even uncountable) number of 
times -- may
find ZFC + V=L to be a natural theory.

The example (*) is Pi^1_4.  By Sigma^1_2 absoluteness, we should not expect a
Pi^1_2 example.  The difficulty of coming up with a Pi^1_3 example is that
mathematical theorems tend to be Pi^1_n instead of Sigma^1_n, and all true
Pi^1_3 statements are satisfied in every inner model and (under projective
determinacy) every generic extension of an inner model, and false Pi^1_3
statements that are true in L do not appear very natural.

Sincerely,
Dmytro Taranovsky