[FOM] cofinite quantifier (fwd)

[Thomas Forster]

This has been much on my mind recently, for several reasons. For 
example, this afternoon, the man two doors away from me mentioned
that for every epsilon, it is true for cofinitely many $n$ that 
there is a prime between n and n + n^{0.5 + epsilon}.   Number 
theory is full of assertions with this kind of flavour.  Sounds 
as if it would be easier if we had the cofinite quantifier as 
primitive.  This reminded me of a factoid Adrian Mathias told 
me years ago: the theory of Turing degrees expressed with the 
measure-one quantifier is decidable.   How undecidable is the 
theory of natural numbers with the cofinite quantifier?  Does 
anyone know about this?


     Thomas