A `natural' theorem not provable in PA?

[Timothy Y. Chow]

Dennis Hamilton wrote:
> Consider
>
> M(0) = M(0-M(-1))/2 = M(0-1)/2 = 1/2
>
> And if 0 is not natural enough, then consider
>
> M(1) = M(1-M(0))/2 = M(1/2)/2 = M(1/2 - M(-1/2))/4 = M(0)/4 = 1/8
>
> so M:Z -> Q and no wonder the termination cannot be proven in PA.

I don't understand this objection.  It sounds like an implicit claim that 
statements about rational numbers cannot be expressed in the first-order 
language of arithmetic, which is obviously false.  Maybe I misunderstand?

Tim