Hi,

I was wondering how much (if anything) $\mathcal{L}_{PA}$ can express about individual nonstandard elements in a nonstandard model of PA. For instance, presumably it can say that each has $k$-many predecessors, for each $k\in\mathbb{N}$. But:

(a) I can't see that there is any way that the type of one element in a $\mathbb{Z}$-chain differs from the type of any other in that same chain. Is this correct?

(b) Are the types of elements in separate $\mathbb{Z}$-chains also identical? I mean, clearly they won't be the same as those of elements in the initial segment $\mathbb{N}$ - these will have a finite number of predecessors - but in two of the additional chains?

To me, it looks like these questions are straightforwardly true but I could be wrong. Many thanks,

Kate