Can you give me a property P which is true along the zero-chain but necessarily false along a separated chain that is infinitely long in both directions? I do not believe this is possible but I may be mistaken.

For any number n, n-n=0.

If you have a separate chain that isn't connected to zero, then this isn't true.

However this statement is pretty simple and can be expressed in first order logic. I have no idea why EY believes that it requires second order logic to eliminate the possibility of other chains that aren't derived from zero.