I just ran across the following paper: [Defining Z in Q][1]

It uses another characterization of the integers inside the rationals that none of us listed, perhaps because it is so trivial.  Namely, the integers are the complement of the set $\mathbb{Q}\setminus\mathbb{Z}$!  Apparently this basic fact is put to good use in the paper.


  [1]: http://arxiv.org/PS_cache/arxiv/pdf/1011/1011.3424v1.pdf