Question
I just had a popular (tree) cut down it was probably about 20 feet tall and the base was probably 24"-30" diameter.  The tree was removed and the stump was grinded but the roots are still growing.  The tree removal company went down 6" in the ground and grinded the roots at surface.  But the roots are still very much a live.  I need to do something quick because the roots are getting close to the slab of my house.  

Answer
Helen,
You will get some regrowth from the roots that were not ground out, but they will not lengthen or survive for very long.
If you see any small trees growing from the roots that are left, just cut it at the base. Eventually these roots will die and rot away.
Good luck.
Ed Gulliksen