“The farthest limits of the west” would of course mean Spain. Clement could have been simply extrapolating from Romans 15, and it would suit his purpose to give the impression of Paul’s worldwide reach. But he was a central figure in the Roman church within a generation of Paul’s day. He is writing at the most about thirty years after Paul’s death. It is far more likely that he knew more solid and reliable traditions about Paul than we, discounting him, can invent on our own.
