FOM: Independent axiomatizations

[wtait@ix.netcom.com]

A follow-up on Neil Tennant's posting on decidability sets of axioms and 
my response (12/11). The old-fashioned criteria for a set S of axioms 
required independence: P not deducible from S\{P} for each P in S. Every 
set S of axioms is logically equivalent to an independent set S' (easily 
proved in the case of a countable language); but clearly S' is not 
decidable in S. There is probably no decidable independent set of axioms 
for PA---Is there an easy proof of this? (Please don't make it too easy!)