[FOM] Proof "from the book" of the incompleteness theorem
aa at tau.ac.il
Sun Aug 29 02:43:17 EDT 2004
> How about:
> The set of arithmetic theorems of any formal system is recursively
> enumerable, while the set of arithmetic truths is not. So any sound formal
> system must fail to prove some arithmetic truth.
This is indeed the first proof (out of 3) that Smullyan presents
in his book on the incompleteness theorems. The trouble with this
proof is that it misses one of the most important aspects of Godel's
proof: the actual construction of a *true* sentence which the system
fails to prove, and a *proof* that it is true. The most interesting
debates concerning the implications of Godel's theorem are connected
with this aspect (in particular: the debate about Lucas-Penrose argument).
More information about the FOM