FOM: 70:Efficient Formulas and Schemes

[Raatikainen Panu A K]

Stephen Fenner is right in pointing out that reasons I gave for my 
claim did not really constitute a proof. I think, however, that it can 
be competed very easily (please correct me if I am wrong):

(Below, by "a theory" I always mean a finitely axiomatizable 
theory) 

First, note that for every sentence/theory there exists an efficient 
sentence/theory that is logically equivalent with it. Next, note that 
one can recursively enumerate all theorems of  f.o. logic that have 
the form S <-> T. 
Assume then that one could recursively enumerate all efficient 
sentences/theories. By simultaneously enumerating them and all 
logical equivalences, one could eventually conclude, for an 
arbitrary* consistent theory or sentence, that it is logically 
equivalent with an efficient sentence/theory more complex than Pa 
& -Pa, and hence consistent. OK?

(* this does not include the finitely many trivial cases that are 
consistent and less complex than Pa & -Pa.)

* * *

About Fenner's conjecture: It was also my initial conjecture too (in 
some 3 years ago), but later I started to doubt it - and I still have a 
hunch that it is false. However, my first idea of how to prove such 
things did not work, and other issue kept me busy, so I forgot the 
whole theme without ever deciding the question. Only the recent 
ideas of Friedman made me to think the issue again.

Panu Raatikainen