Nam Nguyen wrote: > On 24/10/2013 6:58 AM, Peter Percival wrote: >> Nam Nguyen wrote: >> >>> >>> The game of symbol manipulation is there to stay with FOL=, nonetheless. >>> >>> It doesn't matter what philosophical motivation you might have had, it's >>> part of the definition of reasoning with rules of inference in FOL with >>> identity: either you'd conform to it, or betray it. >>> >>> Godel betrayed it, and so have we. >> >> Gödel's incompleteness theorem isn't about FOL=. > > You're wrong of course. > > What Godel wrote on his seminar paper: > > "This situation is not due in some way to the special nature of the > systems set up, but holds for a very extensive class of formal > systems". > > Note his "formal systems" would refer to what's known today as > First Order Logic with equality, and note that his "class of formal > systems" would include the "set theory of Zermelo-Fraenkel" (his words), > which is today known as ZF (of First Order Logic with equality). >
You are confusing first order logic with identity (which is surely what "FOL=" stands for) with first order theories. FOL= has only logical axioms (if it has any at all), that's why it's called logic. A first order theory may have non-logical axioms. That "very extensive class of formal systems" are theories with non-logical axioms. It is the non-logical axioms that lead to incompleteness, FOL= is complete..
-- The world will little note, nor long remember what we say here Lincoln at Gettysburg