[FOM] Inconsistent Systems

[Arnon Avron]

Harvey,


> Let alpha be the sentence (forall x,y)(x in y).
> 
> We argue in CA(no). Let A = {x: x in x implies alpha).
> 
> LEMMA 1. CA(no) proves: A in A implies alpha.
> 
> LEMMA 2. CA(no) proves: A in A.
> 
> LEMMA 3. CA(no) proves: alpha.
> 
> LEMMA 4. CA(no) proves every formula in the language of CA(no).
> 
> I leave it to the experts whether this is convincing, whether this is new,
> and what implications it has for various foundational and philosophical
> enterprises? And what are the next things to look at?
> 

This argument is known in the literature as "Curry Paradox". It shows
that at least  the triviality of Naive set theory (triviality in the sense
of that every formula is a theorem) is not caused by "strange" rules
concerning negation.

  Now the contraction principle (A->(A->B))->(A->B)  is crucial for this 
paradox. Hence the paradox might be avoided in logics in which this is 
not a theorem. Two known logics which reject it are Lukasiewich 
infinite-valued logic and linear logic. There have been indeed
attempts to develop non-trivial set theories with full CA
on these logics. I think that the earliest were due to Chang.

Arnon Avron