[FOM] Inconsistent Systems

Arnon Avron aa at tau.ac.il
Thu Sep 12 16:53:55 EDT 2013


> 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

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