[FOM] Inconsistent Systems
Arnon Avron
aa at tau.ac.il
Thu Sep 12 16:53:55 EDT 2013
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
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