[FOM] Inconsistent Systems

[Joao Marcos]

> 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 is known as *Curry's Paradox*:
http://en.wikipedia.org/wiki/Curry's_paradox

Best, JM