In [4] Curry raised the possibility that his system proposed in ξ15C of [3] might be inconsistent. In this paper this inconsistency is proved using a method also employed in [1].
From Curry's axiom ⊦LH, it follows that
holds for arbitrary X.
The other results from that are required are
Modus Ponens, and the Deduction Theorem for implication:
Assuming ⊦HA, we define as in [1]:
and let
where Y is the paradoxical (or fixed point) combinator.
We have X = G
2
X, so, by (2), H X ⊦ X ⊃ G
2
X which is HX ⊦ X ⊃. H2
X ⊃ G
1
X. Clearly HX ⊦ H(G
1
X) and, by (5), HX ⊦ H3
X, so that, by (3), H X ⊦ H2
X ⊃. X ⊃ G
1
X and by (5) and Modus Ponens HX ⊦ X ⊃ G
1
X. This is HX ⊦ X ⊃: HX ⊃. X ⊃ A which by (3) and Modus Ponens gives
which gives, by (4), HX ⊦ X ⊃ A. Now, by (1) and (DT),
which is ⊦G
1
X. But we have ⊦H2
X so, by the (DT), ⊦H2
X ⊃ G
1
X which is ⊦G
2
X. Thus we have proved ⊦X.