I. The present paper contains a formal proof of the following theorem of the elementary system ⊨ [ 1 0 ] c:This theorem means that if there were a theorem of ⊢0 ( 3 2 ) 0 stating that there is no contradiction in ⊢2 ( 5 4 ) 2, we would have a contradiction in ⊢2 ( 5 4 ) 2. (This is in accordance with the second theorem of Gödel.)Our proof is based on the calculus of the system ⊨ [ 1 0 ] c and of the metasystem ⊢0 ( 3 2 ) 0.Note that we could takeinstead of Ax E .1 ( L ) (.1 ( L ) .0 ( L ) L) Z. We would then have to do with the simple systems and metasystems of Hetper, which do not contain propositional variables or the logico-semantical axiom. Our method applies equally to the systems and metasystems of New Foundation and to the simple systems and metasystems of Hetper.Our proof can be considerably simplified by using a recent result of Hetper concerning ancestral functions. Hetper introduces the following abbreviations:If A ( Z1c), B ( x1cy1cZ1c) are propositions, the expression will be called an ancestral function (we prove without difficulty that this expression is a proposition). The expressions will be called respectively the principal term and the term of derivation of this function.
