Hilbert–Bernays paradox

It is named after David Hilbert and Paul Bernays.

The paradox appears in Hilbert and Bernays' Grundlagen der Mathematik and is used by them to show that a sufficiently strong consistent theory cannot contain its own reference functor.

[1] Although it has gone largely unnoticed in the course of the 20th century, it has recently been rediscovered and appreciated for the distinctive difficulties it presents.

[2] Just as the semantic property of truth seems to be governed by the naive schema: (where single quotes refer to the linguistic expression inside the quotes), the semantic property of reference seems to be governed by the naive schema: Let us suppose however that, for every expression e in the language, the language also contains a name for that expression, and consider a name h for (natural) numbers satisfying: Suppose that, for some number n: Then, surely, the referent of exists, and so does (the referent of )+1.

Since, given the diagonal lemma, every sufficiently strong theory will have to accept something like (H),[clarification needed] absurdity can only be avoided either by rejecting the principle of naive reference (R) or by rejecting classical logic (which validates the reasoning from (R) and (H) to absurdity).