Hilbert–Bernays paradox

The Hilbert–Bernays paradox is a distinctive paradox belonging to the family of the paradoxes of reference. It is named after David Hilbert and Paul Bernays.

History

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. 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.

Formulation

Just as the semantic property of truth seems to be governed by the naive schema:
, 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 numbers satisfying:
Suppose that, for some number n:
Then, surely, the referent of exists, and so does +1. By, it then follows that:
Therefore, by and the principle of indiscernibility of identicals, it is the case that:
But, by two more applications of the indiscernibility of identicals, and yield:
Alas, is absurd, since no number is identical with its successor.

Solutions

Since, given the diagonal lemma, every sufficiently strong theory will have to accept something like, absurdity can only be avoided either by rejecting the principle of naive reference or by rejecting classical logic and. On the first approach, typically whatever one says about the Liar paradox carries over smoothly to the Hilbert–Bernays paradox. The paradox presents instead distinctive difficulties for many solutions pursuing the second approach: for example, solutions to the Liar paradox that reject the law of excluded middle have denied that there is such a thing as the referent of h; solutions to the Liar paradox that reject the law of noncontradiction have claimed that h refers to more than one object.