Hilbert–Bernays Paradox
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The Hilbert–Bernays paradox is a distinctive
paradox A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true or apparently true premises, leads to a seemingly self-contradictor ...
belonging to the family of the paradoxes of reference. It is named after
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad range of fundamental idea ...
and
Paul Bernays Paul Isaac Bernays ( ; ; 17 October 1888 – 18 September 1977) was a Swiss mathematician who made significant contributions to mathematical logic, axiomatic set theory, and the philosophy of mathematics. He was an assistant and close collaborator ...
.


History

The paradox appears in Hilbert and Bernays' ''
Grundlagen der Mathematik ''Grundlagen der Mathematik'' (English: ''Foundations of Mathematics'') is a two-volume work by David Hilbert and Paul Bernays. Originally published in 1934 and 1939, it presents fundamental mathematical ideas and introduced second-order arithme ...
'' 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 Semantics is the study of linguistic Meaning (philosophy), meaning. It examines what meaning is, how words get their meaning, and how the meaning of a complex expression depends on its parts. Part of this process involves the distinction betwee ...
property of
truth Truth or verity is the Property (philosophy), property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth, 2005 In everyday language, it is typically ascribed to things that aim to represent reality or otherwise cor ...
seems to be governed by the naive schema: :(T) The sentence ′''P''′ is true if and only if ''P'' (where single quotes refer to the linguistic expression inside the quotes), the semantic property of reference seems to be governed by the naive schema: :(R) If ''a'' exists, the referent of the name ′''a''′ is identical with ''a'' 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: :(H) is identical with ′(the referent of )+1′ Suppose that, for some number ''n'': :(1) The referent of is identical with ''n'' Then, surely, the referent of exists, and so does (the referent of )+1. By (R), it then follows that: :(2) The referent of ′(the referent of )+1′ is identical with (the referent of )+1 Therefore, by (H) and the principle of indiscernibility of identicals, it is the case that: :(3) The referent of h is identical with (the referent of h)+1 But, by two more applications of the indiscernibility of identicals, (1) and (3) yield: :(4) ''n'' is identical with ''n''+1 Alas, (4) is absurd, since no number is identical with its successor.


Solutions

Since, given the
diagonal lemma In mathematical logic, the diagonal lemma (also known as diagonalization lemma, self-reference lemma or fixed point theorem) establishes the existence of self-referential sentences in certain formal theories. A particular instance of the diagonal ...
, every sufficiently strong theory will have to accept something like (H), absurdity can only be avoided either by rejecting the principle of naive reference (R) or by rejecting
classical logic Classical logic (or standard logic) or Frege–Russell logic is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy. Characteristics Each logical system in this c ...
(which validates the reasoning from (R) and (H) to absurdity). On the first approach, typically whatever one says about the
Liar paradox In philosophy and logic, the classical liar paradox or liar's paradox or antinomy of the liar is the statement of a liar that they are lying: for instance, declaring that "I am lying". If the liar is indeed lying, then the liar is telling the trut ...
''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 In logic, the law of excluded middle or the principle of excluded middle states that for every proposition, either this proposition or its negation is true. It is one of the three laws of thought, along with the law of noncontradiction and t ...
(which is ''not'' used by the Hilbert–Bernays paradox) have denied that there is such a thing as the referent of h; solutions to the
Liar paradox In philosophy and logic, the classical liar paradox or liar's paradox or antinomy of the liar is the statement of a liar that they are lying: for instance, declaring that "I am lying". If the liar is indeed lying, then the liar is telling the trut ...
that reject the law of noncontradiction (which is likewise ''not'' used by the Hilbert–Bernays paradox) have claimed that h refers to more than one object.


References

{{DEFAULTSORT:Hilbert-Bernays paradox Mathematical paradoxes Self-referential paradoxes