Diagonal lemma

In mathematical logic, the diagonal lemma (also known as diagonalization lemma, self-reference lemma[1] or fixed point theorem) establishes the existence of self-referential sentences in certain formal theories of the natural numbers—specifically those theories that are strong enough to represent all computable functions. The sentences whose existence is secured by the diagonal lemma can then, in turn, be used to prove fundamental limitative results such as Gödel's incompleteness theorems and Tarski's undefinability theorem.[2]

BackgroundEdit

Let   be the set of natural numbers. A first-order theory   in the language of arithmetic represents[3] the computable function   if there exists a "graph" predicate   in the language of   such that for each  

 

Here   is the numeral corresponding to the natural number  , which is defined to be the  th successor of presumed first numeral   in  .

The diagonal lemma also requires a systematic way of assigning to every formula   a natural number   (also written as  ) called its Gödel number. Formulas can then be represented within   by the numerals corresponding to their Gödel numbers. For example,   is represented by  

The diagonal lemma applies to theories capable of representing all primitive recursive functions. Such theories include first-order Peano arithmetic and the weaker Robinson arithmetic. A common statement of the lemma (as given below) makes the stronger assumption that the theory can represent all computable functions, which still applies to first-order Peano arithmetic.

Statement of the lemmaEdit

Lemma[4] — Let   be a first-order theory in the language of arithmetic and capable of representing all computable functions, and   be a formula in   with one free variable. Then there exists a formula   such that

 

Intuitively,   is a self-referential formula:   says that   has the property  . The sentence   can also be viewed as a fixed point of the operation assigning to each formula   the formula  . The formula   constructed in the proof is not literally the same as  , but is provably equivalent to it in the theory  .

ProofEdit

Let   be the function defined by:

 

for each formula   with only one free variable   in theory  , and   otherwise. Here   denotes the Gödel number of formula  . The function   is computable (which is ultimately an assumption about the Gödel numbering scheme), so there is a formula   representing   in  . Namely

 

which is to say

 

Now, given an arbitrary formula   with one free variable  , define the formula   as:

 

Then, for all formulas   with one free variable:

 

which is to say

 

Now substitute   with  , and define the formula   as:

 

Then the previous line can be rewritten as

 

which is the desired result.

(The same argument in different terms is given in [Raatikainen (2015a)].)

HistoryEdit

The lemma is called "diagonal" because it bears some resemblance to Cantor's diagonal argument.[5] The terms "diagonal lemma" or "fixed point" do not appear in Kurt Gödel's 1931 article or in Alfred Tarski's 1936 article.

Rudolf Carnap (1934) was the first to prove the general self-referential lemma,[6] which says that for any formula F in a theory T satisfying certain conditions, there exists a formula ψ such that ψ ↔ F(°#(ψ)) is provable in T. Carnap's work was phrased in alternate language, as the concept of computable functions was not yet developed in 1934. Mendelson (1997, p. 204) believes that Carnap was the first to state that something like the diagonal lemma was implicit in Gödel's reasoning. Gödel was aware of Carnap's work by 1937.[7]

The diagonal lemma is closely related to Kleene's recursion theorem in computability theory, and their respective proofs are similar.

See alsoEdit

NotesEdit

  1. ^ Hájek, Petr; Pudlák, Pavel (1998) [first printing 1993]. Metamathematics of First-Order Arithmetic. Perspectives in Mathematical Logic (1st ed.). Springer. ISBN 3-540-63648-X. ISSN 0172-6641. In modern texts these results are proved using the well-known diagonalization (or self-reference) lemma, which is already implicit in Gödel's proof.
  2. ^ See Boolos and Jeffrey (2002, sec. 15) and Mendelson (1997, Prop. 3.37 and Cor. 3.44 ).
  3. ^ For details on representability, see Hinman 2005, p. 316
  4. ^ Smullyan (1991, 1994) are standard specialized references. The lemma is Prop. 3.34 in Mendelson (1997), and is covered in many texts on basic mathematical logic, such as Boolos and Jeffrey (1989, sec. 15) and Hinman (2005).
  5. ^ See, for example, Gaifman (2006).
  6. ^ Kurt Gödel, Collected Works, Volume I: Publications 1929–1936, Oxford University Press, 1986, p. 339.
  7. ^ See Gödel's Collected Works, Vol. 1, Oxford University Press, 1986, p. 363, fn 23.

ReferencesEdit