Diagonal lemma

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In mathematical logic, the diagonal lemma or fixed point theorem establishes the existence of self-referential sentences in formal theories of the natural numbers, if those theories are strong enough to represent all computable functions. Such sentences can be used to prove fundamental results such as Gödel's incompleteness theorems and Tarski's indefinability theorem.[1]

Contents

[edit] Background

Let N be the set of natural numbers. A theory T represents the function f : N→N if there exists a formula δ(x,y) in the language of T such that for each n, T proves

∀ y [f(n) = y ↔ δ(n,y)].

Here n is the numeral corresponding to the natural number n; it is defined to be the closed term 1+ ··· +1 (n ones).

The diagonal lemma applies to theories capable of representing the functions that make up primitive recursive arithmetic. Such theories include Peano arithmetic and the weaker Robinson arithmetic.

The diagonal lemma also requires that there be a systematic way of assigning to every formula θ a natural number #(θ) called its Gödel number. Formulas can then be represented within the theory by the numerals corresponding to their Gödel numbers.

[edit] Statement of the lemma

Let T be a first-order theory in the language of arithmetic and capable of representing all computable functions. Let ψ be a formula in the theory T with one free variable. The diagonal lemma states that there is a sentence φ in T such that φ ↔ ψ(#(φ)) is provable in T.[2]

Intuitively, φ is a self-referential sentence saying that φ has the property ψ. The sentence φ can also be viewed as a fixed point of the operation assigning to each formula θ the sentence ψ(#(θ)). The sentence φ constructed in the proof is not literally the same as ψ(#(φ)), but is provably equivalent to it in the theory T.

[edit] Proof

Let f: N→N be a function such that:

f(#(θ)) = #(θ(#(θ))

for any formula θ in the theory T having one free variable. If n is not the Gödel number of a formula, then f(n) = 0. The function f is computable, so there is a formula δ representing f in T. Thus for each formula θ, T proves

∀ y [ y = #(θ(#(θ)) ↔ δ(#(θ),y)].

Now define the formula β(x) as:

β(x) = ∀ y [δ(x,y) → ψ(y)].

Let φ be the sentence β(#(β)). Then we can prove in T that:

(*) φ ↔ ∀ y [ δ(#(β),y) → ψ(y)] ↔ ∀ y [ (y = #(β(#(β))) → ψ(y)].   

We analyze two cases.
1. Assuming φ holds, substitute #(β(#(β)) for y in the rightmost formula in (*), and obtain:

(#(β(#(β)) = #(β(#(β))) → ψ(#(β(#(β))),

Since φ = β(#(β)), it follows that ψ(#(φ)) holds.
2. Conversely, assume that ψ(#(β(#(β))) holds. Then the final formula in (*) must be true, and φ is also true.

Thus φ ↔ ψ(#(φ)) is provable in T, as desired.

[edit] History

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

The lemma is called "diagonal" because it bears some resemblance to Cantor's diagonal argument. The terms "diagonal lemma" or "fixed point" do not appear in Kurt Gödel's epochal 1931 article, or in Tarski (1936). Carnap (1934) was the first to prove that for any formula ψ in a theory T satisfying certain conditions, there exists a formula φ such that φ ↔ ψ(#(φ)) 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.[3]

[edit] See also

[edit] Notes

  1. ^ See Boolos and Jeffry (2002, sec. 15) and Mendelson's (1997, Prop. 3.37 and Cor. 3.44 ).
  2. ^ 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).
  3. ^ See Gödel's Collected Works, Vol. 1, p. 363, fn 23.

[edit] References

  • George Boolos and Richard Jeffrey, 1989. Computability and Logic, 3rd ed. Cambridge University Press. ISBN 0-521-38026-X ISBN 0-521-38923-2
  • Rudolf Carnap, 1934. Logische Syntax der Sprache. (English translation: 2003. The Logical Syntax of Language. Open Court Publishing.)
  • Hinman, Peter, 2005. Fundamentals of Mathematical Logic. A K Peters. ISBN 1-568811-262-0
  • Mendelson, Elliott, 1997. Introduction to Mathematical Logic, 4th ed. Chapman & Hall.
  • Raymond Smullyan, 1991. Gödel's Incompleteness Theorems. Oxford Univ. Press.
  • --------, 1994. Diagonalization and Self-Reference. Oxford Univ. Press.
  • Alfred Tarski, 1936, "The Concept of Truth in Formal Systems" in Corcoran, J., ed., 1983. Logic, Semantics, Metamathematics: Papers from 1923 to 1938. Indianapolis IN: Hackett.