In mathematical logic, the diagonal lemma 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.
- (∀y) [f(n) = y ↔ δ(n,y)].
Here n is the numeral corresponding to the natural number n, which is defined to be the closed term 1+ ··· +1 (n ones), and f(n) is the numeral corresponding to f(n).
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.
The diagonal lemma applies to theories capable of representing all primitive recursive functions. Such theories include 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.
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 language with one free variable. The diagonal lemma states that there is a sentence φ such that φ ↔ ψ(#(φ)) is provable in T.
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.
Let f: N→N be the function defined by:
- f(#(θ)) = #(θ(#(θ)))
for each T-formula θ in one free variable, and f(n) = 0 otherwise. The function f is computable, so there is a formula δ representing f in T. Thus for each formula θ, T proves
- (∀y) [ δ(#(θ),y) ⇔ y = f(#(θ))],
which is to say
- (∀y) [ δ(#(θ),y) ⇔ y = #(θ(#(θ)))].
Now define the formula β(z) as:
- β(z) = (∀y) [δ(z,y) ⇒ ψ(y)],
- β(#(θ)) ⇔ (∀y) [ y = #(θ(#(θ))) ⇒ ψ(y)],
which is to say
- β(#(θ)) ⇔ ψ(#(θ(#(θ))))
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)].
Working in T, 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.
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.
- See Boolos and Jeffrey (2002, sec. 15) and Mendelson (1997, Prop. 3.37 and Cor. 3.44 ).
- For details on representability, see Hinman 2005, p. 316
- 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).
- See Gödel's Collected Works, Vol. 1, p. 363, fn 23.
- 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 1-56881-262-0
- Mendelson, Elliott, 1997. Introduction to Mathematical Logic, 4th ed. Chapman & Hall.
- Raymond Smullyan, 1991. Gödel's Incompleteness Theorems. Oxford Univ. Press.
- Raymond Smullyan, 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.