In mathematical logic, Gödel's β function is a function used to permit quantification over finite sequences of natural numbers in formal theories of arithmetic. The β function is used, in particular, in showing that the class of arithmetically definable functions is closed under primitive recursion, and therefore includes all primitive recursive functions. The β function was introduced without the name in the proof of the first of Gödel's incompleteness theorems (Gödel 1931). The β function lemma given below is an essential step of that proof. Gödel gave the β function its name in (Gödel 1934).
Definition The β {\displaystyle \beta } function takes three natural numbers as arguments. It is defined as
β ( x 1 , x 2 , x 3 ) = r e m ( x 1 , 1 + ( x 3 + 1 ) ⋅ x 2 ) = r e m ( x 1 , ( x 3 ⋅ x 2 + x 2 + 1 ) ) , {\displaystyle \beta (x_{1},x_{2},x_{3})=\mathrm {rem} (x_{1},1+(x_{3}+1)\cdot x_{2})=\mathrm {rem} (x_{1},(x_{3}\cdot x_{2}+x_{2}+1)),}
where r e m ( x , y ) {\displaystyle \mathrm {rem} (x,y)} denotes the remainder after integer division of x {\displaystyle x} by y {\displaystyle y} (Mendelson 1997:186).
Special schema without parameters The β function is arithmetically definable in an obvious way, because it uses only arithmetic operations and the remainder function which is arithmetically definable. It is therefore representable in Robinson arithmetic and stronger theories such as Peano arithmetic. By fixing the first two arguments appropriately, one can arrange that the values obtained by varying the final argument from 0 to n run through any specified (n+1)-tuple of natural numbers (the β lemma described in detail below). This allows simulating the quantification over sequences of natural numbers of arbitrary length, which cannot be done directly in the language of arithmetic, by quantification over just two numbers, to be used as the first two arguments of the β function. For example, if f is a function defined by primitive recursion on a recursion variable n, say by f(0) = c and f(n+1) = g(n, f(n)), then to express f(n) = y one would like to say: there exists a sequence a0, a1, ..., an such that a0 = c, an = y and for all i < n one has g(i, ai) = ai+1. While that is not possible directly, one can say instead: there exist natural numbers a and b such that β(a,b,0) = c, β(a,b,n) = y and for all i < n one has g(i, β(a,b,i)) = β(a,b,i+1).
General schema with parameters The primitive recursion schema as given may be replaced by one which makes use of fewer parameters. Let w {\displaystyle w} be an elementary pairing function, and π 1 , π 2 {\displaystyle \pi _{1},\pi _{2}} be its projection functions for inversion. Theorem: Any function constructible via the clauses of primitive recursion using the standard primitive recursion schema is constructible when the schema is replaced with the following.
f ′ ( x , 0 ) = g ′ ( x ) {\displaystyle f'(x,0)=g'(x)}
f ′ ( x , y + 1 ) = h ′ ( f ′ ( x , y ) ) {\displaystyle f'(x,y+1)=h'(f'(x,y))}
This is proven by providing two intermediate schemata for primitive recursion, starting with a function defined via the standard schema, and translating the definition into terms of each intermediate schema and finally into terms of the above schema. The first intermediate schemata is as follows:
f 1 ( x , 0 ) = g 1 ( x ) {\displaystyle f_{1}(x,0)=g_{1}(x)}
f 1 ( x , y + 1 ) = h 1 ( x , y , f 1 ( x , y ) ) {\displaystyle f_{1}(x,y+1)=h_{1}(x,y,f_{1}(x,y))}
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