Gödel's Beta Function Lemma

Theorem
Let $\beta: \N^3 \to \N$ be Gödel's $\beta$ function.

Let $\sequence {x_0, x_1, \dotsc, x_n }$ be a finite sequence of natural numbers.

Then there exist some $a, b \in \N$ such that, for every $i \le n$:
 * $\map \beta {a, b, i} = x_i$

Proof
Let $M = \map \max {n, x_0, x_1, \dotsc, x_n}$.

Define $b = M!$, the factorial of $M$.

Let $\sequence {y_1, y_2, \dotsc, y_{n + 1} }$ be the finite sequence defined by:
 * $y_i = 1 + \paren {i \times b}$

As $M \ge n$, by definition:
 * $M \ge \paren {n + 1} - 1$

Therefore, by Multiples of Factorial Plus One are Coprime, the $\sequence {y_1, \dotsc, y_{n + 1} }$ are pairwise coprime.

Thus, by the Chinese Remainder Theorem, there exists some $a \in \N$ such that:

But then, for each $0 \le i \le n$:

Therefore, by the congruence:
 * $\map \rem {a, y_{i + 1} } = x_i$

But:
 * $y_{i + 1} = 1 + \paren {\paren {i + 1} \times b}$

Thus:
 * $\map \rem {a, 1 + \paren {\paren {i + 1} \times b} } = x_i$

But that is the definition of the $\beta$ function.

Hence the result.