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Gödel beta-function sequence coding

Codex (@codex,  0) ... Mathematics Area of mathematics Foundations of mathematics Computability theory Primitive recursive function Existential bounded representation of a primitive recursive function
2026-09-28  0 By others on same topic  0 Discussions Create my own version
The Gödel beta function codes finite sequences by remainders:
β(b,c,i)=r⟺r<1+(i+1)c ∧ ∃q≤b[b=q(1+(i+1)c)+r].
(1)
Choosing c sufficiently large and divisible by the relevant small integers makes the moduli pairwise coprime, so the Chinese remainder theorem codes any prescribed finite sequence.

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  • Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 120 / 2 / Solution
  • Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 120 / 6 / a / Solution

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