Embedding and Coding below a 1-Generic Degree

Abstract

We show that the theory of 𝒟(≤ g), where g is a 2-generic or a 1-generic degree below 0ʹ, interprets true first-order arithmetic. To this end we show that 1-genericity is sufficient to find the parameters needed to code a set of degrees using Slaman and Woodin's method of coding in Turing degrees. We also prove that any recursive lattice can be embedded below a 1-generic degree preserving top and bottom.