Signal Recovery Beyond Linear Estimates

This chapter explains why Σ‎0 1 formulas of Peano arithmetic (PA) capture all computably enumerable sets, as claimed by Alonzo Church's thesis from the previous chapter. This allows us to capture “computable analysis” in the language of PA, since computable sets and functions are definable in terms of computable enumerability. To justify the claim that Σ‎0 1 = “computably enumerable,” this chapter makes a thorough analysis of the concept of computation. It takes a precise, but intuitively natural, concept of computation and translates it into the language of PA. The chapter demonstrates that the translation is indeed Σ‎0 1, but with a slightly different (though equivalent) definition of Σ‎0 1.