Never used this site before so not sure the best way to get help. However, I’ve been looking at many-one reductions in relations to sentences in logic. I wondered if there was a way to prove the following:

TH(N) = {ϕ : ϕ is a first order sentence in the language of arithmetic and N |= ϕ} (Where N is the standard model of arithmetic – , x, <, +, 0, and the successor; S)

Let γ(x) be some formula with exactly one free variable, namely x. Then assume X = {n : γ(n)} to be the arithmetical subset of the natural numbers defined by γ(x).

Is there a way to show either of X or TH(N) many one-reduce to each other?