Incompleteness and Undecidability

Every theory we will consider will be in a language that has a name ṅ for each n ∈ ω. For example, the name for 2 ∈ ω in the language LΩ of arithmetic is the term “1 + 1”. The results stated will only apply to countable languages. This requirement is implicit in the hypothesis of the existence of an enumeration of all formulas with one free variable, for example. We also assume that formulas and terms are literally finite sequences of numbers, so that given any set S of L-formulas, we may reasonably ask whether S is recursive or not. Assume that formulas and terms are formed in the standard way, so that functions such as 〈φ, ψ〉 7→ φ ∧ ψ are recursive. Assume also that the map n 7→ ṅ is recursive. Definition 1.1. A relation R ⊆ ω is T -representable iff there exists an L-formula φ such that