Building models of strongly minimal theories

Calibrating Computational Complexity via Definability: The Work of Julia F. Knight

“…One structure studied extensively in logic is the standard model of arithmetic ℕ, whose domain is 𝜔, the set of natural numbers. 1 We'll fix the language of ℕ as 𝐿 ℕ = {𝑆, +, ⋅, <, 0} so ℕ comes equipped with the functions giving the successor operation 𝑥 → 𝑥 + 1, addition, multiplication, the usual order relation, and a constant for 0.…”

“…Several of Knight's early papers (e.g., [13]) explored "omitting" collections of formulas in this sense. 1 The symbol 𝜔 comes from viewing the set of natural numbers as the first infinite ordinal.…”

“…A formula of this kind is Σ 1 1 (resp., Π 1 1 ) if it has only existential (resp., universal) set quantifiers. 6 The hyperarithmetical sets coincide with the Δ 1 1 sets, those that are definable in ℕ by both Σ 1 1 and Π 1 1 formulas. Computable infinitary formulas are also the "right" logic for understanding computable structures (and hyperarithmetical ones too).…”

“…Knight, together with a large team, also studied the class of high rank structures using the tools described earlier. They showed that the index set of the class of high rank structures is as complicated as possible, specifically Σ 1 1 -complete [3]. 4.3.…”

“…Knight would continue to explore ideas connected to those in §2.1. If a theory 𝑇 (such as 𝑇𝐴) has a computable model, the set of Σ 0 𝑛 statements in 𝑇 is Σ 0 𝑛 uniformly in 𝑛. Andrews, Lempp, and Schweber called such theories Solovay theories (see their paper [1] for all references in this section). Earlier, Knight and Solovay proved that a set computes a copy of some model of every Solovay theory if and only if it computes a nonstandard model of 𝑇𝐴.…”