Complete Theories

Mathematical Logic · Axiom Academy

Theories that decide every sentence: the pinnacle of logical completeness connecting syntax, semantics, and decidability. 1. Definition of Complete Theory Completeness means there are no "undecided" sentences—the theory takes a stand on every statement. This is a very strong property that most theories do not possess. Complete theories have a beautiful equivalent characterization in terms of models: This means that if a theory is complete, all its models "look the same" from the perspective of first-order logic. They may have different sizes or internal structures, but they cannot be distinguished by any first-order sentence. 3. Classic Examples of Complete Theories Example 1: Dense Linear Orders (DLO) The theory of dense linear orders without endpoints is complete. The language has a single binary relation < and the axioms state: Strict linear order: transitivity, irreflexivity, trichotomy Density: ∀x∀y (x < y → ∃z (x < z ∧ z < y)) No endpoints: ∀x ∃y (y < x) and ∀x ∃y (x < y) Examples: ℚ (rationals), ℝ (reals), and any countable dense linear order without endpoints are all elementarily equivalent! Example 2: Algebraically Closed Fields (ACF) For each prime p (or 0 for characteristic 0), the theory ACFₚ of algebraically closed fields of characteristic p is complete. This means the complex numbers ℂ and the algebraic closure of ℚ satisfy exactly the same first-order sentences in the language of fields! 4. How to Show a Theory is Complete

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