O-Minimality

Origin. van den Dries, Knight, Pillay, Steinhorn (1986). Tame geometry over the reals. Definable sets are finite unions of intervals. No pathological sets. Foundation for real algebraic geometry.

Models. Expansion of (ℝ, <). Definable subsets of ℝ are finite unions of points and intervals. "Order-minimal": simplest beyond order alone.

Formalism.

O-minimal structure: M = (ℝ, <, ...) where: Every definable X ⊆ ℝ is finite union of:

Points {a}
Open intervals (a,b)
Half-infinite intervals (-∞,a), (a,∞)
ℝ itself

Definable: Defined by first-order formula with parameters. X = {x ∈ ℝ | M ⊨ φ(x, a₁,...,aₙ)}

Examples of o-minimal structures: (ℝ, <, +, ·): real closed fields (ℝ, <, +, ·, exp): exponential field (ℝ_an): real analytic functions (ℝ_an,exp): analytic + exp

Cell decomposition: Definable sets in ℝⁿ decompose into cells. Cells: products of points and intervals. Finite decomposition.

Definable functions: Piecewise continuous. Eventually monotone. Finite number of local extrema.

Dimension theory: Well-behaved dimension. dim(X) defined topologically. Agrees with algebraic dimension.

Symbols.

Symbol	Unicode	Name	Meaning
(ℝ, <)	—	Ordered reals	Base structure
φ	U+03C6	Formula	Defining formula
dim	—	Dimension	Set dimension
∪	U+222A	Union	Finite union

Metatheory. Cell decomposition theorem. Definable choice. Dimension theory. Tameness theorems.

Applies to. Real algebraic geometry. Diophantine geometry. Analysis. Model theory. Differential equations.

Limitations. Only reals (not complex). Restricts definable sets. Special structure needed. Decidability varies.