Questions: Definability and Applications to Algebraic Geometry

Question 1

Which of the following sets is definable in the structure (ℝ, +, ×, 0, 1) by a first-order formula?

• A. The Cantor set (a nowhere-dense uncountable subset of [0,1])
• B. The unit circle {(x, y) : x² + y² = 1}
• C. The set of transcendental numbers
• D. The set of prime natural numbers within ℝ

Answer: (B)

The unit circle is defined by the polynomial equation x² + y² = 1, which is a quantifier-free first-order formula in the language of ordered fields. It is therefore definable — in fact, semialgebraic. The Cantor set cannot be defined by any first-order formula in this structure: o-minimality of the reals guarantees that every definable subset of ℝ is a finite union of intervals and points, ruling out Cantor-like sets. The transcendental numbers and the primes also fail to be definable in (ℝ, +, ×, 0, 1) — the primes require the natural number predicate, which is not first-order definable in this structure.

Question 2

A geometer wants to show that the image of a semialgebraic set in ℝ³ under projection onto ℝ² is again semialgebraic. Which model-theoretic result directly justifies this?

• A. The compactness theorem, which ensures that any consistent set of formulas has a model
• B. The Tarski-Seidenberg theorem (quantifier elimination for real closed fields), which shows that existential quantification over a semialgebraic set yields a semialgebraic set
• C. O-minimality of (ℝ, <), which restricts definable subsets of the line to finite unions of intervals
• D. The upward Löwenheim-Skolem theorem, which guarantees models of all infinite cardinalities

Answer: (B)

Projection corresponds to existential quantification: the image of S ⊆ ℝ³ under projection to ℝ² is {(x,y) : ∃z ((x,y,z) ∈ S)}. If S is semialgebraic (defined by a quantifier-free formula), quantifier elimination (Tarski-Seidenberg) converts the existential formula to an equivalent quantifier-free one, which is by definition semialgebraic. This single observation — that the image of a semialgebraic set is semialgebraic — is a cornerstone of real algebraic geometry, and it is a direct consequence of quantifier elimination.

Question 3

In an o-minimal structure, definable subsets of the real line can include Cantor-set-like constructions — nowhere-dense sets with uncountably many points.

• T. True
• F. False

Answer: False

O-minimality is precisely the condition that forbids this. A structure (M, <, …) is o-minimal if every definable subset of M using one variable is a finite union of points and open intervals. This rules out the Cantor set, which is uncountable but nowhere dense and cannot be expressed as a finite union of intervals and points. O-minimality is a 'tameness' condition: it ensures that definable sets have the topological complexity of semi-algebraic sets, not of arbitrary Borel sets.

Question 4

In the real closed field (ℝ, +, ×, 0, 1), the projection of a semialgebraic set onto a lower-dimensional space is guaranteed to be semialgebraic, by the Tarski-Seidenberg theorem.

• T. True
• F. False

Answer: True

This is one of the most geometrically useful consequences of quantifier elimination for real closed fields. Projection corresponds to existential quantification, and quantifier elimination converts any existential formula over a semialgebraic set into a quantifier-free formula — which is the definition of semialgebraic. Without this result, real algebraic geometry would lose one of its most basic closure properties: the constructible sets could fail to be closed under projection, making geometric arguments far more difficult.

Question 5

What is a definable set in the sense of model theory, and why does the concept of o-minimality matter for applying model-theoretic results to geometry?

Model answer: A definable set in a structure M is the solution set of a first-order formula with parameters from M — the collection of all n-tuples satisfying some formula φ(x₁, …, xₙ). In a geometric structure like (ℝ, +, ×, 0, 1), definable sets are exactly the semialgebraic sets: finite Boolean combinations of polynomial equations and inequalities. O-minimality matters because it imposes a tameness condition — every definable subset of the line is a finite union of intervals and points — which forces definable sets in higher dimensions to decompose into finitely many 'cells' with controlled topological complexity. This finiteness makes definable geometry tractable: one can count connected components, bound dimensions, and prove uniform finiteness theorems over definable families, enabling applications like the Pila-Wilkie rational point counting theorem used in diophantine geometry.

Without tameness, model-theoretic definability would not translate into geometric control. O-minimality is the bridge: it takes the abstract model-theoretic notion of definability and converts it into a concrete geometric regularity condition, explaining why model theory can prove things about algebraic varieties that purely algebraic methods find difficult to reach.