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The Marker-Steinhorn Theorem

Pablo Andújar Guerrero

TL;DR

The paper provides a gap-free proof of the Marker-Steinhorn Theorem in o-minimal structures, clarifying and consolidating prior arguments that contained gaps. It introduces definable preorders on a family of partial definable functions associated with a type and reduces definability questions to the definability of cuts in a linear preorder, enabling a direct, uniform definability argument. The core technical contribution constructs and analyzes objects such as $P$, $B$, $B^*$, $P^*$, and the index sets $I^*(0)$, $I^*(1)$, $I^*(2)$ to force definability of the relevant sets, circumventing earlier case-by-case complexity and reliance on regular cell decomposition. The work also discusses potential strengthening results and poses questions about extending the approach beyond the field case to broader o-minimal theories, highlighting the significance for external definability and tame extensions in model theory.

Abstract

We give a proof of the Marker-Steinhorn Theorem which fills a gap in previous proofs of the result.

The Marker-Steinhorn Theorem

TL;DR

The paper provides a gap-free proof of the Marker-Steinhorn Theorem in o-minimal structures, clarifying and consolidating prior arguments that contained gaps. It introduces definable preorders on a family of partial definable functions associated with a type and reduces definability questions to the definability of cuts in a linear preorder, enabling a direct, uniform definability argument. The core technical contribution constructs and analyzes objects such as , , , , and the index sets , , to force definability of the relevant sets, circumventing earlier case-by-case complexity and reliance on regular cell decomposition. The work also discusses potential strengthening results and poses questions about extending the approach beyond the field case to broader o-minimal theories, highlighting the significance for external definability and tame extensions in model theory.

Abstract

We give a proof of the Marker-Steinhorn Theorem which fills a gap in previous proofs of the result.
Paper Structure (9 sections, 3 theorems, 4 equations)

This paper contains 9 sections, 3 theorems, 4 equations.

Table of Contents

  1. Introduction
  2. The Marker-Steinhorn Theorem
  3. Conventions
  4. The gap
  5. Proof of the Marker-Steinhorn Theorem
  6. Preorders
  7. The proof
  8. Acknowledgements
  9. Funding

Key Result

Lemma 2.1

Let $(B,\preceq)$ be a definable linear preorder and $(P,Q)$ be a cut. Fix $m=\dim B$, and suppose that every definable subset $C \subseteq B$ satisfying that $C\cap P$ is cofinal in $P$ has dimension $m$. Let $\{C_x : x\in X\}$ be a definable family of pairwise disjoint subsets of $B$, each of dime

Theorems & Definitions (11)

  • Example 1.1
  • Lemma 2.1
  • proof
  • Claim 2.1.1
  • Lemma 2.2
  • proof
  • Theorem 2.3: Marker-Steinhorn Theorem mark_stein_94
  • proof
  • Claim 2.3.1
  • Claim 2.3.2
  • ...and 1 more