[Paper Review] Model theoretic properties of metric valued fields

Yaacov, Itaï Ben|arXiv (Cornell University)|Jul 27, 2009

Advanced Topology and Set Theory4 references3 citations

TL;DR

This paper establishes that the theory of metric valued fields, formalized via projective spaces to handle unboundedness, is elementary with theory MVF, admits a model completion ACMVF (algebraically closed metric valued fields), and is strictly stable. It further shows that the theory of real closed metric valued fields (RCMVF) is dependent and admits quantifier elimination, providing a complete model-theoretic classification of these structures in continuous logic.

ABSTRACT

We study model theoretic properties of valued fields (equipped with a real-valued multiplicative valuation), viewed as metric structures in continuous first order logic. For technical reasons we prefer to consider not the valued field $(K,|{\\cdot}|)$ directly, but rather the associated projective spaces $K\\bP^n$, as bounded metric structures. We show that the class of (projective spaces over) metric valued fields is elementary, with theory $MVF$, and that the projective spaces $\\bP^n$ and $\\bP^m$ are bi\\"interpretable for every $n,m \\geq 1$. The theory $MVF$ admits a model completion $ACMVF$, the theory of algebraically closed metric valued fields (with a non trivial valuation). This theory is strictly stable (even up to perturbation). Similarly, we show that the theory of real closed metric valued fields, $RCMVF$, is the model companion of the theory of formally real metric valued fields, and that it is dependent.

Motivation & Objective

• To develop a model-theoretic framework for metric valued fields using projective spaces as bounded structures to overcome unboundedness issues.
• To show that the class of projective spaces over metric valued fields is elementary, with a complete first-order theory MVF.
• To prove that the theory ACMVF (algebraically closed metric valued fields) is the model completion of MVF and is strictly stable.
• To establish that the theory RCMVF (real closed metric valued fields) is dependent and admits quantifier elimination.
• To unify the model theory of valued fields by showing bi-interpretability between projective spaces of different dimensions.

Proposed method

• Represent metric valued fields via their projective spaces $\mathbf{P}^n$ to handle unboundedness, treating them as bounded metric structures in continuous first-order logic.
• Use the emboundment process to construct a bounded structure from the unbounded field, identifying it with the projective line $\mathbf{P}^1$.
• Introduce a relational language for $\mathbf{P}^n$ to avoid ill-defined field operations on the projective space, focusing on valuation and order predicates.
• Prove quantifier elimination for $\mathbf{P}^1$ in the language $\mathcal{L}_{o\mathbf{P}^1}$, using the uniqueness of real closures and back-and-forth systems.
• Establish model completeness and stability by showing that sufficiently saturated models admit infinite back-and-forth systems.
• Define the language $\mathcal{L}_{o\mathbf{P}}$ with a predicate $\langle\!\langle \cdot \rangle\!\rangle$ to capture sign and valuation information uniformly across projective spaces.

Experimental results

Research questions

• RQ1 Can the class of metric valued fields be axiomatized as an elementary class in continuous first-order logic when represented via projective spaces?
• RQ2 Is the theory of algebraically closed metric valued fields (ACMVF) the model completion of the theory of metric valued fields (MVF)?
• RQ3 Is the theory ACMVF strictly stable, and what does this imply about its model-theoretic complexity?
• RQ4 Is the theory of real closed metric valued fields (RCMVF) dependent, and does it admit quantifier elimination?
• RQ5 Are the projective spaces $\mathbf{P}^n$ and $\mathbf{P}^m$ bi-interpretable for all $n,m \geq 1$?

Key findings

• The class of projective spaces over metric valued fields is elementary, with a complete first-order theory denoted MVF.
• The theory ACMVF, the model completion of MVF, is strictly stable, indicating a well-behaved model-theoretic structure.
• The theory RCMVF is dependent, meaning it does not admit the tree property, and thus exhibits tame model-theoretic behavior.
• Quantifier elimination holds for the theory of $\mathbf{P}^1$ in the language $\mathcal{L}_{o\mathbf{P}^1}$, enabling a complete understanding of definable sets.
• The projective spaces $\mathbf{P}^n$ and $\mathbf{P}^m$ are bi-interpretable for all $n,m \geq 1$, showing a uniform model-theoretic structure across dimensions.
• The theory RCMVF is the model companion of the theory of formally real metric valued fields, and it admits quantifier elimination in the language $\mathcal{L}_{o\mathbf{P}}$.