On VC-density in VC-minimal theories
Vincent Guingona
TL;DR
This paper proves that in VC-minimal theories, formulas with two variables have VC-codensity at most two, and in certain cases, the VC-codensity equals the number of free variables, providing new bounds and proofs.
Contribution
It establishes new bounds on VC-codensity for formulas in VC-minimal theories and offers a novel proof for the general case when acl=dcl.
Findings
Formulas with two variables have VC-codensity ≤ 2 in VC-minimal theories.
In VC-minimal theories with acl=dcl, VC-codensity equals the number of free variables.
Provides a new proof for the VC-codensity bound in these theories.
Abstract
We show that any formula with two free variables in a VC-minimal theory has VC-codensity at most two. Modifying the argument slightly, we give a new proof of the fact that, in a VC-minimal theory where acl = dcl, the VC-codensity of a formula is at most the number of free variables.
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Taxonomy
TopicsAdvanced Topology and Set Theory · Black Holes and Theoretical Physics · Mathematical Dynamics and Fractals