Externally definable sets and dependent pairs
Artem Chernikov, Pierre Simon
TL;DR
This paper proves that externally definable sets in NIP theories have honest definitions, offers insights into stable embeddedness, and applies these results to dependent pairs, answering a question on indiscernible sequences.
Contribution
It provides a new proof of Shelah's expansion theorem and extends understanding of dependent pairs in NIP theories.
Findings
Externally definable sets have honest definitions in NIP theories.
A weak notion of stable embeddedness is established.
The results answer a question on naming indiscernible sequences.
Abstract
We prove that externally definable sets in first order NIP theories have honest definitions, giving a new proof of Shelah's expansion theorem. Also we discuss a weak notion of stable embeddedness true in this context. Those results are then used to prove a general theorem on dependent pairs, which in particular answers a question of Baldwin and Benedikt on naming an indiscernible sequence.
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Taxonomy
TopicsAdvanced Topology and Set Theory · Computability, Logic, AI Algorithms · Philosophy and Theoretical Science