Tree indiscernibilities, revisited

TL;DR

This paper revisits the concepts of indiscernibility in model theory, providing formal definitions, proofs of modeling theorems, and applications including clarifying the equivalence of certain tree properties in logic.

Contribution

It introduces precise definitions of two types of indiscernibility, proves their modeling theorems, and applies these results to clarify key logical equivalences.

Findings

Established formal definitions of nd -indiscernibility.

Proved nd -modeling theorems.

Verified a step in the equivalence of TP, TP1, and TP2.

Abstract

We give definitions that distinguish between two notions of indiscernibility for a set $\{a_\eta \mid \eta \in \W\}$ that saw original use in \cite{sh90}, which we name \textit{$\s$-} and \textit{$\n$-indiscernibility}. Using these definitions and detailed proofs, we prove $\s$- and $\n$-modeling theorems and give applications of these theorems. In particular, we verify a step in the argument that TP is equivalent to TP$_1$ or TP$_2$ that has not seen explication in the literature. In the Appendix, we exposit the proofs of \citep[{App. 2.6, 2.7}]{sh90}, expanding on the details.