Abstract

We give definitions that distinguish between two notions of indiscernibility for a set {aη∣η∈ω>ω}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\{a_{\eta} \mid \eta \in ^{\omega>}\omega\}}$$\end{document} that saw original use in Shelah [Classification theory and the number of non-isomorphic models. North-Holland, Amsterdam, 1990], which we name s- and str−indiscernibility. Using these definitions and detailed proofs, we prove s- and str-modeling theorems and give applications of these theorems. In particular, we verify a step in the argument that TP is equivalent to TP1 or TP2 that has not seen explication in the literature. In the Appendix, we exposit the proofs of Shelah [Classification theory and the number of non-isomorphic models. North-Holland, Amsterdam, 1990, App. 2.6, 2.7], expanding on the details.