Tree indiscernibilities, revisited

We give definitions that distinguish between two notions of indiscernibility for a set { a η ∣ η ∈ ω > ω } that saw original use in Shelah [Classification theory and the number of non-isomorphic models (revised edition). 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 TP 1 or TP 2 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 (revised edition). North-Holland, Amsterdam, 1990 , App. 2.6, 2.7], expanding on the details.