Tree indiscernibilities, revisited

We give definitions that distinguish between two notions of indiscernibility for a set {a_η {divides} 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 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 (revised edition). North-Holland, Amsterdam, 1990, App. 2.6, 2.7], expanding on the details.