Tree indiscernibilities, revisited

• Published in: Archive for mathematical logic Vol. 53; no. 1-2; pp. 211 - 232
• Main Authors: Kim, Byunghan, Kim, Hyeung-Joon, Scow, Lynn
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2014; Springer Nature B.V
• Subjects: Algebra; Applied mathematics; Archives; Mathematical analysis; Mathematical Logic and Foundations; Mathematics; Mathematics and Statistics; Set theory; Theorems; Type Property; Partial Type; Tree Property; Induction Step; Transitive Property
• ISSN: 0933-5846; 1432-0665
• DOI: 10.1007/s00153-013-0363-6

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.