Tree indiscernibilities, revisited

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.Comment: submitte

Dense codense predicates and the NTP2

“…The following fact (the ‘array modeling’ theorem) has been observed (sometimes implicitly) by many researchers (for example, in ). A rigorous proof can be found in [, Theorem 5.5]. Theorem Given any set of tuples , there exists an indiscernible array based on .…”

“…Several variations of the notion indiscernible array exist in the literature [2,7,8,15]. In this paper, we shall follow the definition given in [15]. (The same notion is called strongly indiscernible array in [2,7,8].)…”

“…The following fact (the 'array modeling' theorem) has been observed (sometimes implicitly) by many researchers (for example, in [1,7,8,15]). A rigorous proof can be found in [15,Theorem 5.5]. j<ω ϕ(x,ā 0, j ) has at most finitely many realizations.…”

Indiscernibles, EM-Types, and Ramsey Classes of Trees

“…In [23] the below notion is named lokal wie. The same notion is referred to as based on in [16,10,18] . Below we promote a synthesis of the two names: (1) We say the J-indexed set (b i : i ∈ J) is L * -locally based on the a i (L * -locally based on I) if for any finite set of L-formulas, ∆, and for any finite tuple (t 1 , .…”

Generic trivializations of geometric theories

“…By [19,Lemma 5.8], we may assume that ( a η : η ∈ ω <ω ) is str-indiscernible. It follows, that for any choice of η ∈ ω <ω and i 0 < .…”