Simple monadic theories and indiscernibles

Abstract

Aiming for applications in monadic second-order model theory, we study first-order theories without definable pairing functions. Our main results concern forking-properties of sequences of indiscernibles. These turn out to be very well-behaved for the theories under consideration.

Simple monadic theories and partition width

“…We have shown in 8 that, in structures which do not admit coding, indiscernible sequences are well‐behaved. Before giving the precise statements, let us introduce some notation for sequences and recall some basic facts concerning indiscernible sequences.…”

“…Lemma 2.8 (8) Let κ be an infinite cardinal, Δ a set of formulae of size |Δ| ⩽ κ, and A , B ⊆ M sets. If \documentclass{article}\usepackage{amsmath,amssymb,amsfonts}\begin{document}\pagestyle{empty}${\rm ti}^n_\Delta (A/B) > 2^\kappa$\end{document} then there exist a formula \documentclass{article}\usepackage{amsmath,amssymb,amsfonts}\begin{document}\pagestyle{empty}$\varphi (\bar{x},\bar{y}) \in \Delta$\end{document}, a number m < ω, and tuples \documentclass{article}\usepackage{amsmath,amssymb,amsfonts}\begin{document}\pagestyle{empty}$\bar{a}^v \in A^n$\end{document} and \documentclass{article}\usepackage{amsmath,amssymb,amsfonts}\begin{document}\pagestyle{empty}$\bar{b}^v \in B^m$\end{document}, for v < κ + , such that …”

“…In 1 Baldwin and Shelah started an investigation of the monadic theories of structures without pairing function. Continuing their work we studied indiscernible sequences in 8. Let us briefly summarise these results.…”

“…Building on techniques developed in 7, 8 we approached Seese's conjecture by considering a weaker statement about first‐order theories and applying standard tools from first‐order model theory. Instead of grids we consider first‐order definable pairing functions and we will prove in Theorem 6.3 below that every structure where there is no pairing function is tree‐like (as defined below).…”

“…Sections 3 to 5 constitute the main technical body of the article. Section 2 summarises the results of 8 about indiscernible sequences. Starting from these, we present more detailed structure theorems for indiscernible sequences in structures without definable pairing functions in Section 2.…”