General non-structure theory and constructing from linear orders

2023/05/03 by Saharon Shelah, Shelah, Saharon · 1 citation
Mathematics · #03E75 #03G05 #Advanced Topology and Set Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO) #Primary 03C55 #Secondary 03E05

paper · pdf · doi:10.48550/arxiv.2305.02003

openalex publication_date 2023/05/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The theme of the first two sections, is to prepare the framework of how from a ``complicated'' family of so called index models I ∈ K₁ we build many and/or complicated structures in a class K₂. The index models are characteristically linear orders, trees with κ+1 levels (possibly with linear order on the set of successors of a member) and linearly ordered graphs; for this we formulate relevant complicatedness properties (called bigness). In the third section we show stronger results concerning linear orders. If for each linear order I of cardinality λ> ℵ₀ we can attach a model M_I ∈ K_λ in which the linear order can be embedded such that for enough cuts of I, their being omitted is reflected in M_I, then there are 2^λ non-isomorphic cases. We also do the work for some applications.

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• Twins: non-isomorphic models forced to be isomorphic. Part I

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