Coherent adequate sets and forcing square

John Krueger

Displaying similar documents to “Coherent adequate sets and forcing square”

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We sort out to a large extent when a (first order complete theory) T has a superlimit model in a cardinal λ. Also we deal with related notions of being limit.

Borel sets with large squares

Saharon Shelah (1999)

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For a cardinal μ we give a sufficient condition $\oplus_{μ}$ (involving ranks measuring existence of independent sets) for: $\otimes_{μ}$ if a Borel set B ⊆ ℝ × ℝ contains a μ-square (i.e. a set of the form A × A with |A| =μ) then it contains a $2^{ℵ_{0}}$ -square and even a perfect square, and also for $\otimes_{μ}^{'}$ if $ψ \in L_{ω_{1}, ω}$ has a model of cardinality μ then it has a model of cardinality continuum generated in a “nice”, “absolute” way. Assuming $M A + 2^{ℵ_{0}} > μ$ for transparency, those three conditions ( $\oplus_{μ}$ , $\otimes_{μ}$ and $\otimes_{μ}^{'}$ ) are equivalent, and from this we...

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Joram Hirschfeld (1976)

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Silvia Barbina, Domenico Zambella (2010)

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We give a self-contained introduction to universal homogeneous models (also known as rich models) in a general context where the notion of morphism is taken as primitive. We produce an example of an amalgamation class where each connected component has a saturated rich model but the theory of the rich models is not model-complete.

Killing GCH everywhere by a cofinality-preserving forcing notion over a model of GCH

Sy-David Friedman, Mohammad Golshani (2013)

Fundamenta Mathematicae

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Starting from large cardinals we construct a pair V₁⊆ V₂ of models of ZFC with the same cardinals and cofinalities such that GCH holds in V₁ and fails everywhere in V₂.