We investigate some natural questions about the class of posets which can be embedded into ⟨ω,≤*⟩. Our main tool is a simple ccc forcing notion $H_E$ which generically embeds a given poset E into ⟨ω,≤*⟩ and does this in a "minimal" way (see Theorems 9.1, 10.1, 6.1 and 9.2).
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For any given ε > 0 we construct an ε-exhaustive normalized pathological submeasure. To this end we use potentially exhaustive submeasures and barriers of finite subsets of ℕ.
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We consider the question of whether 𝒫(ω) is a subalgebra whenever it is a quotient of a Boolean algebra by a countably generated ideal. This question was raised privately by Murray Bell. We obtain two partial answers under the open coloring axiom. Topologically our first result is that if a zero-dimensional compact space has a zero-set mapping onto βℕ, then it has a regular closed zero-set mapping onto βℕ. The second result is that if the compact space has density at most ω₁, then it will map onto βℕ if it contains a zero-set that maps onto βℕ.
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We prove there is a countable dense homogeneous subspace of ℝ of size ℵ₁. The proof involves an absoluteness argument using an extension of the $L_{ω₁ω}(Q)$ logic obtained by adding predicates for Borel sets.
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We examine the properties of existentially closed ($𝓡 ^{ω}$-embeddable) II₁ factors. In particular, we use the fact that every automorphism of an existentially closed ($𝓡 ^{ω}$-embeddable) II₁ factor is approximately inner to prove that Th(𝓡) is not model-complete. We also show that Th(𝓡) is complete for both finite and infinite forcing and use the latter result to prove that there exist continuum many nonisomorphic existentially closed models of Th(𝓡).
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