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).

We use Tsirelson’s Banach space ([2]) to define an ${F}_{\sigma}$ P-ideal which refutes a conjecture of Mazur and Kechris (see [12, 9, 8]).

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.

We shall show that Open Coloring Axiom has different influence on the algebra $\mathcal{P}\left(\mathbb{N}\right)/fin$ than on ${\mathbb{N}}^{\mathbb{N}}/fin$. The tool used to accomplish this is forcing with a Suslin tree.

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...

We examine the properties of existentially closed (${}^{\omega}$-embeddable) II₁ factors. In particular, we use the fact that every automorphism of an existentially closed (${}^{\omega}$-embeddable) II₁ factor is approximately inner to prove that Th() is not model-complete. We also show that Th() 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().

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 ${L}_{\omega \u2081\omega}\left(Q\right)$ logic obtained by adding predicates for Borel sets.

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