We show that K-generic projective planes have Morley rank either two or infinity. We also show give a direct argument that such planes are not Desarguesian.

We study the following conjecture. Conjecture. Let $T$ be an $\omega$-stable theory with continuum many countable models. Then either i) $T$ has continuum many complete extensions in $L_1$, or ii) some complete extension of $T$ in $L_1$ has continuum many $L_1$-types without parameters. By Shelah's proof of Vaught's conjecture for $\omega$-stable theories, we know that there are seven types of $\omega$-stable theory with continuum many countable models. We show that the conjecture is true for all but one of these seven (...) cases. In the last case we show the existence of continuum many $L_2$-types. (shrink)

The notion of quasi-minimal structures was defined by B. Zil'ber as a natural generalization of minimal structures. Inspired by his work, we study here basic model theoretic properties of quasiminimal structures. Main result is the construction of ω-saturated quasi-minimal models under ω-stability assumption.

数理論理学の基本となる命題・述語論理から、チューリング機械・形式手法・ブール代数といった「少し先」の内容までを丁寧に解説。.

We study the following conjecture. Conjecture. Let T be an ω-stable theory with continuum many countable models. Then either i) T has continuum many complete extensions in L1(T), or ii) some complete extension of T in L1 has continuum many L1-types without parameters. By Shelah's proof of Vaught's conjecture for ω-stable theories, we know that there are seven types of ω-stable theory with continuum many countable models. We show that the conjecture is true for all but one of these seven (...) cases. In the last case we show the existence of continuum many L2-types. (shrink)

We show that is a quasi-minimal torsion-free divisible abelian group. After discussing the axiomatization of the theory of this structure, we present its ω-saturated quasi-minimal model.