We prove that the existence of the integral closure of a countable commutative ring R in a countable commutative ring S is equivalent to Arithmetical Comprehension (over RCA0). We also show that i) the Lying Over ii) the Going Up theorem for integral extensions of countable commutative rings and iii) the Going Down theorem for integral extensions of countable domains R ⊂ S, with R normal, are p...

In a recent paper, novel class of generalized compact sets (briefly, g-Tg -compact sets) in topological spaces Tg -spaces) has been studied. this the concept is further studied and, other derived concepts called countable, sequential, and local compactness (countable, -compactness) -spaces are also relatively. The study reveals that -compactness implies countable g-Tg-compactness, sequential pr...

Nagata conjectured that everyM -space is homeomorphic to a closed subspace of the product of a countably compact space and a metric space. This conjecture was refuted by Burke and van Douwen, and A. Kato, independently. However, we can show that there is a c.c.c. poset P of size 2 such that in V P Nagata’s conjecture holds for each first countable regular space from the ground model (i.e. if a ...

In [Vau61] Vaught conjectured that a countable first order theory has countably many or 2א0 many countable models. Here, the following special case is proved. Theorem. If T is a superstable theory of finite rank with < 2א0 many countable models, then T has countably many countable models. The basic idea is to associate with a theory a ∧ − definable group G (called the structure group) which con...

in this paper we find sufficient conditions on primitive inverse topological semigroup s under which: the inversion inv : (h(s)) (h(s)) is continuous; we show that every topologically periodic countable compact primitive inverse topological semigroups with closed h-classes is topologically isomorphic to an orthogonal sum p i2= bi (gi) of topological brandt extensions bi (gi) of countably compac...

Vaught’s conjecture says that for any countable (complete) first-order theory T, the number of non-isomorphic countable models of T is either countable or 2, where ω is the first infinite cardinal. Vaught’s conjecture for ω-stable theories of modules was proved by Garavaglia [6, Theorem 6]. Buechler proved that Vaught’s conjecture is correct for modules of U-rank 1 [2]. It has been shown that V...

The main theorem of this article is that every countable model of set theory 〈M,∈ 〉, including every well-founded model, is isomorphic to a submodel of its own constructible universe 〈L ,∈ 〉. In other words, there is an embedding j : M → L that is elementary for quantifier-free assertions. It follows from the proof that the countable models of set theory are linearly pre-ordered by embeddabilit...

We show that the isomorphism relation for countable Boolean algebras is Borel complete, i.e., the isomorphism relation for arbitrary countable structures is Borel reducible to that for countable Boolean algebras. This implies that Ketonen’s classification of countable Boolean algebras is optimal in the sense that the kind of objects used for the complete invariants cannot be improved in an esse...