In this paper, countable compactness and the Lindel¨of propertyare defined for L-fuzzy sets, where L is a complete de Morgan algebra. Theydon’t rely on the structure of the basis lattice L and no distributivity is requiredin L. A fuzzy compact L-set is countably compact and has the Lindel¨ofproperty. An L-set having the Lindel¨of property is countably compact if andonly if it is fuzzy compact. ...

In this paper, we prove the following statements: (1) There exists a Tychonoff star countable discrete closed, pseudocompact space having a regular-closed subspace which is not star countable. (2) Every separable space can be embedded into an absolutely star countable discrete closed space as a closed subspace. (3) Assuming 20 = 21 , there exists a normal absolutely star countable discrete clos...

A subgroup A of the ^-primary group G is nice if p(G/A) = {pG, A }/A for all ordinals a. We consider the following countability condition: there exists a collection G of nice subgroups of G such that (0) oee. (1) 6 is closed with respect to group-theoretic union in G. (2) If A^e and if H is a subgroup of G such that {Af H}/A is countable, there exists i ? £ G such that 5 2 {-4, H\ and such that...

Many people believe that there is a Dutch Book argument establishing that the principle of countable additivity is a condition of coherence. De Finetti himself did not, but for reasons that are at first sight perplexing. I show that he rejected countable additivity, and hence the Dutch Book argument for it, because countable additivity conflicted with intuitive principles about the scope of aut...

The field of descriptive set theory is mainly concerned with studying subsets of the space of all countable binary sequences. In this paper we study the generalization where countable is replaced by uncountable. We explore properties of generalized Baire and Cantor spaces, equivalence relations and their Borel reducibility. The study shows that the descriptive set theory looks very different in...

One of the most basic measures of the complexity of a given partially ordered structure is the quantity of partial orderings embeddable in this structure. In the structure of the Turing degrees, DT , this problem is investigated in a series of results: Mostowski [15] proves that there is a computable partial ordering in which every countable partial ordering can be embedded. Kleene and Post [10...

We prove that any infinite sequence of countable series-parallel orders contains an increasing (with respect to embedding) infinite subsequence. This result generalizes Laver’s and Corominas’ theorems concerning better-quasi-order of the classes of countable chains and trees. Let C be a class of structures and ≤ an order on C. This class is well-quasi-ordered with respect to ≤ if for any infini...

A tame theory is a decidable first-order theory with only countably many countable models, and all complete types recursive. It is shown here that the recursive complexity of countable homogeneous models of tame theories is unbounded in the hyperarithmetic hierarchy. Recursive model theory considers model-theoretic objects and constructions from the point of view of recursion-theoretic complexi...

We relate the existence problem of universal objects to the properties of corresponding enriched categories (lifts and expansions). Particularly, extending earlier results, we prove that for every countable set F of finite structures there exists a (countably) universal structure U for the class Forbh(F) (of all countable structures omitting a homomorphism from all members of F). In fact U is t...