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Abstract: In the realm of Lindelöf metric spaces the following results are obtained in ZF: (i) If X = ( X , d ) is a Lindelöf metric space then it is both densely Lindelöf and almost Lindelöf. In addition, under the countable axiom of choice CAC, the three notions coincide. (ii) The statement “every separable metric space is almost Lindelöf” implies that every infinite subset of R has a countably infinite subset). (iii) The statement “every almost Lindelöf metric space X = ( X , d ) is quasi totally bounded implies CAC ω. (iv) The proposition “every quasi totally bounded metric space is separable” lies, in the deductive hierarchy of choice principles, strictly between the countable union theorem CUC and CAC. Likewise, the statement “every pre‐Lindelöf (or Lindelöf) metric space is separable” lies strictly between CAC ( R ) and CAC. [ABSTRACT FROM AUTHOR]

We present various results regarding the decidability of certain sets of sentences by Simple Type Theory ( [ABSTRACT FROM AUTHOR]

In set theory without the axiom of choice [ABSTRACT FROM AUTHOR]

A set x is Dedekind infinite if there is an injection from ω into x; otherwise x is Dedekind finite. A set x is power Dedekind infinite if [ABSTRACT FROM AUTHOR]

Given a topological Ramsey space [ABSTRACT FROM AUTHOR]

We define the notion of Euler characteristic for definable quotients in an arbitrary o-minimal structure and prove some fundamental properties. [ABSTRACT FROM AUTHOR]

Quasiminimal structures play an important role in non-elementary categoricity. In this paper we explore possibilities of constructing quasiminimal models of a given first-order theory. We present several constructions with increasing control of the properties of the outcome using increasingly stronger assumptions on the theory. We also establish an upper bound on the Hanf number of the existence of arbitrarily large quasiminimal models. [ABSTRACT FROM AUTHOR]

We present a direct proof of the consistency of the existence of a five element basis for the uncountable linear orders. Our argument is based on the approach of König, Larson, Moore and Veličković and simplifies the original proof of Moore. [ABSTRACT FROM AUTHOR]

We investigate questions involving Aronszajn trees, square principles, and stationary reflection. We first consider two strengthenings of [ABSTRACT FROM AUTHOR]

This paper introduces the theme of killing-them-softly between set-theoretic universes. The main theorems show how to force to reduce the large cardinal strength of a cardinal to a specified desired degree. The killing-them-softly theme is about both forcing and the gradations in large cardinal strength. Thus, I also develop meta-ordinal extensions of the hyper-inaccessible and hyper-Mahlo degrees. This paper extends the work of Mahlo to create new large cardinals and also follows the larger theme of exploring interactions between large cardinals and forcing central to modern set theory. [ABSTRACT FROM AUTHOR]

Let [ABSTRACT FROM AUTHOR]

Our main results are: (A) It is consistent relative to a large cardinal that [ABSTRACT FROM AUTHOR]

By a celebrated theorem of Morley, a theory T is ℵ1-categorical if and only if it is κ-categorical for all uncountable κ. In this paper we are taking the first steps towards extending Morley's categoricity theorem 'to the finite'. In more detail, we are presenting conditions, implying that certain finite subsets of certain ℵ1-categorical T have at most one n-element model for each natural number [ABSTRACT FROM AUTHOR]

Sufficient conditions are given for the computation of an arc that accesses a point on the boundary of an open subset of the plane from a point within the set. The existence of a not-computably-accessible but computable point on a computably compact arc is also demonstrated. [ABSTRACT FROM AUTHOR]

Given a group ( G, ·), G⊆ M m, definable in a first-order structure \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {M}=(M,\ldots )$\end{document} equipped with a dimension function and a topology satisfying certain natural conditions, we find a large open definable subset V⊆ G and define a new topology τ on G with which ( G, ·) becomes a topological group. Moreover, τ restricted to V coincides with the topology of V inherited from M m. Likewise we topologize transitive group actions and fields definable in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {M}$\end{document}. These results require a series of preparatory facts concerning dimension functions, some of which might be of independent interest. [ABSTRACT FROM AUTHOR]