In this paper we prove that it consistent to have a two-point set in a model of ZF in which the real line cannot be well-ordered. We prove two results related to a construction of Chad of a two-point set inside the countable union of concentric circles. We show that if the reals are the countable union of countable sets, then every well-orderable set of reals is countable. However, it is consis...

While there are several arguments on either side, it is far from clear as to whether or not countable additivity is an acceptable axiom of subjective probability. I focus here on de Finetti’s central argument against countable additivity and provide a new Dutch book proof of the principle, to argue that if we accept the Dutch book foundations of subjective probability, countable additivity is a...

Let T be a 1-sorted structure. The functor TotFamT yielding a family of subsets of T is defined by: (Def. 1) TotFamT = 2 carrier of T . The following proposition is true (1) For every set T and for every family F of subsets of T holds F is countable iff F c is countable. Let us note that Q is countable. The scheme FraenCoun11 concerns a unary predicate P, and states that: {{n};n ranges over ele...

In this paper, we show that a regular space with a locally countable weak-base is g-metrizable. Secondly, we establish the relationships between spaces with a locally countable weak-base (resp. spaces with a locally countable weak-base consisting of א0-subspaces) and metric spaces (resp. locally separable metric spaces) by means of compact-covering maps, 1-sequence-covering maps, compact maps, ...