We show that the statement CCFC = “the character of a maximal free filter F of closed sets in a T1 space (X, T ) is not countable” is equivalent to the Countable Multiple Choice Axiom CMC and, the axiom of choice AC is equivalent to the statement CFE0 = “closed filters in a T0 space (X, T ) extend to maximal closed filters”. We also show that AC is equivalent to each of the assertions: “every c...

Possible-world semantics are provided for Parikh’s relevance-sensitive model for belief revision. Having Grove’s system-of-spheres construction as a base, we consider additional constraints on measuring distance between possible worlds, and we prove that, in the presence of the AGM postulates, these constraints characterize precisely Parikh’s axiom (P). These additional constraints essentially ...

Design-relevant information is sometimes vague or linguistic (technically speaking, f-granular) (e.g., design a slim shaft with light weight where dimensions are critical than weight). This puts obstacle in practicing the information axiom (minimize the information content of the design) of axiomatic design theory, a theory that establishes a science base for design. In such a situation, how on...

In this article we discuss some elementary concepts in axiomatics, which is the study of axioms. The axiomatic system we use is that of a metric space, which will be familiar to many readers. Metric spaces were introduced by the French mathematician Maurice Fréchet in 1906 as a generalization of the real numbers and the “distance function” f (x , y) = |x − y|. The aim is to construct limits, co...

The expansion axiom of matroids requires only the existence of some kind of independent sets, not the uniqueness of them. This causes that the base families of some matroids can be reduced while the unions of the base families of these matroids remain unchanged. In this paper, we define unique expansion matroids in which the expansion axiom has some extent uniqueness; we define union minimal ma...

Several axioms concerning probabilistic beliefs are examined here, and the relations between them are established, using belief spaces that generalize Harsanyi type spaces. Two axioms concerning highorder probabilistic beliefs are investigated in particular. The first is the triviality axiom, which says that one is certain (that is, has a belief of degree one) of one’s own beliefs. The second i...

In the paper we prove that axiom CPA prism, which follows from the Covering Property Axiom CPA and holds in the iterated perfect set model, implies that there exists a Hamel basis which is a union of less than continuum many pairwise disjoint perfect sets. We will also give two consequences of this last fact. 1 The result and its consequences In this paper we will use standard set theoretic ter...