We work in set-theory without choice ZF. Given a closed subset F of [0, 1] which is a bounded subset of l(I) (resp. such that F ⊆ l(I)), we show that the countable axiom of choice for finite subsets of I, (resp. the countable axiom of choiceACN) implies that F is compact. This enhances previous results where ACN (resp. the axiom of Dependent Choices DC) was required. Moreover, if I is linearly ...

We study the relationships between weakened forms of the Kinna-Wagner Selection Principle (KW), the Axiom of Choice (AC), and the Axiom of Multiple Choice (MC). Introduction. In this paper we study the relationships between weakened forms of the Kinna-Wagner Selection Principle (KW), the Axiom of Choice (AC), and the Axiom of Multiple Choice (MC). The forms of these statements that we shall use...

Well-ordering of the Reals presents a major challenge in Set theory. Under the standard Zermelo Fraenkel Set theory (ZF) with the Axiom of Choice (ZFC), a well-ordering of the Reals is indeed possible. However the Axiom of Choice (AC) had to be introduced to the original ZF theory which is then shown equivalent to the well-ordering theorem. Despite the result however, no way has still been foun...

philosophers from one hand take the “al-vahed” axiom for granted, and on the other hand, they assume that the almighty to be of absolute power. now the question is whether: this is at odds with the “al- vahed” axiom? how can we accept at the same time the absolute power of the almighty and the “al-vahed” axiom? we should say here: majority of philosophers, mystics and kalam specialists have cla...

The principle of set theory known as the Axiom of Choice (AC) has been hailed as “probably the most interesting and, in spite of its late appearance, the most discussed axiom of mathematics, second only to Euclid’s axiom of parallels which was introduced more than two thousand years ago”1 It has been employed in countless mathematical papers, a number of monographs have been exclusively devoted...