It is shown that for compact metric spaces the following statements are pairwise equivalent: “X is Loeb”, “X is separable”, “X has a we ordered dense subset”, “X is second countable”, and “X has a dense set G = ∪{Gn : n ∈ ω}, ∣Gn∣ < ω, with limn→∞ diam = 0”. Further, it is shown that the statement: “Compact metric spaces are weakly Loeb” is not provable in ZF0 , the Zermelo-Fraenkel set theory without the axiom of regularity, and (...) that the countable axiom of choice for families of finite sets CACfin does not imply the statement “Compact metric spaces are separable”. (shrink)

We study the role the axiom of choice plays in the existence of some special subsets of ℝ and its power set ℘.

We provide a general criterion for Fraenkel–Mostowski models of $${\textsf{ZFA}}$$ (i.e. Zermelo–Fraenkel set theory weakened to permit the existence of atoms) which implies “every linearly ordered set can be well ordered” ( $${\textsf{LW}}$$ ), and look at six models for $${\textsf{ZFA}}$$ which satisfy this criterion (and thus $${\textsf{LW}}$$ is true in these models) and “every Dedekind finite set is finite” ( $${\textsf{DF}}={\textsf{F}}$$ ) is true, and also consider various forms of choice for well-ordered families of well orderable sets in these (...) models. In Model 1, the axiom of multiple choice for countably infinite families of countably infinite sets ( $${\textsf{MC}}_{\aleph _{0}}^{\aleph _{0}}$$ ) is false. It was the open question of whether or not such a model exists (from Howard and Tachtsis “On metrizability and compactness of certain products without the Axiom of Choice”) that provided the motivation for this paper. In Model 2, which is constructed by first choosing an uncountable regular cardinal in the ground model, a strong form of Dependent choice is true, while the axiom of choice for well-ordered families of finite sets ( $${\textsf{AC}}^{{\textsf{WO}}}_{{\textsf{fin}}}$$ ) is false. Also in this model the axiom of multiple choice for well-ordered families of well orderable sets fails. Model 3 is similar to Model 2 except for the status of $${\textsf{AC}}^{{\textsf{WO}}}_{{\textsf{fin}}}$$ which is unknown. Models 4 and 5 are variations of Model 3. In Model 4 $${\textsf{AC}}_{\textrm{fin}}^{{\textsf{WO}}}$$ is true. The construction of Model 5 begins by choosing a regular successor cardinal in the ground model. Model 6 is the only one in which $$2{\mathfrak {m}} = {\mathfrak {m}}$$ for every infinite cardinal number $${\mathfrak {m}}$$. We show that the union of a well-ordered family of well orderable sets is well orderable in Model 6 and that the axiom of multiple countable choice is false. (shrink)

In $\mathsf {ZF}$ (i.e., Zermelo–Fraenkel set theory minus the axiom of choice ( $\mathsf {AC}$ )), we investigate the open problem of the deductive strength of the principle UFwob(ω): “There exists a free ultrafilter on ω with a well-orderable base”, which was introduced by Herzberg, Kanovei, Katz, and Lyubetsky [(2018), Journal of Symbolic Logic, 83(1), 385–391]. Typical results are: (1) “ $\aleph _{1}\leq 2^{\aleph _{0}}$ ” is strictly weaker than $\mathsf {UF_{wob}}(\omega )$ in $\mathsf {ZF}$. (2) “There exists a free (...) ultrafilter on $\omega $ ” does not imply “ $\aleph _{1}\leq 2^{\aleph _{0}}$ ” in $\mathsf {ZF}$, and thus (by (1)) neither does it imply $\mathsf {UF_{wob}}(\omega )$ in $\mathsf {ZF}$. This fills the gap in information in Howard and Rubin [Mathematical Surveys and Monographs, American Mathematical Society, 1998], as well as in Herzberg et al. (2018). (3) Martin’s Axiom ( $\mathsf {MA}$ ) implies “no free ultrafilter on $\omega $ has a well-orderable base of cardinality $<2^{\aleph _{0}}$ ”, and the latter principle is not implied by $\aleph _{0}$ -Martin’s Axiom ( $\mathsf {MA}(\aleph _{0})$ ) in $\mathsf {ZF}$. (4) $\mathsf {MA} + \mathsf {UF_{wob}}(\omega )$ implies $\mathsf {AC}(\mathbb {R})$ (the axiom of choice for non-empty sets of reals), which in turn implies $\mathsf {UF_{wob}}(\omega )$. Furthermore, $\mathsf {MA}$ and $\mathsf {UF_{wob}}(\omega )$ are mutually independent in $\mathsf {ZF}$. (5) For any infinite linearly orderable set X, each of “every filter base on X can be well ordered” and “every filter on X has a well-orderable base” is equivalent to “ $\wp (X)$ can be well ordered”. This yields novel characterizations of the principle “every linearly ordered set can be well ordered” in $\mathsf {ZFA}$ (i.e., Zermelo–Fraenkel set theory with atoms), and of $\mathsf {AC}$ in $\mathsf {ZF}$. (6) “Every filter on $\mathbb {R}$ has a well-orderable base” implies “every filter on $\omega $ has a well-orderable base”, which in turn implies $\mathsf {UF_{wob}}(\omega )$, and none of these implications are reversible in $\mathsf {ZF}$. (7) “Every filter on $\omega $ can be extended to an ultrafilter with a well-orderable base” is equivalent to $\mathsf {AC}(\mathbb {R}),$ and thus is strictly stronger than $\mathsf {UF_{wob}}(\omega )$ in $\mathsf {ZF}$. (8) “Every filter on $\omega $ can be extended to an ultrafilter” implies “there exists a free ultrafilter on $\omega $ which has no well-orderable base of cardinality ${<2^{\aleph _{0}}}$ ”. The former principle does not imply “there exists a free ultrafilter on $\omega $ which has no well-orderable base” in $\mathsf {ZF}$, and the latter principle is true in the Basic Cohen Model. (shrink)

We establish the following results: 1. In ZF (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC), for every set I and for every ordinal number α ≥ ω, the following statements are equivalent: (a) The Tychonoff product of| α| many non-empty finite discrete subsets of I is compact. (b) The union of| α| many non-empty finite subsets of I is well orderable. 2. The statement: For every infinite set I, every closed subset of the Tychonoff product [0, 1] (...) I which consists of functions with finite support is compact, is not provable in ZF set theory. 3. The statement: For every set I, the principle of dependent choices relativised to I implies the Tychonoff product of countably many non-empty finite discrete subsets of I is compact, is not provable in ZF⁰ (i.e., ZF minus the Axiom of Regularity). 4. The statement: For every set I, every ℵ₀-sized family of non-empty finite subsets of I has a choice function implies the Tychonoff product of ℵ₀ many non-empty finite discrete subsets of I is compact, is not provable in ZF⁰. (shrink)

In set theory without the Axiom of Choice, we investigate the set-theoretic strength of the principle NDS which states that there is no function f on the set ω of natural numbers such that for everyn ∈ ω, f ≺ f, where for sets x and y, x ≺ y means that there is a one-to-one map g : x → y, but no one-to-one map h : y → x. It is a long standing open problem whether NDS implies (...) AC. In this paper, among other results, we show that NDS is a strong axiom by establishing that ACLO ↛ NDS in ZFA set theory. The latter result provides a strongly negative answer to the question of whether “every Dedekind-finite set is finite” implies NDS addressed in G. H. Moore “Zermelo’s Axiom of Choice. Its Origins, Development, and Influence” and in P. Howard–J. E. Rubin “Consequences of the Axiom of Choice”. We also prove that ACWO ↛ NDS in ZF and that “for all infinite cardinals m, m + m = m” ↛ NDS in ZFA. (shrink)

In the setting of ZF, i.e., Zermelo–Fraenkel set theory without the Axiom of Choice , we study partitions of Russell-sets into sets each with exactly n elements , for some integer n. We show that if n is odd, then a Russell-set X has an n -ary partition if and only if |X | is divisible by n. Furthermore, we establish that it is relative consistent with ZF that there exists a Russell-set X such that |X | is not divisible (...) by any finite cardinal n > 1. (shrink)

A Hausdorff space is called effectively Hausdorff if there exists a function F—called a Hausdorff operator—such that, for every with,, where U and V are disjoint open neighborhoods of x and y, respectively. Among other results, we establish the following in, i.e., in Zermelo–Fraenkel set theory without the Axiom of Choice (): is equivalent to “For every set X, the Cantor cube is effectively Hausdorff”. This enhances the result of Howard, Keremedis, Rubin and Rubin [13] that is equivalent to “Hausdorff (...) spaces are effectively Hausdorff” in. The Boolean Prime Ideal Theorem and the statement “For every infinite set X, the Stone space of the Boolean algebra is effectively Hausdorff” are mutually independent. In particular, the latter statement is not provable in. The Axiom of Choice for non‐empty subsets of () is equivalent to each of “Separable Hausdorff spaces are effectively Hausdorff” and “The Cantor cube is effectively Hausdorff”. The Principle of Dependent Choices in conjunction with the Axiom of Choice for continuum sized families of non‐empty subsets of does not imply the axiom of choice for partitions of. The latter independence result fills the gap in information in Howard and Rubin's book “Consequences of the Axiom of Choice”. The axiom of countable choice for non‐empty subsets of is equivalent to each of “Denumerable Hausdorff spaces are effectively Hausdorff”, “Denumerable T3 spaces are completely normal” and “Denumerable Tychonoff spaces are Urysohn”. (shrink)

In the _World Library of Educationalists_, international experts themselves compile career-long collections of what they judge to be their finest pieces – extracts from books, key articles, salient research findings, major theoretical and practical contributions – so the world can read them in a single manageable volume, allowing readers to follow the themes of their work and see how it contributes to the development of the field. Mary James has researched and written on a range of educational subjects which encompass (...) curriculum, pedagogy and assessment in schools, and implications for teachers´ professional development, school leadership and policy frameworks. She has written many books and journals on assessment, particularly assessment for learning and is an expert on teacher learning, curriculum, leadership for learning and educational policy. Starting with a specially written introduction in which Mary gives an overview of her career and contextualises her selection, the chapters are divided into three parts: Educational Assessment and Learning Educational Evaluation and Curriculum Development Educational Research and the Improvement of Practice Through this book, readers can follow the different strands that Mary James has researched and written about over the last three decades, and clearly see her important contribution to the field of education. (shrink)

The noted psychologist, Doreen Kimura, has argued that we should not expect to find equal numbers of men and women in various professions because there is a natural sexual inequality of intellect. In rebuttal I argue that each of these mutually supporting theses is insufficiently supported by the evidence to be accepted. The social and ethical dimensions of Kimura's work, and of the scientific study of the nature-nurture controversy in general, are briefly discussed.

The purpose of this essay is to analyze succinctly the concept of faith through the very penetrating gaze of the Danish philosopher Sören Kierkegaard. Faith is analyzed by Kierkegaard from absurdity, resignation and the paradox of faith. In the same way, his concept of faith is carved with a chisel that would complete all his philosophy, that is, to touch human reality away from any abstract thought, trying at all times that his examination be seen on a personal dimension in (...) his rigorously individual coret. (shrink)

The orthodox explanation of the syntax of lines 453–4 is that repeated by the most recent commentator, F. Bömer, p.343): ‘neque adhuc epota parte ist Abl. absol.; der Gegenstand, mit dem Ceres den Jungen überschüttet, ist mixta … polenta.‘ The ablative absoluteis in itself unexceptionable, but the proliferation of three ablatives in two verses is awkward writing. As transmitted, line 454 is the product of a copyist who, as is often the habit of copyists, was confininghis attention to the verse (...) on which he was engaged and still had ‘tosta … polenta’from line 450 echoing in his head. Unless I am much mistaken, Ovid wrote. (shrink)

Книга адресована специалистам в области теории познания, обществоведам, преподавателям философских дисциплин, аспирантам и студентам философских факультетов вузов, всем тем, кто интересуется данной проблемой.

In 394 Agesilaus, treading in the footsteps of Xerxes, came from Asia by way of Thrace and Macedon into Thessaly, threw off the attacks of the Thessalian cavalry, proceeded without further trouble into Boeotia, and met the enemy at Coronea, where a great battle was fought. The question ought to have been asked before now, why was he not held up at Thermopylae?

In the middle of the third book of his Histories, Tacitus records some movements in the western provinces occasioned by the events of 69. The troops in Britain, he says, were partial to Vespasian, who had served there with distinction in the reign of Claudius, and a party was formed in his interest.