Bartoszynski, T. and S. Shelah, Closed measure zero sets, Annals of Pure and Applied Logic 58 93–110. We study the relationship between the σ-ideal generated by closed measure zero sets and the ideals of null and meager sets. We show that the additivity of the ideal of closed measure zero sets is not bigger than covering for category. As a consequence we get that the additivity of the ideal of closed measure zero sets is equal to the additivity of the (...) ideal of meager sets. (shrink)

We conclude the discussion of additivity, Baire number, uniformity, and covering for measure and category by constructing the remaining 5 models. Thus we complete the analysis of Cichon's diagram.

When asked in 1962 on what he was working Kazimierz Ajdukiewicz replied: Several years ago Polish Scientific Publishers suggested that I pre pare a new edition of The Logical Foundations of Teaching, which I wrote 1 before 1939 as a contribution to The Encyclopaedia of Education. It was a small booklet covering elementary information about logical semantics and scientific methodology, information which in my opinion was necessary as a foundation of teaching and as an element of the education of (...) any teacher. When I recently set to preparing the new edition, I rewrote practically everything, and a booklet of some 100 pages swelled into a bulky volume almost five times bigger. The issues have remained practically the same, but they are now analysed much more thoroughly and the threshold of difficulty is much higher now. The main stress has been laid on the methods used in the empirical sciences, and within that field, on the theory of measurement and the methods of statistical inference. I am now working on the last chapter of the book, concerned with explanation procedures and theory construction in the empirical sciences. When that book, which I intend to entitle Pragmatic Logic, is com pleted I intend to prepare for the press Vol. 2 of my minor writings, 2 Language and Cognition, which will cover some of my post-war pa pers. (shrink)

We construct a model of ZFC satisfying the Dual Borel Conjecture in which there is a set of size ℵ₁ that does not have measure zero.

. ACT, CONTENT, AND OBJECT OF THE PRESENTATION It is one of the best known positions of psychology, hardly contested by anyone, that every mental phenomenon ...

We prove that the following are consistent with ZFC. 1. 2 ω = ℵ ω 1 + K C = ℵ ω 1 + K B = K U = ω 2 (for measure and category simultaneously). 2. 2 ω = ℵ ω 1 = K C (L) + K C (M) = ω 2 . This concludes the discussion about the cofinality of K C.

We prove that the cofinality of the smallest covering of R by meager sets is bigger than the additivity of measure.

The volume contains almost thirty papers by Kazimierz Twardowski , the founder of the Lvov-Warsaw School. The papers are published in English for the first time. The papers concern fundamental problems of philosophy: the methods of philosophizing, the boundary of psychology and semiotics, the conceptual apparatus of metaphysics, ethical skepticism, the question of free will and ethical obligation, the aesthetics of music and so on. The systematic considerations are complemented by concise but excellent sketches of the philosophical views of (...) Socrates, Aquinas, Leibniz, Spencer, Nietzsche, and Bergson. (shrink)

Autor porusza zagadnienie struktury Peryta, która posiada tę właściwość, iż środkiem świata jest miejsce założone przez obserwatora. W ten sposób zmienia się charakter zjawisk geometrycznych, w tym perspektywy.

The article reviews the book Die Möglichkeit des Guten. Ethik im 21. Jahrhundert [The Possibility of the Good. Ethics in the 21st Century], by Wilhelm Vossenkuhl.

The contemporary development of socialist economy is featured not only by high dynamics, but also by essential structural and qualitative changes. The economic policy and the socialist planning in particular tendtowards optimalization of economic growth, then to engage for its proper directions, structure and rythmicality. The every year growing national product ought to be, as it is possible, most suitable and useful for implementation of threefold objective: 1° the better satisfaction with the necessities of population, 2° the quicker modernization of (...) the national economy, and 3° the effective development of the foreign trade as well as the international cooperation.The superiority of a set of social criteria is becoming the general assumption of economic policy. A vision of the future human being and the mature society is determining prospective objectives and aims of the development as well is indicating ways and means for their realization by economic policy.The economic growth is not objective by itself, but has to serve for a comprehensive social advancement. At the same time the factors of social advancement advantageously impact the labour productivity and otherwisethey perform a role of additional stimulatives of the economic growth.The objectives of social advancement should be realized gradually basing on staging patterns of systematic raise of the living standards of the population. A hierarchy and a sequence of satisfaction with the both individual and collective necessities demand a skillful application of the criteria of the economic and social choice. We are using here the principle of social and economic rationality.Hence the factors of social advancement, involving the scientific and technical progress, belang to the fundumatel elements of the present economic and social policy. At the same time objectives and means of modern economic and social advancement are here strictly interlaced. (shrink)

We answer a question of Just, Miller, Scheepers and Szeptycki whether certain diagonalization properties for sequences of open covers are provably closed under taking finite or countable unions.

We address ZFC inequalities between some cardinal invariants of the continuum, which turned out to be true in spite of strong expectations given by [11].

We show that it is consistent with ZFC that the intersection of some family of less than ultrafilters have measure zero. This answers a question of D. Fremlin.

Abstract.We will construct several models where there are no strongly meager sets of size 2ℵ0.

This essay explores social forces which contributed to regaining independence by the Polish people and sovereignty by the Polish state after 45 years of Soviet domination There were four major factors or forces of historical change: workers' resistance (big-industry working class); intellectual opposition (dissidents); grass-roots movement (families, households and their microeconmuc activity); and the Catholic Church (in the late phase of the Polish People's Republic). The preliminary thesis is that Poland succeeded in transcending communism and Soviet domination as quickly as (...) it did thanks to its civil society traditions. In particular, universities and their intellectual influence on the young generations of Poles helped nurture the political opposition Equally, the grass-roots movement of Polish family households undermined the unrealistic, strange system of national (planned) economy, which otherwise could have lasted much longer in Poland, as it did in all of Eastem Europe. The conclusion follows that the historically formed cultural capital of the Polish people was the decisive factor in the nation's liberation from totalitarian rule. (shrink)

A set X⊆ℝ is strongly meager if for every measure zero set H, X+H ≠ℝ. Let [Formula: see text] denote the collection of strongly meager sets. We show that assuming [Formula: see text], [Formula: see text] is not an ideal.

We study the cardinal invariants of measure and category after adding one random real. In particular, we show that the number of measure zero subsets of the plane which are necessary to cover graphs of all continuous functions may be large while the covering for measure is small.

In this paper we present two consistency results concerning the existence of large strong measure zero and strongly meager sets. RID=""ID="" Mathematics Subject Classification (2000): 03e35 RID=""ID="" The first author was supported by Alexander von Humboldt Foundation and NSF grant DMS 95-05375. The second author was partially supported by Basic Research Fund, Israel Academy of Sciences, publication 658.

We will construct several models where there are no strongly meager sets of size 2ℵ0.

We show that it is consistent with MA + ¬CH that the Forcing Axiom fails for all forcing notions in the class of ωω–bounding forcing notions with norms of [17].

In the paper some varieties E s n of BCK-algebras with condition defined by W. H. Cornish [1] are considered. A characterization of simple algebras in E s 2 is given and some properties of simple algebras in E s 3 are indicated.

Artykuł jest próbą wyodrębnienia i ogólnego opisu aspektów filozoficznych, zawartych w teorii duchowości ekologicznej papieża Franciszka oraz porównania ich z myślą tomistyczną, rozwijaną przez jej polskich przedstawicieli w czasach współczesnych. Aspekty te dostrzeżono na gruncie wielu dyscyplin filozoficznych, ponieważ w tego typu duchowości chodzi o stosunek człowieka do najogólniej pojętej rzeczywistości realnej, czyli do samego siebie, przyrody, kultury i religii. W toku analizy obu stanowisk zauważono liczne analogie, a także pewne rozbieżności w sposobie widzenia przyrody, osoby ludzkiej, kultury oraz religii, (...) które przynajmniej częściowo można tłumaczyć niejednoznacznością niektórych sformułowań papieskich. -------------- Zgłoszono: 19/08/2020. Zrecenzowano: 26/09/2020. Zaakceptowano do publikacji: 18/10/2020. (shrink)

“The concept of Logos at Heraclitus. The analyze of B1 fragment”. “Logos” is the most important word in the Heraclitus’ philosophy. One can tell, that Heraclitusis the philosopher of logos. The way of interpretation of this notion influences the comprehention of all work of the Greek thinker.The word appears eleven times in the following fragments : B 1, B 2, B 31, B 39, B 45, B 50, B 72, B 87, B 108, B 115 [numeration of Diels-Kranz]. Logos contains (...) several notions, and it can not be reduced to one, as for exemple reason, speach, fire or god. The analyse of B1, where the logos apperas for the first time, shows the complexity of this term. (shrink)

This paper is devoted to showing certain connections between normal modal logics and those strictly regular modal logics which have as a theorem. We extend some results of E. J. Lemmon (cf. [66]). In particular we prove that the lattice of the strictly regular modal logics with the axiom is isomorphic to the lattice of the normal modal logics.