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This book provides students of mathematics with the minimum amount of knowledge in logic and set theory needed for a profitable continuation of their studies. There is a chapter on statement calculus, followed by eight chapters on set theory.

Set theory.

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The monograph contains the most important elements of the Zermelo-Fraenkel axiomatic set theory with the axiom of choice: axiomatics, definitions of basic concepts, theories of binary relations, partial ordering, equivalence, functions, ordinal numbers and cardinal numbers.It was created on the basis of lectures conducted over many years by the author for philosophy students at the University of Łódź. Therefore, it does not require a thorough mathematical background; it is enough to have some logical "skills" in theorem proving, or, indeed, a knowledge of such logical constants as Boolean functions and quantifiers. It can be used not only by mathematicians and mathematics students, but also by humanists wishing to consolidate their knowledge of sets, often used in various formalization procedures. The more so because some topics have a philosophical character, including discussions about the axiom of regularity and the concept of founding a set, equivalence relation, ordinal number or axiom of choice.

Axiomatic set theory.

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The book aims to formalise tableau methods for the logics of propositions and names. The methods described are based on Set Theory. The tableau rule was reduced to an ordered n-tuple of sets of expressions where the first element is a set of premises, and the following elements are its supersets.

Set theory. --- Philosophy --- Poland.

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Descriptive set theory and definable proper forcing are two areas of set theory that developed quite independently of each other. This monograph unites them and explores the connections between them. Forcing is presented in terms of quotient algebras of various natural sigma-ideals on Polish spaces, and forcing properties in terms of Fubini-style properties or in terms of determined infinite games on Boolean algebras. Many examples of forcing notions appear, some newly isolated from measure theory, dynamical systems, and other fields. The descriptive set theoretic analysis of operations on forcings opens the door to applications of the theory: absoluteness theorems for certain classical forcing extensions, duality theorems, and preservation theorems for the countable support iteration. Containing original research, this text highlights the connections that forcing makes with other areas of mathematics, and is essential reading for academic researchers and graduate students in set theory, abstract analysis and measure theory.

Descriptive set theory. --- Forcing (Model theory) --- Model theory --- Set theory

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This textbook is Volume 1 of a two-volume set on the axiomatics of economic design. Its central argument is that economic institutions are not God-given: they are man-made. Their ultimate goal is to promote social welfare. The book describes the axiomatic approach to design. It consists in the formulation of criteria of desirability of solution mappings, and of the examination of their logical implications when imposed in various combinations. Its goal is to identify as precisely as possible the line that separates those combinations of desiderata that are compatible and those that are not. The end product of axiomatic work are menus of choices for practitioners to choose from when they have to make a decision. The first volume offers pedagogical coverage of the axiomatic approach to economic design, in the form of answers to questions posed by a young person curious about it. It introduces readers to what motivates economic design. It continues with the mathematical representation of a class of allocation problems. The bulk of the volume is to present structured inventories of the field of axioms, arranged by format first, and content next. These chapters are followed by a user's manual on the axiomatic method. Lastly, the volume discusses how economic design can be aided by other disciplines, in particular philosophy, mathematics, and computer science.

Axiomatic set theory. --- Economics --- Methodology.