By Paul R. Halmos

ISBN-10: 0828401543

ISBN-13: 9780828401548

From the Preface: "Algebraic common sense [addresses] a few of the difficulties of mathematical common sense and the speculation of polyadic Boolean algebras, with which this quantity is usually involved. It is meant to be a good approach of treating algebraic good judgment in a unified demeanour. the cloth is obtainable to a normal mathematical viewers; no monstrous wisdom of algebra or common sense is required ... with the exception of a moderate Boolean starting place, the amount is largely self-contained."

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This generalization, called the method of generalized quantifiers, has proved to be close to the so-called "omitting types technique" and to forcing. In particular, models obtained by forcing have proved to be a particular case of models obtained by a certain generalized quantifier. A search for a deeply motivated system of axioms of set theory and a desire to learn the power of such a system were the essential motives of Mostowski's research in the foundations of set theory and the results which he obtained were of enormous importance for the rise and development of that domain.

Lukasiewicz, who was the first to conceive an idea of a logic different from the usual one, hoped that one day several logics will emerge which will actually be used, as are for instance the non-Euclidean geometries. Most of the non-classical logics invented so far are not being actually used although several of them are being studied in the meta-mathematical fashion on the basis of the two-valued logic. It looks as if the intuitionistic logic were the only one in the case of which Lukasiewicz's plan has still some chance of realization.

7. The example shows that the groups of automorphisms of complicated theories cannot be quite arbitrary. Namely, if the theory is sufficiently rich, then the group of automorphisrns of any of its models is isomorphic with the group ofaut omorphisms of a certain linearly ordered set. It follows hence that considerations should be restricted to groups of automorphisms of linearly ordered sets. For such groups the answer is positive, which means that for any linearly ordered set

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Algebraic logic by Paul R. Halmos


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