A Formalization of Set Theory Without Variables, Volume 41

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American Mathematical Soc. - Mathematics - 318 pages
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The Formalism 𝓛 of Predicate Logic
The Formalism 𝓛+ a Definitional Extension of 𝓛
The Formalism 𝓛x without Variables and the Problem of Its Equipollence with 𝓛
The Relative Equipollence of 𝓛 and 𝓛x and the Formalization of Set Theory in 𝓛x
Some Improvements of the Equipollence Results
Implications of the Main Results for Semantic and Axiomatic Foundations of Set Theory
Extension of Results to Arbitrary Formalisms of Predicate Logic and Application to the Formalization of the Arithmetics of Natural and Real Numbers
Applications to Relation Algebras and to Varieties of Algebras

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Page 276 - A. Levy, A hierarchy of formulas in set theory, Memoirs Amer. Math. Soc. No. 57 (1965).
Page xv - The mathematics of the present work is rooted in the calculus of relations (or the calculus of relatives, as it is sometimes called) that originated in the work of A. De Morgan, CS Peirce, and E. Schroder during the second half of the nineteenth century.
Page xv - Schroder, who extended Peirce's work in a very thorough and systematic way in Schroder [1895], were interested in the expressive powers of the calculus of relations and the great diversity of laws that could be proved. They were aware that many elementary statements about (binary) relations can be expressed as equations in this calculus. (By an "elementary statement...
Page xv - Schroder seems to have been the first to consider the question whether all elementary statements about relations are expressible as equations in the calculus of relations, and in Schroder [1895], p.
Page xii - It is therefore quite surprising that x proves adequate for the formalization of practically all known systems of set theory, and hence for the development of all of classical mathematics. As a language suitable for the formalization of most set-theoretical systems, we take the first-order logic with equality and one nonlogical binary predicate E. (For technical reasons we use "i...

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