## A Formalization of Set Theory Without Variables, Volume 41 |

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### Contents

The Formalism 𝓛 of Predicate Logic | 1 |

The Formalism 𝓛+ a Definitional Extension of 𝓛 | 23 |

The Formalism 𝓛x without Variables and the Problem of Its Equipollence with 𝓛 | 45 |

The Relative Equipollence of 𝓛 and 𝓛x and the Formalization of Set Theory in 𝓛x | 95 |

Some Improvements of the Equipollence Results | 147 |

Implications of the Main Results for Semantic and Axiomatic Foundations of Set Theory | 169 |

Extension of Results to Arbitrary Formalisms of Predicate Logic and Application to the Formalization of the Arithmetics of Natural and Real Numbers | 191 |

Applications to Relation Algebras and to Varieties of Algebras | 231 |

### Common terms and phrases

analogous apply arbitrary atomic axiom schemata axiom set axiomatic binary operation binary relation comprehension schema conclusion conjugated quasiprojections consequence construction decision problems deduction theorem defined definitional extension definitionally equivalent denoted derivable discussion dually undecidable elements equational formalism equational theory equipollent in means expression and proof fact finite axiomatizability finitely based first-order first-order logic formulas function given hence induction instance involved lemmas logical axioms logical constants logically equivalent logically provable MABX means of expression means of proof metalogical natural numbers nonlogical constants obtained obviously one-one operation symbols particular Peano arithmetic possible definition predicate logic problem proper classes prove QAB hx RA's recursive referred relation algebras replaced schema semantical notions semantically complete sentential logic sequence set of sentences set theory set-theoretical systems structure subformalism subsystem systems of set Tarski theorem translation mapping unary variant vocabulary weak Q-system

### Popular passages

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...