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Predicative Foundations of Arithmetic
 Journal of Philosophical Logic
, 1995
"... Predicative mathematics in the sense originating with Poincaré andWeylbegins by taking the natural number system for granted, proceeding immediately to real analysis and related fields. On the other hand, from a logicist or settheoretic standpoint, this appears problematic, for, as the story is usu ..."
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Cited by 10 (3 self)
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Predicative mathematics in the sense originating with Poincaré andWeylbegins by taking the natural number system for granted, proceeding immediately to real analysis and related fields. On the other hand, from a logicist or settheoretic standpoint, this appears problematic, for, as the story is usually told, impredicative
Chiron: A set theory with types, undefinedness, quotation, and evaluation
, 2007
"... Chiron is a derivative of vonNeumannBernaysGödel (nbg) set theory that is intended to be a practical, generalpurpose logic for mechanizing mathematics. Unlike traditional set theories such as ZermeloFraenkel (zf) and nbg, Chiron is equipped with a type system, lambda notation, and definite and ..."
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Cited by 6 (5 self)
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Chiron is a derivative of vonNeumannBernaysGödel (nbg) set theory that is intended to be a practical, generalpurpose logic for mechanizing mathematics. Unlike traditional set theories such as ZermeloFraenkel (zf) and nbg, Chiron is equipped with a type system, lambda notation, and definite and indefinite description. The type system includes a universal type, dependent types, dependent function types, subtypes, and possibly empty types. Unlike traditional logics such as firstorder logic and simple type theory, Chiron admits undefined terms that result, for example, from a function applied to an argument outside its domain or from an improper definite or indefinite description. The most noteworthy part of Chiron is its facility for reasoning about the syntax of expressions. Quotation is used to refer to a set called a construction that represents the syntactic structure of an expression, and evaluation is used to refer to the value of the expression that a construction
Undecidability in diagonalizable algebras
 The Journal of Symbolic Logic
"... For a formal theory T, the diagonalizable algebra (a.k.a. Magari algebra) of T, denoted DT, is the Lindenbaum sentence algebra of T endowed with the unary operator T arising from the provability predicate of T: (the equivalence class of) a sentence ' is sent by T to (the equivalence class of) the T ..."
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Cited by 1 (0 self)
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For a formal theory T, the diagonalizable algebra (a.k.a. Magari algebra) of T, denoted DT, is the Lindenbaum sentence algebra of T endowed with the unary operator T arising from the provability predicate of T: (the equivalence class of) a sentence ' is sent by T to (the equivalence class of) the Tsentence expressing that T proves '. It was shown in Shavrukov [6] that the diagonalizable algebras of PA and ZF, as well as the diagonalizable algebras of similarly related pairs of 1sound theories, are not isomorphic. Neither are these algebras rstorder equivalent (Shavrukov [7, Theorem 2.11]). In the present paper we establish a su cient condition, which we name B ( ) coherence, for the diagonalizable algebras of two theories to be isomorphic. It is then immediately seen that DZF = DGB, which answers a question of Smorynski [11]. We also construct nonidentity automorphisms of diagonalizable algebras of all theories under consideration. The techniques we useareacombination of those developed in the
BERNAYS AND SET THEORY
"... Abstract. We discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higherorder reflection principles. Paul Isaak Bernays (1888–1977) is an important figure in the development of mathematical logic, being the main bridge between Hilbert and Göd ..."
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Abstract. We discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higherorder reflection principles. Paul Isaak Bernays (1888–1977) is an important figure in the development of mathematical logic, being the main bridge between Hilbert and Gödel in the intermediate generation and making contributions in proof theory, set theory, and the philosophy of mathematics. Bernays is best known for the twovolume 1934,1939 Grundlagen der Mathematik [39, 40], written solely by him though Hilbert was retained as first author. Going into many reprintings and an eventual second edition thirty years later, this monumental work provided a magisterial exposition of the work of the Hilbert school in the formalization of firstorder logic and in proof theory and the work of Gödel on incompleteness and its surround, including the first complete proof of the Second Incompleteness Theorem. 1 Recent reevaluation of Bernays ’ role actually places him at the center of the development of mathematical logic and Hilbert’s program. 2 But starting in his forties, Bernays did his most individuated, distinctive mathematical work in set theory, providing a timely axiomatization and later applying higherorder reflection principles, and produced a stream of
Classical Predicative LogicEnriched Type Theories ✩
, 906
"... A logicenriched type theory (LTT) is a type theory extended with a primitive mechanism for forming and proving propositions. We construct two LTTs, named LTT0 and LTT ∗ 0, which we claim correspond closely to the classical predicative systems of second order arithmetic ACA0 and ACA. We justify this ..."
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A logicenriched type theory (LTT) is a type theory extended with a primitive mechanism for forming and proving propositions. We construct two LTTs, named LTT0 and LTT ∗ 0, which we claim correspond closely to the classical predicative systems of second order arithmetic ACA0 and ACA. We justify this claim by translating each secondorder system into the corresponding LTT, and proving that these translations are conservative. This is part of an ongoing research project to investigate how LTTs may be used to formalise different approaches to the foundations of mathematics. The two LTTs we construct are subsystems of the logicenriched type theory LTTW, which is intended to formalise the classical predicative foundation presented by Herman Weyl in his monograph Das Kontinuum. The system ACA0 has also been claimed to correspond to Weyl’s foundation. By casting ACA0 and ACA as LTTs, we are able to compare them with LTTW. It is a consequence of the work in this paper that LTTW is strictly stronger than ACA0. The conservativity proof makes use of a novel technique for proving one LTT conservative over another, involving defining an interpretation of the stronger system out of the expressions of the weaker. This technique should be applicable in a wide variety of different cases outside the present work. Key words: type theory, logicenriched type theory, predicativism, Hermann Weyl, second order arithmetic