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Higher Order Logic
 In Handbook of Logic in Artificial Intelligence and Logic Programming
, 1994
"... Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Definin ..."
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Cited by 21 (0 self)
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Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Defining data types : : : : : : : : : : : : : : : : : : : : : 6 2.4 Describing processes : : : : : : : : : : : : : : : : : : : : : 8 2.5 Expressing convergence using second order validity : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.6 Truth definitions: the analytical hierarchy : : : : : : : : 10 2.7 Inductive definitions : : : : : : : : : : : : : : : : : : : : : 13 3 Canonical semantics of higher order logic : : : : : : : : : : : : 15 3.1 Tarskian semantics of second order logic : : : : : : : : : 15 3.2 Function and re
STMM: A Set Theory for Mechanized Mathematics
 JOURNAL OF AUTOMATED REASONING
, 2000
"... Although set theory is the most popular foundation for mathematics, not many mechanized mathematics systems are based on set theory. ZermeloFraenkel (zf) set theory and other traditional set theories are not an adequate foundation for mechanized mathematics. stmm is a version of vonNeumannBerna ..."
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Although set theory is the most popular foundation for mathematics, not many mechanized mathematics systems are based on set theory. ZermeloFraenkel (zf) set theory and other traditional set theories are not an adequate foundation for mechanized mathematics. stmm is a version of vonNeumannBernaysGödel (nbg) set theory that is intended to be a Set Theory for Mechanized Mathematics. stmm allows terms to denote proper classes and to be undened, has a denite description operator, provides a sort system for classifying terms by value, and includes lambdanotation with term constructors for function application and function abstraction. This paper describes stmm and discusses why it is a good foundation for mechanized mathematics.
A Set Theory with Support for Partial Functions
 STUDIA LOGICA
, 2000
"... Partial functions can be easily represented in set theory as certain sets of ordered pairs. However, classical set theory provides no special machinery for reasoning about partial functions. For instance, there is no direct way of handling the application of a function to an argument outside its dom ..."
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Cited by 10 (2 self)
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Partial functions can be easily represented in set theory as certain sets of ordered pairs. However, classical set theory provides no special machinery for reasoning about partial functions. For instance, there is no direct way of handling the application of a function to an argument outside its domain as in partial logic. There is also no utilization of lambdanotation and sorts or types as in type theory. This paper introduces a version of vonNeumannBernaysGödel set theory for reasoning about sets, proper classes, and partial functions represented as classes of ordered pairs. The underlying logic of the system is a partial firstorder logic, so classvalued terms may be nondenoting. Functions can be specified using lambdanotation, and reasoning about the application of functions to arguments is facilitated using sorts similar to those employed in the logic of the imps Interactive Mathematical Proof System. The set theory is intended to serve as a foundation for mechanized mathematics systems.
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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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
BERNAYS AND SET THEORY
"... We discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higherorder reflection principles. ..."
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We discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higherorder reflection principles.
Chiron: A Set Theory with Types, Undefinedness, Quotation, and Evaluation*
, 2007
"... Abstract Chiron is a derivative of vonNeumannBernaysG"odel (nbg) set theorythat is intended to be a practical, generalpurpose logic for mechanizing mathematics. Unlike traditional set theories such as ZermeloFraenkel( ..."
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Abstract Chiron is a derivative of vonNeumannBernaysG&quot;odel (nbg) set theorythat is intended to be a practical, generalpurpose logic for mechanizing mathematics. Unlike traditional set theories such as ZermeloFraenkel(
Weak Systems of Set Theory Related to HOL
, 1994
"... this paper all these theories will be assumed to have the axiom of infinity This is clearly a much simpler syntax that HOL: TST is what you get from HOL once you decide to implement functions and ordered pairs as sets: a lot of the type structure collapses. It doesn't even matter very much how ..."
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this paper all these theories will be assumed to have the axiom of infinity This is clearly a much simpler syntax that HOL: TST is what you get from HOL once you decide to implement functions and ordered pairs as sets: a lot of the type structure collapses. It doesn't even matter very much how you implement functions and ordered pairs as sets: any implementation of functions and ordered pairs as sets gives rise to an interpretation of HOL in TST. Accordingly we have the following: