Hyperformulas and Classifications
نویسندگان
چکیده
The word existence is repeatedly used in arguments in mathematics, and there are several possibilities for the explicit definition of the existence of mathematical objects. Sometimes the word is used in a strict and rigorous sense, but in other occasions, it may have a more moderate meaning. One account of existence says that the existence of a mathematical object is affirmed when the consistency of the existence is demonstrated. This sort of existence is named fictitious existence and analyzed in terms of hyperformulas in Kakuda [4]. This paper is a direct descendant of [4]. We shall develop the theory of hyperformulas with a new definition, and discuss the semantics for them by using some concepts which come from Barwise and Seligman’s theory of Information Flow [1]. Hyperformulas are extensions of formulas with infinite propositional connectives. The definition of them in this paper is slightly different from [4]. A hyperformula is a set {Φi}i∈I of sets Φi = {φi}j∈Ji of formulas, with corresponding interpretation ∨ i∈I ∧ j∈Ji φ j i , while hyperformulas in [4] was a set of formulas. In this paper, the consistency and the consequence relation H for hyperformulas are introduced as natural extensions of those of formulas, and propositional connectives (with infinite conjunction and disjunction) for hyperformulas are defined in the standard way for the connectives for formulas in
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