Hereditarily finite sets and identity trees
نویسندگان
چکیده
منابع مشابه
The Hereditarily Finite Sets
The theory of hereditarily finite sets is formalised, following the development of Świerczkowski [2]. An HF set is a finite collection of other HF sets; they enjoy an induction principle and satisfy all the axioms of ZF set theory apart from the axiom of infinity, which is negated. All constructions that are possible in ZF set theory (Cartesian products, disjoint sums, natural numbers, function...
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We axiomatize hereditarily finite sets in constructive type theory and show that all models of the axiomatization are isomorphic. The axiomatization takes the empty set and adjunction as primitives and comes with a strong induction principle. Based on the axiomatization, we construct the set operations of ZF and develop the basic theory of finite ordinals and cardinality. We construct a model o...
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Hereditarily finite (HF) set theory provides a standard universe of sets, but with no infinite sets. Its utility is demonstrated through a formalisation of the theory of regular languages and finite automata, including the Myhill-Nerode theorem and Brzozowski’s minimisation algorithm. The states of an automaton are HF sets, possibly constructed by product, sum, powerset and similar operations.
متن کاملMathematical Logic A hierarchy of hereditarily finite sets
This article defines a hierarchy on the hereditarily finite sets which reflects the way sets are built up from the empty set by repeated adjunction, the addition to an already existing set of a single new element drawn from the already existing sets. The structure of the lowest levels of this hierarchy is examined, and some results are obtained about the cardinalities of levels of the hierarchy.
متن کاملEnumeration of the adjunctive hierarchy of hereditarily finite sets
Hereditarily finite sets (sets which are finite and have only hereditarily finite sets as members) are basic mathematical and computational objects, and also stand at the basis of some programming languages. We solve an open problem proposed by Kirby in 2008 concerning a recurrence relation for the cardinality an of the n-th level of the adjunctive hierarchy of hereditarily finite sets; in this...
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ژورنال
عنوان ژورنال: Journal of Combinatorial Theory, Series B
سال: 1983
ISSN: 0095-8956
DOI: 10.1016/0095-8956(83)90068-0